{"text":"\\section{Introduction}\nThe Rydberg $nP_{j}$ states of Rb afford the capability of studying long-range molecular interactions in macrodimers~\\cite{Boisseau2002,Hollerith2019,Hollerith2021}, Rydberg-ground pairs~\\cite{Niederpruem2016a, Niederpruem2016b}, and, recently predicted and observed, Rydberg-ion mixtures~\\cite{Duspayev2021, Deiss2021, Zuber2022}. Furthermore, their couplings with states of different parities are useful in understanding dipole-dipole interactions~\\cite{Li2005,Ravets2014} and employing quantum electrometry of resonant rf waves via Autler-Townes splitting, observable through electromagnetically-induced-transparency (EIT) spectroscopy~\\cite{Holloway2014, Anderson2022}. Hyperfine interactions of the nuclear magnetic moment and electric quadrupole moment with the angular momentum of the valence electron typically are not observable in $nP_{j}$ Rydberg states through laser-based spectroscopic methods due to limitations in frequency resolution (energy splittings are on the order of kHz), although hyperfine effects have been experimentally presented in Cs~\\cite{Ripka2022}. Millimeter-wave spectroscopy of Rydberg molecular states involving $nP_{j}$ atoms could provide insights in the role of hyperfine coupling on the adiabatic potentials of the molecules, for the spectroscopic measurement is, in principle, only limited by the Rydberg-state lifetime and the rf-field interaction time. As a consequence, knowledge of the hyperfine structure (HFS) is essential for predicting these quantum behaviors.\n\n\\par While precision measurements for the hyperfine-coupling constants in $nP$ Rydberg states have been provided before in~\\cite{Belin1974,Belin1976a,Belin1976b,Farley1977} on $^{133}$Cs and $^{87}$Rb for $n<13$, no coupling constant has been provided for $^{85}$Rb using principal quantum numbers greater than $n=8$, at which point the hyperfine interaction does not scale with $n$. In~\\cite{Li2003}, the $nP_{1\/2}$ HFS is observable for both $^{85}$Rb and $^{87}$Rb (see their Fig. 2). However, the thermal atomic beam used contributed to a significant amount of Doppler broadening, and a measurement was not provided. \n\\par In the present work, we perform mm-wave resonance spectroscopy in the Ka- and U-bands on ultracold $^{85}$Rb. Thus, in the absence of any Doppler and transit-time broadening, we obtain Fourier-limited spectral lines of the \\gfix{$\\ket{nS_{1\/2}, F=3, m_F}\\rightarrow \\ket{nP_{1\/2}, F', m_{F'}}$} transitions and use the splitting between the $F'=2$ and $F'=3$ \\gfix{hyperfine} peaks in order to arrive at an $n$-independent, HFS-coupling-constant $A_{\\text{hfs}}$ measurement for $nP_{1\/2}$ Rydberg states. The spectroscopic series involves $n=42-44$ and $46$. \\gfix{Careful} cancellation of stray magnetic fields to $< 5$~mG is necessary to observe symmetric, Fourier-limited spectral features for both peaks. Our uncertainty budget, as a result, takes into account the role of the background magnetic field on our measurement. Additionally, we provide a systematic uncertainty arising from electric dipole-dipole interactions between $nS_{1\/2}$ and $nP_{1\/2}$ atoms. \n\\par An alkali metal like $^{85}$Rb features a single valence electron of total angular momentum $\\textbf{J}$, spin $\\textbf{S}$, and orbital angular momentum $\\textbf{L}$. The nucleus of the given isotope features an intrinsic angular momentum $\\textbf{I}$ associated with the net magnetic moments of all contained nucleons. For $^{85}$Rb, the nuclear spin quantum number is $I=5\/2$. In general, the hyperfine shift \\gfix{of a $nP_{j}$ level with hyperfine quantum number $F'$} is, in atomic units, \n\\begin{multline}\n \\Delta_{\\text{hfs}} = \\frac{A_{\\text{hfs}}}{[n-\\delta_{lj}(n)]^3}\\langle \\textbf{I}\\cdot\\textbf{J}\\rangle\\\\\n + \\frac{B_{\\text{hfs}}}{[n-\\delta_{lj}(n)]^3}\\bigg<\\frac{3(\\textbf{I}\\cdot\\textbf{J})^2+\\frac{3}{2}\\textbf{I}\\cdot\\textbf{J}-IJ(I+1)(J+1)}{2IJ(2I-1)(2J-1)}\\bigg>,\n\\end{multline}\nwhere $\\delta_{lj}(n)$ is the $nlj$-dependent quantum defect~\\cite{Li2003}, the first term describes the magnetic dipole-dipole interaction between the nucleus and Rydberg electron, and the second term quantifies the nuclear electric-quadrupole interaction. A third term, immeasurable in this type of experiment, involves magnetic-octupole interactions between the two particles~\\cite{SteckRb}. For $nP_{1\/2}$ states, only $A_{\\text{hfs}}$ is nonzero. \n\\par \\gfix{Due to the large extent of the Rydberg electron wave function, short-range interactions scale as $[n-\\delta_{lj}(n)]^{-3}$~\\cite{GallagherBook}.} Thus, the measured splitting $\\nu_{\\text{hfs}}$ between $F'=2$ and $F'=3$ can be expressed as \\begin{equation}\n \\nu_{\\text{hfs}}=\\frac{3A_{\\text{hfs}}}{[n-\\delta_{lj}(n)]^3},\n \\label{eq:HFSFormula}\n\\end{equation}\nwhere the units of $A_{\\text{hfs}}$ are GHz. \n\\par In our experiment, a slow atomic beam of $^{85}$Rb prepared by a continuously operating 2D$^{+}$ MOT~\\cite{Dieckmann1998} is captured and cooled via polarization gradients (PG) in the $\\sigma^{+}$-$\\sigma^{-}$ configuration~\\cite{Dalibard1989} for $14.2~$ms. We leave the 2D$^{+}$ MOT laser beams and all repumping beams on throughout the duration of the experiment. The D2-molasses cooling light is switched off for 80~$\\mu$s before $5-\\mu$s-long optical excitation beams are switched on. These beams produce $nS_{1\/2}$ Rydberg atoms used for the mm-wave spectroscopy, where $n=42$-44 and 46. A $40-\\mu$s mm-wave pulse drives \\gfix{the $\\ket{nS_{1\/2},F=3, m_F}\\rightarrow\\ket{nP_{1\/2},F'=2 \\, {\\rm{or}} \\, 3, m_F}$} transitions necessary for determining the hyperfine splitting. At the end of the mm-wave exposure time, an electric field is smoothly ramped up to $100-150$~V\/cm in $1~\\mu$s for state-selective field ionization (SSFI) of the $nS_{1\/2}$ and $nP_{1\/2}$ levels~\\cite{GallagherBook}. $^{85}$Rb$^{+}$ counts are detected with a micro-channel-plate detector (MCP). A timing sequence for the experimental cycle is given in Fig.~\\ref{fig:timingandlevel}(a). \n\n\\begin{figure}[htb]\n \\centering\n \\includegraphics[width=0.45\\textwidth]{HFSPaperFig1.pdf}\n \\caption{(Color online) Timing sequence of an experimental cycle is shown in (a). \"Optical excitation\" refers to the simultaneous 780-nm and 480-nm pulses. The 2D$^+$ MOT and repumping laser beams are always on. In (b), we show the level diagram of $^{85}$Rb states relevant to the experiment (not drawn to scale). Atoms are excited off-resonantly from the upper hyperfine level of the $5S_{1\/2}$ state into the $nS_{1\/2}$ Rydberg state during the \"optical excitation\" pulse. There is a statistical mixture of $F=2$ and $F=3$ Rydberg states after the optical excitation, but the number of atoms in the $F=2$ states is too small to achieve an appropriate signal-to-noise ratio during the spectroscopic mm-wave pulse. Thus, the mm-wave frequency scan range is set to only probe the atoms in $F=3$ state.} \n \\label{fig:timingandlevel}\n\\end{figure}\n\n\\par Optical excitation from the upper hyperfine level of the ground state is provided in the form of an off-resonant, two-photon transition using $780$- and $480$-nm pulses, described in the quantum-state diagram of Fig.~\\ref{fig:timingandlevel}(b). A 780-nm external-cavity diode laser (ECDL) is tuned 100~MHz above the upper-most hyperfine level of the $5P_{3\/2}$ state, while a 960-nm ECDL, amplified and doubled to make $480$-nm light, is tuned to make up the resonance with the Rydberg state. Polarizations of the optical excitation beams and atomic sample, as well as the blue-detuning of the 780-nm laser from the upper-most hyperfine level of the $5P_{3\/2}$ state result in significantly more Rydberg-atom population in \\gfix{$\\ket{nS_{1\/2},F=3, m_F}$ than $\\ket{nS_{1\/2},F=2, m_F}$}. Therefore, we perform our mm-wave spectroscopy only on the $F=3$ hyperfine levels for all $n$ studied.\n\n\\begin{figure}[htb]\n \\centering\n \\includegraphics[width=0.53\\textwidth]{HFSPaperFig2.pdf}\n \\caption{Single-photon resonance spectrum of the $\\ket{44S_{1\/2},F=3}\\rightarrow\\ket{44P_{1\/2},F'}$ transition using mm-waves. The spectrum shown is an arithmetic mean of 8 individual spectra. Each individual spectrum is averaged over 400 experimental cycles. On the frequency axis, we show the $44P_{1\/2}$ hyperfine shifts for each $F'$ level with respect to the center-of-gravity transition frequency, $\\nu_{0}=45.113624~$GHz, i.e., the frequency of the \\gfix{$\\ket{44S_{1\/2},F=3, m_F}\\rightarrow\\ket{44P_{1\/2}}$ } transition with the $44P_{1\/2}$ hyperfine structure removed. Each scatter point corresponds to a frequency step size of 2~kHz. Signal error bars for the scatter points indicate the standard error of the mean (SEM) over the 8 individual points acquired. In this spectrum, the total detected count rate is below two ions per experimental cycle. The solid curves are the double- and individual-Lorentzian fit functions from which the peak centers are acquired to measure the HFS splittings. Measured linewidths are $21(1)~$kHz for both peaks. } \n \\label{fig:spectrum}\n\\end{figure}\n\n\n\\begin{figure*}[t]\n \\centering\n \\includegraphics[width=0.6\\textwidth]{HFSPaperFig3.png}\n \\caption{ This figure explains the static electromagnetic field zeroing process necessary for a HFS measurement. In (a), we show a map of dc Stark shifts on the $\\ket{44S_{1\/2}}\\rightarrow\\ket{44P_{1\/2}}$ transition as a function of applied potential $\\phi_{z}$ on plate electrodes in order to find the voltage that cancels shifts from stray electric fields along the $z$-direction. The differential dc polarizability between the two states is $\\alpha_{44P_{1\/2}}-\\alpha_{44S_{1\/2}} = 9.564$~kHz\/(V\/m)$^{2}$. Here, the mm-wave frequency steps are not resolved enough to observe the HFS splitting. (b) Verification that the hyperfine splitting is not affected by electric fields smaller than $60~$mV\/cm. The $\\ket{43S_{1\/2}, F=3}\\rightarrow\\ket{43P_{1\/2}, F'}$ spectra are plotted as a function of applied electric field $E_x$ in the $x$-direction with stray fields canceled in the other two directions. (c) Calculated Zeeman splitting of the $\\ket{43S_{1\/2},F=3}\\rightarrow\\ket{43P_{1\/2},F'}$ transitions for $F'=2$ and 3. This calculation is for the case that the applied magnetic field is perpendicular ($y$-direction) to the mm-wave polarization. In (d), we show an experimental analogue to our calculation. } \n \\label{fig:HFSPaperFig3}\n\\end{figure*}\n\n\\par The mm-waves are synthesized by an Agilent MXG Analog Signal Generator (Model N5183A) that is referenced to an SRS Model FS725 Rubidium Frequency Standard. For spectroscopy of $n=42$-44, the synthesized mm-waves are frequency doubled by a SAGE Model SFA-192KF-S1 active X2 multiplier and broadcast from $\\simeq$40~cm to the $nS_{1\/2}$ Rydberg atoms with a horn antenna. We do not double the mm-waves at $39.121~$GHz for the $\\ket{46S_{1\/2},F=3, m_F}\\rightarrow\\ket{46P_{1\/2},F', m_F}$ spectrum and directly connect the synthesizer to a standard-gain horn antenna, located $\\simeq$30~cm from the spectroscopic interaction region, with a 20-dBi directivity. \n\n\\par Spectra of the $\\ket{nS_{1\/2},F = 3, m_F}\\rightarrow\\ket{nP_{1\/2},F', m_F}$ transition were acquired for each $n$ in the $n=42-44$ and 46 series. A double Lorentzian was fit to an arithmetic average of eight experimental scans of the mm-wave frequency over the two hyperfine lines. In order to determine $\\nu_{\\text{hfs}}$ from our data, we take the difference between the line centers of the Lorentzian fit functions. The uncertainties in the line centers were added in quadrature and used as the uncertainty in the HFS splitting, $\\delta\\nu_{\\text{hfs}}$. Fig.~\\ref{fig:spectrum} shows a typical spectrum obtained in this study. The linewidths are at the level of the Fourier limit ($0.89\/40~\\mu$s$=22~$kHz), meaning the Rabi frequencies of the transitions are in the range of $10~$kHz, preventing any observable ac Stark shifts.\n\n\n\n\nOnce we obtain $\\nu_{\\text{hfs}}$, we use the $\\delta_{0}$ and $\\delta_{2}$ quantum-defect values for Rb $nP_{1\/2}$ measured in~\\cite{Li2003} and the Rydberg-Ritz equation~\\cite{GallagherBook} to derive a measurement for $A_{\\text{hfs}}$ using Eq.~\\ref{eq:HFSFormula}. These quantities are $\\delta_{0}=2.6548849(10)$ and $\\delta_{2}=0.2900(6)$. Because the uncertainties in $\\delta_{0}$ and $\\delta_{2}$ lead to shifts much smaller than our measurement uncertainties, we neglect them in our uncertainty budget. Thus, $\\delta A_{\\text{hfs}}\/A_{\\text{hfs}}=\\delta\\nu_{\\text{hfs}}\/\\nu_{\\text{hfs}}$. Table~\\ref{measurements} lists $\\nu_{\\text{hfs}}$ and $A_{\\text{hfs}}$ for a given $n$ in the range $n = 42-44$ and 46. A weighted average and uncertainty over all $n$ provides a final value for $A_{\\text{hfs}}$ and a statistical uncertainty, also included in the table.\n\n\\begin{table}[htbp]\n\\caption{\\label{measurements} Summary of HFS splittings and derived $A_{\\text{hfs}}$ using Eq.~\\ref{eq:HFSFormula} and $\\delta_{0}=2.6548849(10)$,~$\\delta_{2}=0.2900(6)$~\\cite{Li2003}. }\n\\begin{ruledtabular}\n\\begin{tabular}{c|c|c}\n $n$ & $\\nu_{\\text{hfs}}$~(kHz) & $A_{\\text{hfs}}$~(GHz)\\\\\n \\hline\n 42 & 72.7(6) & 1.476(12) \\\\ \n 43 & 65.3(6) & 1.429(13)\\\\\n 44 & 60.1(5) & 1.416(12)\\\\\n 46 & 54(1) & 1.466(27) \\\\\n\\end{tabular}\n\\begin{tabular}{c|c}\n $A_{\\text{hfs}}$, weighted average~(GHz) & 1.443 \\\\\n Statistical uncertainty~(GHz)& 0.007 \\\\\n\\end{tabular}\n\\end{ruledtabular}\n\\end{table}\n\n\n\n\\par Symmetry of our observed spectral lines indicates that background electric- and magnetic-field inhomogeneities are negligible. A set of six, orthogonal plate electrodes situated in our science chamber is used to cancel electric fields below 50~mV\/cm by observing shifts in $\\ket{nS_{1\/2}}\\rightarrow\\ket{nP_{1\/2}}$ spectra as a function of applied electric field; a map of these spectra is shown along the $z$-axis in Fig.~\\ref{fig:HFSPaperFig3}(a) for $n=44$. Fig.~\\ref{fig:HFSPaperFig3}(b) displays a more resolved map for $n=43$ with an applied field along the $x$-direction. Electric fields contribute no systematic shift in the HFS splitting because the $nP_{1\/2}$ Rydberg states lack a tensor polarizability that would otherwise cause distortions in the $F'=2$ and $F'=3$ peaks as a result of $|m_{F'}|$ splititngs. Therefore, both $F'$ states and all $|m_{F'}|$ undergo the same dc Stark shifts leaving $\\nu_{\\text{hfs}}$ insensitive to stray electric fields. This insensitivity is verified in Fig.~\\ref{fig:HFSPaperFig3}(b) for $n=43$, where we apply electric field $E_{x}$ magnitudes up to $60~$mV\/cm and scan over both hyperfine peaks of the $43P_{1\/2}$ state. Inhomogeneous broadening from position-dependent electric fields within the atom cloud is the only possible dc Stark effect, which is negligible as exhibited by the line symmetries and linewidths near the Fourier limit of $22$~kHz. Excessive magnetic fields within the interaction region on the other hand do distort measurements of $\\nu_{\\text{hfs}}$ from the Zeeman splittings of the $m_F$ and $m_{F'}$ sublevels, as seen in the following paragraph. \n\n\\begin{figure}[htb]\n \\centering\n \\includegraphics[width=0.5\\textwidth]{HFSPaperFig4.pdf}\n \\caption{ Measured $\\nu_{\\text{hfs}}$ of $43P_{1\/2}$ for given applied magnetic fields in all three spatial dimensions. The SEM of all nine $\\nu_{\\text{hfs}}$ is used as the systematic $\\delta\\nu_{\\text{hfs}}$ from potential stray magnetic fields. Magnetic field uncertainties arise from noise in our current sources.} \n \\label{fig:magneticfield}\n\\end{figure}\n\n\\par Three pairs of externally located Helmholtz coils apply homogeneous magnetic fields to eliminate Zeeman broadening and splitting of the $F'$ states~\\cite{Ramos2019}. Expected and observed behaviors of the Zeeman splittings for $n=43$ are shown in Figs.~\\ref{fig:HFSPaperFig3}(c) and (d), respectively, for the case of a magnetic field perpendicular to the mm-wave polarization. Our stray magnetic fields are reduced down to a magnitude no greater than $5$~mG. In order to quantify the possible systematic uncertainties from any leakage within this range, we take the standard error of the mean (SEM) in a sample of splittings at $n=43$ by offsetting our compensation magnetic fields within 10~mG of the cancellation values in all three directions $x,~y,$~\\&~$z$ independently. A similar analysis was done in the context of measuring the $nS_{1\/2}$ HFS for $^{85}$Rb Rydberg states~\\cite{Ramos2019}. This distribution is presented in Fig.~\\ref{fig:magneticfield}. Our SEM yields $\\delta\\nu_{\\text{hfs}}= 0.6$~kHz at $n=43$ and $\\delta A_{\\text{hfs}}=13$~MHz.\n\n\n\n\\par We also take into account shifts from dipole-dipole interactions between one atom with an internal state of $nS_{1\/2}$ and another with that of $nP_{1\/2}$. In Fig.~\\ref{fig:counts}, we exhibit that the shift in $\\nu_{\\text{hfs}}$ does not exceed $1$~kHz for $n=44P_{1\/2}$, as the maximum ion-count rate, and therefore density, is increased by a factor four by prolonging the optical excitation time up to $15$~$\\mu$s. All measurements in Table~\\ref{measurements} were taken with fewer than three detected total counts. An upper-limit of the $C_{3}$ coefficient is estimated to be $1.7~$GHz~$\\mu$m$^{3}$ for $n=44$ by finding and fitting adiabatic potentials of Rydberg-Rydberg molecules~\\cite{Han2018}. This estimate implies that the atomic spacing is $R\\gtrsim 120~\\mu$m and the systematic uncertainty in $A_{\\text{hfs}}$ from dipole-dipole interactions has an upper limit of $27$~MHz. Higher-order Rydberg-Rydberg interactions, such as van der Waals shifts between two atoms of the same internal state are at the order of 1~mHz or less for these $n$ and therefore are not included in our overall uncertainty budget~\\cite{Ramos2019}.\n\n\\begin{figure}[htb]\n \\centering\n \\includegraphics[width=0.53\\textwidth]{HFSPaperFig5.pdf}\n \\caption{In this series, the $44P_{1\/2}$ HFS is measured for three different bins of total detected ion counts from field-ionized Rydberg atoms. The increase in count rate is achieved by starting the optical excitation 5-10~$\\mu$s earlier to prolong the laser pulse duration. Because the atomic density is rising proportionally with the count rate, the dipole-dipole shifts, if they are significant, should increase as well. There is no apparent dipole-dipole shift over 1~kHz, implying that the atomic spacing must be at a minimum of $120~\\mu$m. } \n \\label{fig:counts}\n\\end{figure}\n\n\\par We present our uncertainty budget in Table~\\ref{uncertainties}. Adding the three sources in quadrature, we find the overall uncertainty to be $\\delta A_{\\text{hfs}}=31$~MHz. \n\n\\begin{table}[htbp]\n\\caption{\\label{uncertainties} Uncertainty budget for a measurement of $A_{\\text{hfs}}$. }\n\\begin{ruledtabular}\n\\begin{tabular}{c|c}\n Source & $\\delta A_{\\text{hfs}}$ (GHz)\\\\\n\\hline\n Dipole-dipole interactions & 0.027 \\\\\n Stray magnetic fields & 0.013 \\\\\n Statistical uncertainty & 0.007 \\\\\n\\end{tabular}\n\\end{ruledtabular}\n\\end{table}\n\n\\par In summary, we measured the hyperfine coupling constant $A_{\\text{hfs}}$ for Rydberg-$nP_{1\/2}$ states of $^{85}$Rb using mm-wave spectroscopy with Fourier-limited linewidths. Our precision in $A_{\\text{hfs}}$ is mainly limited by the estimated lower limit of atomic spacing within our Rydberg cloud that may lead to dipole-dipole interactions. In addition to our measurement's applicability for investigating ultracold Rydberg chemistry~\\cite{Boisseau2002,Niederpruem2016a,Niederpruem2016b,Hollerith2019,Hollerith2021,Duspayev2021,Deiss2021,Zuber2022} and dynamic electric-field sensing of rf waves with thermal Rydberg atoms~\\cite{Holloway2014,Anderson2022,Ripka2022}, the HFS of $nP_{1\/2}$ states can possibly be included in models for quantum simulation~\\cite{Khazali2022}.\n\n\\section*{ACKNOWLEDGMENTS}\nThis work was supported by NSF Grants No. PHY-2110049 and PHY-1806809 and NASA Grant No. NNH13ZTT002N. R.C. acknowledges support from the Rackham Predoctoral Fellowship. \n\n\\bibliographystyle{apsrev4-1}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} {"text":"\\section{Introduction}\nThis paper is the third part of a study of combinatorial quantization of\nChern-Simons\ntheory\\cite{BR1,BR2}. Several ideas developed here are in fact products of\nthose introduced by\nV.V.Fock and A.A.Rosly in their study of Poisson structures on the moduli\nspace of flat\nconnections\\cite{FR}. To understand the motivation of these papers we must\nrecall some general facts\nabout 3D Chern-Simons theory. Chern-Simons theory is a gauge theory in 3\ndimensions defined by the\naction principle \\begin{equation} {\\cal S}_{CS}= \\frac{k}{4 \\pi} tr( \\int_{{\\cal M}} {\\cal\nA}d{\\cal A}+{2 \\over\n3}{\\cal A}^3 ) \\end{equation} where ${\\cal M}$ is a 3-manifold, $k$ a positive integer\nand ${\\cal A}$ a\nconnection associated to a semisimple Lie algebra ${\\cal G}.$ If we first\nsuppose that the manifold\nlocally looks like a cylinder $\\Sigma \\times R$, $R$ considered as the the\ntime direction, we can\nconsider the Chern-Simons theory in an Hamiltonian point of view. We will\ndenote by $A$ the two\nspace-components of the gauge field taken to be the dynamical variables of the\ntheory, and the time\ncomponent $A_0$ will become a Lagrange multiplier. With these notations the\naction can be written:\n\\begin{equation}\n{\\cal S}_{CS}= \\frac{k}{4 \\pi} tr( \\int_{{\\cal M}} (- A \\partial_0 A + 2 A_0 F\n)dt).\n\\end{equation}\nThe first\nterm gives the Poisson structure:\n\\begin{equation}\n\\{ A^a_i(x,y), A^b_j(x',y') \\} =\n\\frac{-2\\pi}{k}\\delta^{ab}\\epsilon_{ij}\\delta^{(2)}((x,y),(x',y')).\n\\label{Poisson}\n\\end{equation}\nThe hamiltonian is a combination of constraints and the\nsecond term imposes as a constraint that the curvature of the connection $A$ is\nzero:\n\\begin{equation}\nF=dA+A^2=0.\\label{constraint}\n\\end{equation}\nComputing the Poisson brackets of the constraints we obtain that they are first\nclass:\n\\begin{equation}\n\\{ F^a(x,y), F^b(x',y') \\} =\n\\frac{2\\pi}{k}f^{ab}_{c}F^c(x,y)\\delta^{(2)}((x,y)(x',y')).\n\\end{equation}\nwhere the $f^{ab}_{c}$ are the structure constants of the Lie algebra ${\\cal\nG}.$\nThe constraints (\\ref{constraint}) generate the infinitesimal gauge\ntransformations of the gauge field then the phase space of the hamiltonian\nChern-Simons theory is the space of flat connections modulo gauge\ntransformations.\\\\\nQuantizing the latter Poisson structure of gauge fields in the usual way,\n\\begin{equation}\n[ A^a_i(x,y), A^b_j(x',y') ] =\n\\frac{-2\\pi}{k}\\delta^{ab}\\epsilon_{ij}\\delta^{(2)}((x,y)(x',y'))\n\\end{equation}\nwe are led to work with an infinite dimensional algebra, observables becoming\nfunctionals over the\nelements of this algebra. The idea of V.V.Fock and A.A.Rosly is completely\ndifferent.Let us briefly\ndescribe it. \\\\\n The space of flat connections modulo the gauge group having only a finite\nnumber of freedom degrees,\n we reduce\nthe connection to live on a graph encoding the topology of the surface and\nequip\nthis \"graph connection\" with a Poisson structure such that the Poisson\nstructure induced on the gauge\n invariant observables is compatible with that induced from\nthe usual symplectic structure. We obtain a \"simulation\" of hamiltonian\nChern-Simons theory in the\nsense that the operator algebra derived from both descriptions are the same.\\\\\n\nWe will consider a graph dividing the surface into contractile plaquettes. This\ngraph will be\nequiped with an additionnal structure of \"ciliated fat graph\", i.e. a graph\nwith a linear order\nbetween adjacent links at each vertex. Let $x,y$ be neighbour vertices of the\ngraph, we will denote\nby $U_{[x,y]}$ the parallel transport operator associated to the link and the\nconnection. The gauge\ngroup acts in the usual way on this object: \\begin{equation}\nU_{[x,y]} \\rightarrow g_{x}U_{[x,y]}g^{-1}_{y}\n\\end{equation}\nWe can define a Lie Poisson structure on such objects \\cite{FR} owning the\nfollowing fascinating\nproperty: \\\\\nthe space of flat graph connections modulo graph gauge group is Lie Poisson\nisomorphic\nto the space of flat connections modulo gauge group on the surface.\\\\\n\nThe problem has been reduced to a finite dimensional problem by limiting the\ngauge group to act at a finite number of sites.\\\\\nThe quantization of such a Lie Poisson structure leads us to an exchange\nalgebra\non which acts a quantum group. The object of \\cite{BR1,AGS} was to define this\nalgebraic structure.\\\\\n\nIn a second time we found a projector in this algebra imposing \"a posteriori\"\nthe flatness condition,\nthe result was then a two dimensional lattice gauge theory based on a quantum\ngroup. The correlation\nfunctions associated to gauge invariant objects in this theory being related to\nexpectation values in\nChern-Simons theory.\\\\\n\nIn our second paper \\cite{BR2} we further investigated the algebra of gauge\ninvariant elements,\nparticularily the algebra associated to loops. Our aim was to describe a new\napproach to knots\ninvariants, showing in a well defined framework the relation between\nReshetikhin-Turaev invariants and\nChern-Simons theory.\\\\\n\nThe aim of the following paper is double:\\\\\n\\begin{itemize}\n\\item to investigate further the\ncomputation of the correlation functions of our theory on any surface and\nconstruct the derived\ninvariants associated to the mapping class group, and establish a new\ndescription of three manifold\ninvariants using a generalization of our previous construction to the case of\nroots of unity\\\\\n\\item\n to build up a three dimensional lattice q-gauge theory associated to\ntriangulations of any\n3-manifolds which will describe a well defined finite path integral formula for\nChern-Simons theory\nand the way to compute any correlation functions of this theory.\\\\ This work is\na tentativ to revisit\nthe work of E.Witten \\cite{W1} with a well defined formalism allowing a lot of\nnew computations and\nspecially computations of invariants associated to intersecting loops in\nChern-Simons theory.\n\\end{itemize}\n\n\n\\section{Lattice gauge theory based on a quantum group}\n\n\\subsection{Quantum groups and exchange algebras associated to fat graphs}\n\nIn this chapter, after a brief summary of the results of \\cite{AC} \\cite{MS},\nwe will further\ndevelop the notion of quantum group at root of unity in the dual version and\nthen, using this construction, generalize the results on gauge fields algebra\ndeveloped in \\cite{BR1,BR2} in a way quite different from that described in\n\\cite{AGS}.\nWe will consider a Hopf algebra $({\\cal A},m,1,\\Delta,S,\\epsilon).$ To simplify\nwe will take\n${\\cal A}={\\cal U}_q ( sl_2 )$ with $q$ being a complex number different from\n$\\pm 1.$ As usual we will refer to ${\\bf R}$ as the universal $R-$matrix\nassociated to ${\\cal A}$ ( we will often write ${\\bf R}=\\sum_i a_i \\otimes b_i$\n), $u$ the element defined by $u= \\sum_i S(a_i)b_i$ verifying the usual\nproperties and $v$ the ribbon element defined by $v^2=uS(u)$ (for details see\n\\cite{Dr,RT1,RT2}).\\\\\nDepending on whether $q$ is a root of unity or not, the representation theory\nof ${\\cal A}$ is\ncompletely different. We will denote by $Irr({\\cal A})$ the set of equivalence\nclasses of finite\ndimensional irreducible representations of ${\\cal A}.$ In each class\n$\\dot{\\alpha}$ we will pick\nout a representativ $\\alpha$ and, like often in physics, denote equivalently by\n$\\alpha$ or\n$V_{\\alpha}$the representation space associated to $\\alpha.$ We will denote by\n$\\bar{\\alpha}$ (resp.\n$\\tilde{\\alpha}$) the right (resp. the left ) contragredient representation\nbuild up from the\nantipode (resp. the inverse of the antipode) by $\\bar{\\alpha}={}^{t}\\alpha\n\\circ S$ and $0$ the one\ndimensional representation associated to $\\epsilon.$ The tensor product of two\nrepresentations is\ndefined by the coproduct $\\Delta.$ If q is a root of unity the decomposition of\nthe tensor product\nof two irreducible representations can involve indecomposable representations\n(i.e representations\nwhich are not irreducible but cannot ye!\n t be decomposed in a direct sum of stable ${\\cal A}-$modules). We are able to\nintroduce\n$\\Psi_{\\alpha \\beta}^{\\gamma,m}$ and $\\Phi_{\\gamma,m}^{\\alpha \\beta}$\nrespectively projection of the\n tensor product of $\\alpha$ and $\\beta$ on the m-th isotypic component $\\gamma$\nand the inclusion of\n$\\gamma$ in the tensor product $\\alpha \\otimes \\beta.$ Using this notation we\nwill make one more\nrestriction between \"physical\" representations verifying $\\Psi_{\\alpha\n\\bar{\\alpha}}^{0}\\Phi_{0}^{\\alpha \\bar{\\alpha}} \\not= 0$ and the other\nrepresentations, we will\ndenote by $Phys({\\cal A})$ this subset of $Irr({\\cal A})$. We will introduce a\nnew tensor product\nbetween elements of $Phys({\\cal A})$ simply realizing a truncation of the\nprevious one, defined by:\n\\begin{equation} \\alpha \\otimes \\beta = \\bigoplus_{\\gamma \\in Phys({\\cal A})}N^{\\alpha\n\\beta}_{\\gamma} \\gamma.\n\\end{equation} $N$ is the fusion matrix of ${\\cal A}$ and we will also use the notation\n$\\delta(\\alpha \\beta\n\\gamma)$ to be equal to 1 or 0 depending on whether $\\gamma$ occurs or not in\nthe decomposition of\nthe tensor product $\\alpha \\otimes \\beta.$\n If q is generic the tensor product of two elements of $Irr(A)$ can be\ndecomposed in a direct\n sum of elements of $Irr(A).$ Moreover all irreducible representations are\n\"physical\", so we do not\nhave to change the tensor product in this case. We will associate new\nprojection and inclusion\noperators $\\psi_{\\alpha \\beta}^{\\gamma,m}$ and $\\phi_{\\gamma,m}^{\\alpha \\beta}$\nbuild up from the\ntruncated tensor product. We will also use the $6-j$ notation\n$\\sixj{\\alpha}{\\beta}{\\gamma}{\\delta}{\\mu}{\\nu}$ defined in the usual way from\nprojection and\ninclusion operators (see \\cite{KR} for definitions and properties).\\\\ We will\nuse the following\nnotation replacing the coproduct by a truncated coproduct for any element $\\xi$\nof the algebra\n${\\cal A}$: \\begin{equation} {\\buildrel {\\alpha \\otimes \\beta} \\over {\\xi}}= \\sum_{\\gamma\n\\in Phys({\\cal A})}\n\\phi^{\\alpha \\beta}_{\\gamma,m} {\\buildrel \\gamma \\over \\xi}\n\\psi^{\\gamma,m}_{\\alpha \\beta}. \\end{equation} The\nantipode and counity maps do not change through truncation and we will again\nhave: \\begin{equation} {\\buildrel\n\\alpha \\over S(\\xi)}= {}^{t} {\\buildrel \\bar{\\alpha} \\over \\xi} \\end{equation}\nThe first trivial properties of the projection and inclusion operators are:\n\\begin{eqnarray}\n&&\\psi_{\\alpha \\beta}^{{\\gamma'},m'}\\phi_{\\gamma,m}^{\\alpha\n\\beta}=\\delta_{m,m'}\\delta_{\\gamma,{\\gamma'}}\\delta(\\alpha \\beta\n\\gamma)id_{\\gamma}\\\\\n&&\\sum_{\\gamma \\in Phys({\\cal A}),m} \\phi^{\\alpha\n\\beta}_{\\gamma,m}\\psi^{\\gamma,m}_{\\alpha \\beta}={\\buildrel {\\alpha \\otimes\n\\beta} \\over {\\bf 1}}\\\\\n&&\\phi^{\\alpha 0}_{\\beta}=\\phi^{0 \\alpha}_{\\beta}=\\psi_{\\alpha\n0}^{\\beta}=\\phi_{0 \\alpha }^{\\beta}=\\delta_{\\alpha \\beta} id_{\\alpha}.\n\\end{eqnarray}\nThe essential fact is that when the truncation is not trivial (i.e in the root\nof unity case) the representations $((\\alpha \\otimes \\beta)\\otimes \\gamma)$ and\n$(\\alpha \\otimes (\\beta \\otimes \\gamma))$ are no more equal but are equivalent,\nthe intertwiner map between them being ${\\buildrel {\\alpha \\beta \\gamma} \\over\n\\Theta}$ defined by:\n\\begin{equation}\n{\\buildrel {\\alpha \\beta \\gamma} \\over \\Theta}=\\sum_{\\delta,\\nu,\\mu \\in\nPhys({\\cal A})}\n\\sixj{\\gamma}{\\beta}{\\delta}{\\alpha}{\\nu}{\\mu}\\phi_{\\delta}^{\\beta\n\\gamma}\\phi_{\\nu}^{\\alpha \\delta}\\psi^{\\mu \\gamma}_{\\nu}\\psi^{\\alpha\n\\beta}_{\\mu}\n\\end{equation}\n( where we have omited the multiplicities to simplify the notation, it will\noften be the case in the\nfollowing). We will denote by ${\\buildrel {\\alpha \\beta \\gamma} \\over\n{\\Theta^{-1}}}$ its quasi-inverse.\n\nWe will often use the notation ${\\buildrel {\\alpha \\beta \\gamma} \\over\n\\Theta}_{123}=\\sum_{i} {\\buildrel \\alpha \\over {\\theta_i^{(1)}}} \\otimes\n{\\buildrel \\beta \\over\n{\\theta_i^{(2)}}} \\otimes {\\buildrel \\gamma \\over {\\theta_i^{(3)}}},$ and the\ncoproduct notations\n${\\buildrel {(\\alpha \\otimes \\beta) \\gamma \\delta} \\over\n\\Theta}_{1234}=\\sum_{i} {\\buildrel \\alpha\n\\over {\\theta_i^{(11)}}} \\otimes {\\buildrel \\beta \\over {\\theta_i^{(12)}}}\n\\otimes {\\buildrel \\gamma\n\\over {\\theta_i^{(2)}}} \\otimes {\\buildrel \\delta \\over {\\theta_i^{(3)}}}...$\\\\\n\nIn the case where $q$ is generic ${\\buildrel {\\alpha \\beta \\gamma} \\over\n\\Theta}$ is simply\nthe identity but more generally it is possible to collect some interesting\nproperties in the root of\nunity case. Using the pentagonal identity and other trivial identities on $6-j$\nsymbols we can verify:\n\\begin{eqnarray} {\\buildrel {\\alpha \\beta (\\gamma \\otimes \\delta)} \\over\n\\Theta}\\;\\;{\\buildrel {(\\alpha \\otimes\n\\beta) \\gamma \\delta} \\over \\Theta}&=&({\\buildrel \\alpha \\over {\\bf\n1}}\\otimes{\\buildrel {\\beta \\gamma\n\\delta} \\over \\Theta}){\\buildrel {\\alpha (\\beta \\otimes \\gamma) \\delta} \\over\n\\Theta}({\\buildrel\n{\\alpha \\beta \\gamma} \\over \\Theta}\\otimes{\\buildrel \\delta \\over {\\bf\n1}})\\label{pentagon}\\\\\n{\\buildrel {0 \\alpha \\beta} \\over \\Theta}={\\buildrel { \\alpha 0 \\beta} \\over\n\\Theta}&=& {\\buildrel {\n\\alpha \\beta 0} \\over \\Theta}= {\\buildrel {\\alpha \\otimes \\beta} \\over {\\bf\n1}}\\label{theta0} \\end{eqnarray}\n and\nother similar identities for $\\Theta^{-1}.$ Moreover we have the quasi-inverse\nproperties:\n\\begin{equation}\n{\\buildrel {\\alpha \\beta \\gamma} \\over \\Theta}{\\buildrel {\\alpha \\beta \\gamma}\n\\over\n{\\Theta^{-1}}}={\\buildrel {(\\alpha \\otimes \\beta) \\otimes \\gamma} \\over {\\bf\n1}} \\mbox{ and }\n{\\buildrel {\\alpha \\beta \\gamma} \\over {\\Theta^{-1}}}{\\buildrel {\\alpha \\beta\n\\gamma} \\over\n{\\Theta}}={\\buildrel {\\alpha \\otimes (\\beta \\otimes \\gamma)} \\over {\\bf 1}} \\end{equation}\nrecalling that, here, ${\\buildrel {\\alpha \\otimes \\beta} \\over {\\bf 1}} $ is\nsimply a projector.\n Let us now define\nintertwiners between $\\alpha\\otimes\\beta$ and $\\beta\\otimes\\alpha$ using our\nbasic objects\n$\\psi,\\phi$: \\begin{eqnarray} &&P_{12}\\; {\\buildrel \\alpha\\beta \\over R}\\!= \\sum_{\\gamma \\in Phys({\\cal A}),m}\n\\lambda_{\\alpha \\beta \\gamma}^{1 \\over 2}\\; \\phi^{\\beta\n\\alpha}_{\\gamma,m}\\psi_{ \\alpha \\beta}^{\\gamma,m}\\label{Rdef}\\\\\n&& P_{12}\\; \\Rpmff{\\alpha}{\\beta}= \\sum_{\\gamma \\in Phys({\\cal A}),m}\n\\lambda_{\\alpha \\beta \\gamma}^{-{1 \\over 2}}\\; \\phi^{\\beta\n\\alpha}_{\\gamma,m}\\psi_{ \\alpha \\beta}^{\\gamma,m}\\\\\n\\end{eqnarray}\nwhere $R'=\\sigma(R)$ and $\\lambda_{\\alpha \\beta\n\\gamma}=(\\frac{v_{\\alpha}v_{\\beta}}{v_{\\gamma}})$\nwhere $v_{\\alpha}$ is the Drinfeld casimir, equal to $q^{C^{(2)}_{\\alpha}}$,\nwhere\n$C^{(2)}_{\\alpha}$ is the quadratic Casimir. We will denote in the following\n${\\buildrel \\alpha\\beta \\over R}\\!=\\sum_i {\\buildrel\n\\alpha \\over a_i} \\otimes {\\buildrel \\beta \\over b_i}$ and ${\\buildrel \\alpha\\beta \\over R}\\!^{-1}=\\sum_i\n{\\buildrel \\alpha \\over\nc_i} \\otimes {\\buildrel \\beta \\over d_i}$ and use sometimes the notation\n$R^{(+)}=R$ and\n$R^{(-)}=R'^{-1}$ . Using the hexagonal identities on the $6-j$ symbols it can\nbe shown that: \\begin{eqnarray}\n&&\\Rff{(\\alpha \\otimes \\beta)}{\\gamma}={\\buildrel {\\gamma \\alpha \\beta} \\over\n\\Theta}\\Rff{\\alpha}{\\gamma}{\\buildrel {\\alpha \\gamma \\beta} \\over\n{\\Theta^{-1}}}\\Rff{\\beta}{\\gamma}{\\buildrel {\\alpha \\beta \\gamma} \\over\n\\Theta}\\\\\n&&\\Rff{\\alpha}{(\\beta \\otimes \\gamma)}={\\buildrel {\\beta \\gamma \\alpha} \\over\n{\\Theta^{-1}}}\\Rff{\\alpha}{\\gamma}{\\buildrel {\\beta \\alpha \\gamma} \\over\n{\\Theta}}\\Rff{\\alpha}{\\beta}{\\buildrel {\\alpha \\beta \\gamma} \\over\n{\\Theta^{-1}}}\\label{quasitriang}\n\\end{eqnarray} which is simply the analog of the quasitriangularity property of\n$R-$matrices.\\\\ This matrix is\nno more inversible but we have: \\begin{equation} {\\buildrel \\alpha\\beta \\over R}\\! \\Rmff{\\alpha}{\\beta}=\n\\sigma({\\buildrel {\\alpha \\otimes\n\\beta} \\over {\\bf 1}}) \\mbox{ and } \\Rmff{\\alpha}{\\beta}{\\buildrel \\alpha\\beta \\over R}\\! = {\\buildrel\n{\\alpha \\otimes \\beta}\n\\over {\\bf 1}}.\\label{Rinv} \\end{equation} Let us now study the properties of the\nantipodal map and develop the\nanalog of the ribbon properties \\cite{AC}.\n We will denote by ${\\buildrel \\alpha \\over A}$ and\n${\\buildrel \\alpha \\over B}$ the matrices defined by $\\psi^{0}_{ \\bar{\\alpha}\n\\alpha}=\n< . , {\\buildrel \\alpha \\over A} .>$\nand $\\phi_{0}^{\\alpha \\bar{\\alpha}}= (\\lambda \\rightarrow \\lambda \\sum_i\n{\\buildrel \\alpha \\over B}\n{\\buildrel \\alpha \\over{e_i}} \\otimes {\\buildrel {\\bar{\\alpha}} \\over {e^i}}$)\nwhere ${\\buildrel\n\\alpha \\over {e_i}}$ (resp. ${\\buildrel {\\bar{\\alpha}} \\over {e^i}}$) is a\nbasis of the representation\nspace of $\\alpha$ (resp. $\\bar{\\alpha}$) and $<.,.>$ is the duality bracket. To\nchoose the\nnormalisation of $\\phi$ and $\\psi$s we will impose the ambiant isotopy\nconditions: \\begin{equation} \\sum_i\n\\theta^{(1)}_i B S(\\theta^{(2)}_i) A \\theta^{(3)}_i = 1 \\mbox{ and } \\sum_i\nS({\\theta^{-1}}^{(1)}_i)\nA {\\theta^{-1}}^{(2)}_i B S({\\theta^{-1}}^{(3)}_i) = 1 \\end{equation}\n In order to generalize the known properties relative to the antipode, we will\nalso introduce some\nnotations which will be useful in the following: \\begin{eqnarray} &&{\\buildrel {\\alpha\n\\beta} \\over G } =\n\\sum_{i,j} ({\\buildrel {\\alpha} \\over {S(\\theta^{-1 (12)}_i)}}\\otimes\n{\\buildrel {\\beta} \\over\n{S(\\theta^{-1 (11)}_i)}}) ({\\buildrel {\\alpha} \\over\n{S(\\theta^{(2)}_j})}\\otimes {\\buildrel {\\beta}\n\\over {S(\\theta^{(1)}_j})})({\\buildrel \\alpha \\over A} \\otimes {\\buildrel \\beta\n\\over A} )({\\buildrel\n{\\alpha} \\over {\\theta^{(3)}_j}}\\otimes {\\buildrel {\\beta} \\over {\\bf\n1}})({\\buildrel {\\alpha} \\over\n{\\theta^{-1 (2)}_i }}\\otimes {\\buildrel {\\beta} \\over {\\theta^{-1\n(3)}_i)}}\\nonumber\\\\ &&{\\buildrel\n{\\alpha \\beta} \\over D } = \\sum_{i,j} ({\\buildrel {\\alpha \\otimes \\beta} \\over\n{\\theta^{(1)}_i}})\n({\\buildrel {\\alpha} \\over {\\theta^{-1 (1)}_j}}\\otimes {\\buildrel {\\beta} \\over\n{\\theta^{-1 (2)}_j}})(\n{\\buildrel \\alpha \\over B} \\otimes {\\buildrel \\beta \\over B} ) ({\\buildrel\n\\alpha \\over {\\bf 1}}\n\\otimes {\\buildrel {\\beta} \\over {S(\\theta^{-1(3)}_j})}) ({\\buildrel {\\alpha}\n\\over {S(\\theta^{\n(3)}_i)}}\\otimes{\\buildrel {\\beta} \\over {S(\\theta^{ (2)}_i })}) \\nonumber\\\\\n&&{\\buildrel {\\alpha\n\\beta} \\over f }= \\sum_i ({\\buildrel {\\alpha} \\over {S(\\theta^{-1\n(12)}_i})}\\otimes {\\buildrel {\\beta}\n\\over {S(\\theta^{-1 (11)}_i})} ) {\\buildrel {\\alpha \\beta} \\over G }\\;\n {\\buildrel {\\alpha \\otimes \\beta} \\over {(\\theta^{-1 (2)}_i B S(\\theta^{-1\n(3)}_i))} }\n\\end{eqnarray}\nit can be shown that the latter matrices verify:\n\\begin{eqnarray}\n&&{\\buildrel {\\alpha \\beta} \\over {f^{-1}} }\\;{\\buildrel {\\alpha \\beta} \\over G\n}= {\\buildrel {\\alpha \\otimes \\beta} \\over A } \\mbox{ and }\n{\\buildrel {\\alpha \\beta} \\over D }\\;{\\buildrel {\\alpha \\beta} \\over f }=\n{\\buildrel {\\alpha \\otimes \\beta} \\over B }\\label{deltaA}\\\\\n&&\\phi^{\\alpha \\beta}_{\\gamma}={\\buildrel {\\alpha \\beta} \\over\n{f^{-1}}}\\;{}^{t}\\psi^{\\bar{\\gamma}}_{\\bar{\\beta}\\bar{\\alpha}}\\mbox{ and }\n\\psi_{\\alpha \\beta}^{\\gamma}=\n{}^{t}\\phi_{\\bar{\\gamma}}^{\\bar{\\beta}\\bar{\\alpha}}\\;{\\buildrel {\\alpha \\beta}\n\\over f }\n\\end{eqnarray}\nWe endly introduce the element $u$ associated to the square of the antipode,\ndefined by:\n\\begin{equation}\nu=\\sum_{i,j} S(\\theta^{-1 (2)}_i B S(\\theta^{-1 (3)}_i))S(b_j)A a_j\\theta^{-1\n(1)}_i,\n\\end{equation}\nu is invertible and\n\\begin{eqnarray}\n1&=&u\\sum_{i,j} S^{-1}(\\theta^{-1 (1)}_i) S^{-1}(A d_j)c_j\\theta^{-1 (2)}_i B\n\\theta^{-1 (3)}_i=\\\\\n&=&S^{2}(\\sum_{i,j} S^{-1}(\\theta^{-1 (1)}_i) S^{-1}(A d_j)c_j\\theta^{-1 (2)}_i\nB \\theta^{-1 (3)}_i)u\n\\end{eqnarray}\nmoreover we have as usual the essential property\n\\begin{equation}\n\\forall \\xi \\in {\\cal A},S^2(\\xi)=u \\xi u^{-1}\n\\end{equation}\nand the usual corollaries\n\\begin{eqnarray}\n&&S^{2}(u)=u\\\\\n&&uS(u)=S(u)u \\mbox{ is central}\\\\\n&&\\sum_i S(b_i) A a_i=S(A)u=S(u)u\\sum_i S(c_i)Ad_i\\label{contract}\\\\\n&&\\epsilon(u)=1\n\\end{eqnarray}\nWe will denote by $v$ the element satisfying:\n\\begin{eqnarray}\n&&v^2=uS(u)\\\\\n&& S(v)=v \\mbox{ and } \\epsilon(v)=1.\n\\end{eqnarray}\nand by $\\mu$ the element $uv^{-1}.$ Then it can be shown that:\n\\begin{equation}\n{\\buildrel {\\alpha \\otimes \\beta} \\over \\mu} = {\\buildrel {\\alpha \\beta} \\over\n{f^{-1}}}\n{\\buildrel {\\alpha \\beta} \\over {(S \\otimes S)(\\sigma(f))}}\\;\\;({\\buildrel\n{\\alpha} \\over \\mu}\\otimes{\\buildrel {\\beta} \\over \\mu})\n\\end{equation}\nusing the latter notations it can be shown that $\\phi_{0}^{\\alpha\n\\bar{\\alpha}}=\n$ and $\\psi^{0}_{\\bar{\\alpha} \\alpha}=(\\lambda \\rightarrow\n\\lambda \\sum_i\n {\\buildrel {\\bar{\\alpha}} \\over {e^i}} \\otimes \\mu^{-1}S(B){\\buildrel \\alpha\n\\over {e_i}}).$ In the\nfollowing the q-dimension of the representation $\\alpha$ will be defined by\n$[d_{\\alpha}]=tr_{\\alpha}(S(A) \\mu B).$\n\n\n\\medskip\n\nNow, using the latter framework we can give a well defined construction of the\nquantum group in the dual version for any value of $q$. As a vector space this\nalgebra, called $\\Gamma$, is generated by $\\{ {\\buildrel \\alpha \\over {g^i_j}}\n\\mbox{ with $\\alpha \\in Phys({\\cal A})$ and $i,j=1 \\cdots dim(\\alpha)$} \\}$\nand the product is simply defined by:\n\\begin{equation}\n{\\buildrel \\alpha \\over {g}}_1 {\\buildrel \\beta \\over {g}}_2= \\sum_{\\gamma \\in\nPhys({\\cal A})}\\phi_{\\gamma}^{\\alpha \\beta}\\;\\;{\\buildrel \\gamma \\over\n{g}}\\;\\;\\psi^{\\gamma}_{\\alpha \\beta}.\n\\end{equation}\nThis product is not associative but verify:\n\\begin{equation}\n{\\buildrel {\\alpha \\beta \\gamma} \\over \\Theta}_{123}(({\\buildrel \\alpha \\over\n{g}}_1 {\\buildrel \\beta \\over {g}}_2){\\buildrel \\gamma \\over {g}}_3)=\n({\\buildrel \\alpha \\over {g}}_1({\\buildrel \\beta \\over {g}}_2 {\\buildrel \\gamma\n\\over {g}}_3))\n{\\buildrel {\\alpha \\beta \\gamma} \\over \\Theta}_{123}.\n\\label{quasi-associativity}\n\\end{equation}\nMoreover we have the exchange relation:\n\\begin{equation}\n{\\buildrel \\alpha\\beta \\over R}\\!_{12} \\;\\;{\\buildrel \\alpha \\over {g}}_1 {\\buildrel \\beta \\over {g}}_2=\n{\\buildrel \\beta \\over {g}}_2 {\\buildrel \\alpha \\over {g}}_1 \\;{\\buildrel \\alpha\\beta \\over R}\\!_{12}.\n\\label{exchange}\n\\end{equation}\nThis algebra can be equiped with a coproduct and a counity:\n\\begin{equation}\n\\Delta({\\buildrel \\alpha \\over {g^i_j}})=\\sum_i {\\buildrel \\alpha \\over\n{g^i_k}}\\; \\otimes \\;{\\buildrel \\alpha \\over {g^k_j}}\n\\mbox{ and }\\epsilon({\\buildrel \\alpha \\over {g^i_j}})=\\delta^i_j.\n\\end{equation}\nMoreover it can be shown that the antipodal map $S$ defined to be the linear\nmap verifying $S({\\buildrel \\alpha \\over {g^i_j}})={\\buildrel {\\bar{\\alpha}}\n\\over {g^j_i}}$ owns the properties:\n\\begin{eqnarray}\n&& S(g)^i_k A^k_l g^l_j=A^i_j \\label{antipode1}\\\\\n&&(S(A)\\mu)^k_l g^l_j S(g)^i_k=(S(A)\\mu)^i_j \\label{antipode2}\\\\\n&&S({\\buildrel {\\beta} \\over {g_2}})S({\\buildrel {\\alpha} \\over\n{g_1}})={\\buildrel {\\alpha \\beta} \\over {f}}_{12}\\;S({\\buildrel {\\alpha} \\over\n{g_1}} {\\buildrel {\\beta} \\over {g_2}})\\;{\\buildrel {\\alpha \\beta} \\over\n{f^{-1}}}_{12}\\\\\n&&S^{2}({\\buildrel {\\alpha} \\over {g}}) = {\\buildrel {\\alpha} \\over\n{\\mu}}{\\buildrel {\\alpha} \\over {g}}{\\buildrel {\\alpha} \\over {\\mu^{-1}}}\n\\end{eqnarray}\n\n\\medskip\n\nOur aim is now to define as in our first paper the gauge theory associated\nto this gauge symmetry algebra.\n\n\nLet $\\Sigma$ be a compact connected oriented surface\n with boundary $\\partial\\Sigma$ and let ${\\cal T}$ be a triangulation of\n$\\Sigma.$ Let us denote by ${\\cal F}$ the\noriented faces of ${\\cal T},$ by ${\\cal L}$ the set of edges counted\nwith their orientation. If $l$ is an interior link, $-l$ will denote\nthe opposite link.\nWe have ${\\cal L}={{\\cal L}}^{int}\\cup{{\\cal L}}^{\\partial \\Sigma},$ where\n${{\\cal L}}^{int}, {{\\cal L}}^{\\partial \\Sigma}$ are respectively the set of\ninterior edges and boundary edges.\n\nFinally let us also define ${\\cal V}$ to be the set of points (vertices)\nof\nthis triangulation, ${\\cal V}={{\\cal V}}^{int}\\cup {{\\cal V}}^{\\partial \\Sigma}$, where\n${{\\cal V}}^{int},\n{{\\cal V}}^{\\partial \\Sigma}$ are respectively\nthe set of interior vertices and boundary vertices.\n\nIf $l$ is an oriented link it will be convenient to write $l=xy$ where $y$ is\nthe departure point\nof $l$ and $x$ the end point of $l.$ We will write $y=d(l)$ and $x=e(l).$\n\n\n\n\\begin{definition}[gauge symmetry algebra]\nLet us define for $z\\in {\\cal V},$ the Hopf algebra $\\Gamma_z=\\Gamma\\times \\{z\\}$\n and $\\hat\n\\Gamma=\\bigotimes_{z\\in {\\cal V}} \\Gamma_{z}.$ This Hopf algebra was called in\n \\cite{BR1} ``the gauge\nsymmetry algebra.''\n\\end{definition}\nIf $x$ is a vertex we shall write $\\buildrel\\alpha \\over g_x$ to denote the embedding of the\n element $\\buildrel\\alpha \\over g$ in\n$\\Gamma_x.$\n\n\nIn order to define the non commutative analogue of algebra of gauge fields we\nhave to\nendow the triangulation with an additional structure\\cite{FR}, an order between\nlinks incident to\na same vertex, the {\\sl cilium order}.\n\n\\begin{definition}[Ciliation]\nA ciliation of the triangulation is an assignment of a cilium\n$c_z$ to each vertex $z$ which consists in a non zero tangent vector at z.\nThe orientation of the Riemann surface\ndefines a canonical cyclic order of the links admitting $z$ as departure or end\npoint. Let $l_1, l_2$ be links incident to a common vertex $z,$\nthe strict partial cilium order $<_{c} $ is defined by:\n\n $l_1<_{c}l_2$ if $l_1\\not=l_2, -l_2$ and the unoriented links\n $c_z,l_1,l_2$ appear\nin the cyclic order defined\nby the orientation.\n\\end{definition}\nIf $l_1, l_2$ are incident to a same vertex $z$ we define:\n\n$$\\epsilon(l_1,l_2)=\\left\\{ \\begin{array}{ll}\n+1 &\\mbox{if}\\,l_1<_{c} l_2\\\\\n -1& \\mbox{if}\\,l_2<_c l_1\n\\end{array}\n\\right. $$\n\n\n\n\\begin{definition}[Gauge fields algebra]\nThe algebra of {\\it } gauge fields \\cite{AGS}\\cite{BR1} $\\Lambda$ is the non\nassociative algebra\ngenerated by the formal variables $\\buildrel\\alpha \\over u(l)^i_j$ with $l\\in {\\cal L},\\alpha\\in\nPhys({\\cal A}),i,j=1\\cdots dim(\\alpha)$ and satisfying the following relations:\n\n\n\\smallskip\n\n{\\bf Commutation rules}\n\\begin{eqnarray}\n& &{\\buildrel \\alpha\\beta \\over R}\\!_{12} \\buildrel\\alpha \\over u (yx)_1 \\buildrel\\beta \\over u (yz)_2 = \\buildrel\\beta \\over u (yz)_2 \\buildrel\\alpha \\over u (yx)_1\\label{SS}\\\\\n & &\\buildrel\\alpha \\over u (xy)_1 {}(S \\otimes id)({\\buildrel \\alpha\\beta \\over R}\\!_{12}) \\buildrel\\beta \\over u (yz)_2 =\\buildrel\\beta \\over u (yz)_2 \\buildrel\\alpha \\over u\n(xy)_1\\label{ES}\\\\\n& &\\buildrel\\alpha \\over u (xy)_1 \\buildrel\\beta \\over u (zy)_2 (S \\otimes S)({\\buildrel \\alpha\\beta \\over R}\\!_{12}) =\n \\buildrel\\beta \\over u (zy)_2 \\buildrel\\alpha \\over u (xy)_1 \\label{EE}\\\\\n& &\\, \\forall \\,\\,(yx), (yz) \\in {\\cal L}\\, x\\not= z \\,\\,\\,{\\rm and}\\,\\,\\,\nxy<_{c}yz\\nonumber\\\\\n& &\\buildrel\\alpha \\over u(l){\\buildrel \\alpha \\over A}\\buildrel\\alpha \\over u(-l)={\\buildrel \\alpha \\over\nB}\\label{ES=1}\\\\\n& &\\forall \\,\\,l \\in {\\cal L}^i, \\nonumber\\\\\n& &\\buildrel\\alpha \\over u (xy)_1 \\buildrel\\beta \\over u (zt)_2 = \\buildrel\\beta \\over u (zt)_2 \\buildrel\\alpha \\over u (xy)_1 \\label{Udisjoint}\\\\\n& &\\forall\\,\\, x, y, z, t \\mbox{ pairwise distinct in}\\, {\\cal V}\\nonumber\n\\end{eqnarray}\n\n\n{\\bf Decomposition rule}\n\\begin{eqnarray}\n\\buildrel\\alpha \\over u(l)_1 \\buildrel\\beta \\over u(l)_2&=&\\sum_{\\gamma,m}\\phi_{\\gamma,m}^{\\alpha,\\beta}\n\\buildrel\\gamma \\over u(l)\\psi_{\\beta,\\alpha}^{\\gamma,m}{\\buildrel {\\alpha \\beta} \\over {f^{-1}}}\n P_{12}, \\label{UCG}\\\\\n\\buildrel 0\\over u(l)&=&1,\\,\\,\\forall l\\in {\\cal L}.\n\\end{eqnarray}\n\n{\\bf Quasi-associativity}\nLet ${\\cal M}_P$ be a monomial of gauge fields algebra elements with\na certain parenthesing $P.$ For each vertex $x$ of the triangulation\nwe construct a tensor product of representations of ${\\cal A}$ by replacing\neach ${\\buildrel \\alpha \\over {u_l}}$ in the monomial by the vector space\n$\\alpha$ (resp. $\\bar{\\alpha}$ resp. $0$) depending on whether $x$ is the\nendpoint (resp. departure point resp. not element) of the edge $l,$ and\nkeeping the previous parenthesing. Let us consider two different parenthesing\n$P_1$ and $P_2$ of the same monomial. We can construct for both, as described\nbefore, the corresponding vector spaces for each $x$ and deduce the intertwiner\n$\\Theta_x$ relating them. The relation of quasi-associativity is then simply:\n\\begin{equation}\n(\\prod_{x \\in {\\cal V}} \\Theta_x ){\\cal M}_{P_1}={\\cal M}_{P_2}\\label{assocalg}\n\\end{equation}\n\\end{definition}\n\n\\begin{proposition}[ Gauge covariance ] $\\Lambda$ is\na right $\\hat \\Gamma$ comodule defined by the morphism of algebra\n$\\Omega:\\Lambda\\rightarrow \\Lambda\\otimes\\hat\\Gamma$ :\n\\begin{equation}\n\\Omega(\\buildrel\\alpha \\over u(xy))=\\buildrel\\alpha \\over g_x \\buildrel\\alpha \\over u(xy) S(\\buildrel\\alpha \\over g_y).\n\\end{equation}\nThe definition relations of the gauge fields algebra are compatible\nto the coaction of the gauge symmetry algebra.\n\\end{proposition}\n\nThe subalgebra of gauge invariant elements of $\\Lambda$ is denoted\n $\\Lambda^{inv}.$\n\n\\subsection{Invariant measure, holonomies, zero-curvature projector}\n\nIt was shown\n(provided some assumption on the existence of a basis of $\\Lambda$ of a special\ntype) \\cite{AGS}\\cite{BR1}\nthat there exists a unique non zero linear form $h\\in \\Lambda^{\\star}$\nsatisfying:\n\n\\begin{enumerate}\n\\item (invariance) $(h\\otimes id)\\Omega(A)= h(A)\\otimes 1 \\,\\,\\forall A\\in\n\\Lambda$\n\\item (factorisation) $h((A)(B))=h(A)h(B)\\\\\n\\forall A\\in\\Lambda_X, \\forall B\\in\\Lambda_Y,\\forall X, Y\\subset L,\\,\\,\n(X\\cup -X) \\cap (Y\\cup - Y)=\\emptyset$\n \\end{enumerate}\n(we have used the notation $\\Lambda_{X}$ for $X\\subset {\\cal L}$ to denote the\nsubalgebra of $\\Lambda$ generated as an algebra by $\\buildrel\\alpha \\over u_{l}$ with $l\\in X$).\n\n\n\n\nIt can be evaluated on any element using the formula:\n\\begin{equation}\nh(\\buildrel\\alpha \\over u(x,y){}^i_j)=\\delta_{\\alpha,0}\n\\end{equation}\nwhere $0$ denotes the trivial representation of dimension $1,$ i.e $0$ is the\ncounit.\n\nIt was convenient to use the notation $\\int d h$ instead of $h.$\nWe obtained the important formula:\n\\begin{equation}\nh(\\buildrel\\alpha \\over u(y,x)_1 (S \\otimes id)(\\buildrel \\alpha\\alpha \\over R_{12}) v_{\\alpha}^{-1}\\buildrel\\alpha \\over u(x,y)_2)=\n{1\\over [d_{\\alpha}]}P_{12}{\\buildrel \\alpha \\over B}_1.\n\\label{ortho}\n\\end{equation}\n\n\nA path $P$ (resp. a loop $P$) is a path (resp.a loop) in the graph attached\n to the triangulation of $\\Sigma$, given by the collection of its vertices,\n it will also denote equivalently the\ncontinuous curve (resp. loop) in $\\Sigma$ defined by the links of $P.$\nIn this article we will denote by $P=[x_n,x_{n-1},\\cdots,x_1,x_0]$ a link with\ndeparture point $x_0$ and end point $x_n$.\nFollowing the definition for links, the departure point of $P$ is denoted\n$d(P)$ and its\nendpoint $e(P).$ The set of vertices (resp. edges) of the path $P$ is\ndenoted by ${\\cal V}(P)$ (resp. ${\\cal L}(P)$ ), the cardinal of this set\nis called the \"length\" of $P$ and will be denoted by $Length(P).$\n\n\nProperties of path and loops such as self intersections, transverse\nintersections will always be understood as properties satisfied by the\ncorresponding curves on $\\Sigma.$\n\nLet $P=[x_n,...,x_0]$ be a path, we defined\n the sign $\\epsilon(x_i, P)$ to be $-1$ (resp. $1$) if\n$x_{i-1}x_i<_{c}x_ix_{i+1}$ (resp. $x_ix_{i+1}<_{c}x_{i-1}x_i$).\n\n\\begin{definition}[Holonomies and Wilson loops]\nIf $P$ is a simple path $P=[x_n,\\cdots,x_0]$ with $x_0\\not= x_n$, we define\nthe holonomy along $P$ by\n \\begin{equation}\n\\buildrel\\alpha \\over u_P=v_{\\alpha}^{{1\\over 2}\\sum_{i=1}^{n-1}\\epsilon(x_i, P)}(\\buildrel\\alpha \\over u(x_n\nx_{n-1}){\\buildrel \\alpha \\over A} \\buildrel\\alpha \\over u(x_{n-1}x_{n-2}){\\buildrel \\alpha \\over\nA} \\cdots{\\buildrel \\alpha \\over A}\\buildrel\\alpha \\over u(x_{1}x_{0})) .\n\\end{equation}\nWhen $C$ is a simple loop $C=[x_{n+1}=x_0, x_n,\\cdots, x_0],$\nwe define the holonomy along $C$ by\n \\begin{equation}\n\\buildrel\\alpha \\over u_C=v_{\\alpha}^{{1\\over 2}(\\sum_{i=1}^n \\epsilon(x_i, C)-\n\\epsilon(x_0,C))}(\\buildrel\\alpha \\over u(x_0 x_{n}){\\buildrel \\alpha \\over A}\n\\buildrel\\alpha \\over u(x_{n}x_{n-1}){\\buildrel \\alpha \\over A} \\cdots{\\buildrel \\alpha \\over\nA}\\buildrel\\alpha \\over u(x_{1}x_{0})).\n\\end{equation}\nWe define an element of $\\Lambda$, called {\\sl Wilson loop} attached to $C:$\n\\begin{equation}\n\\buildrel\\alpha \\over W_C= tr_{\\alpha}(S({\\buildrel \\alpha \\over A})\\buildrel\\alpha \\over \\mu \\buildrel\\alpha \\over u_C).\\label{Wilsonloop1}\n\\end{equation}\nthe interior parenthesing being irrelevant because the loop is simple\n( the vector space attached to a vertex occurs, first with $\\alpha$, second\nwith $\\bar{\\alpha}$ and\nthe other times in the trivial representation) and moreover we have the\nrelations (\\ref{theta0}). We\nwill also use the notation $\\buildrel\\alpha \\over W_C=\\buildrel\\alpha \\over W_{[x_0,x_n\\cdots,x_1]}.$ \\end{definition}\n\nThe properties shown in our first paper are easily generalized:\n\n\\begin{proposition}[Properties of Wilson loops]\nThe element $\\buildrel\\alpha \\over W_C$ is gauge invariant and moreover\nit does not depend on the departure point of the loop $C.$\nMoreover it verifies the fusion equation:\n\\begin{equation}\n\\buildrel\\alpha \\over W_C \\buildrel\\beta \\over W_C=\\sum_{\\gamma\\in Phys(A)} N_{\\alpha\\beta}^{\\gamma} \\buildrel\\gamma \\over W_C\n\\end{equation}\n\\end{proposition}\n\n\\medskip\n\n{\\sl Proof:}\\,\nThe gauge invariance is quite obvious because of relations (\\ref{antipode1}),\n(\\ref{antipode2}).\nTo show the cyclicity property we must put our Wilson loop in another form\ncalled\n\"expanded form\" in our first paper.\n Using relations (\\ref{contract}),(\\ref{Rinv}) we easily obtain:\n\\begin{eqnarray}\n&&\\buildrel\\alpha \\over W(C)=v_{\\alpha}^{-{1\\over 2}(\\sum_{x\\in C}\\epsilon(x,C))}\ntr_{{\\alpha}^{\\otimes n}}( (S({\\buildrel \\alpha \\over A})\\buildrel\\alpha \\over \\mu){}^{\\otimes\nn}\\prod_{i=n}^1\nP_{ii-1}\\times\\\\ &&\\times(\\prod_{i=n}^{1}\\buildrel\\alpha \\over u(x_{j+1} x_{j})_j (S \\otimes id\n)(R_{jj-1}^{(\\epsilon(x_{j},C))}))\n \\buildrel\\alpha \\over u(x_1 x_0)_{1})\\label{Wilsonloop2}.\\nonumber\n\\end{eqnarray}\nIn this form the cyclicity invariance is obvious using the commutation\nrelations\n(\\ref{ES}).\\\\\nThe fusion relation is less trivial to show.\n We first show a lemma describing the decomposition rules for holonomies:\n\\begin{equation}\n({\\buildrel \\alpha \\over u}_P)_1 ({\\buildrel \\beta \\over u}_P)_2 = \\phi^{\\alpha\n\\beta}_{\\gamma}\\;\\; {\\buildrel \\gamma \\over u}_P \\;\\; \\psi^{\\gamma}_{\\beta\n\\alpha} P_{12} {\\buildrel {\\alpha \\beta} \\over {f^{-1}}}_{21}.\n\\end{equation}\nIndeed, using (\\ref{assocalg})(\\ref{ES}), we easily obtain for a path\n$P=[x,y,z]$:\n\\begin{eqnarray*}\n&&(v_{\\alpha}^{{1\\over 2}\\epsilon(y, P)}(\\buildrel\\alpha \\over u(x y){\\buildrel \\alpha \\over A}\n\\buildrel\\alpha \\over u(yz))_1\\;(v_{\\beta}^{{1\\over 2}\\epsilon(y, P)}(\\buildrel\\beta \\over u(x y){\\buildrel \\beta\n\\over A} \\buildrel\\beta \\over u(yz))_2=\\\\\n&&=\\sum_{i,j,k,l}v_{\\alpha}^{{1\\over 2}\\epsilon(y, P)}v_{\\beta}^{{1\\over\n2}\\epsilon(y, P)}(\\buildrel\\alpha \\over u(x y)_1\\buildrel\\beta \\over u(x y)_2)(S(\\theta^{(1)}_l)A \\theta^{(1)}_i b_j\n\\theta^{-1 (2)}_k \\theta^{(31)}_l)_1\\times\\\\\n&&\\times (S(\\theta^{(2)}_l) S(\\theta^{-1(1)}_k)S(a_j)S(\\theta^{(2)}_i) A\n\\theta^{(3)}_i \\theta^{-1 (3)}_k \\theta^{(32)}_l)_2\\;(\\buildrel\\alpha \\over u(y z)_1\\buildrel\\beta \\over u(y z)_2)=\\\\\n&&=v_{\\alpha}^{{1\\over 2}\\epsilon(y, P)}v_{\\beta}^{{1\\over 2}\\epsilon(y,\nP)}(\\buildrel\\alpha \\over u(x y)_1\\buildrel\\beta \\over u(x y)_2)\\;\n{\\buildrel {\\alpha \\beta} \\over G}_{21} \\Rpmff{\\alpha}{\\beta}\\;(\\buildrel\\alpha \\over u(y z)_1\\buildrel\\beta \\over u(y\nz)_2)\\nonumber\\\\\n\\end{eqnarray*}\nthe last equality is obtained by using successively (\\ref{quasitriang}) and\n(\\ref{pentagon}).\nNow, using (\\ref{deltaA})(\\ref{Rdef}), we obtain the announced result for a two\nlinks path.\nProceeding by induction we can prove it for any simple open path.\\\\\n\nLet us now consider the loop $C$ as formed by two pieces $[xy]$ and $[yx]$, we\nhave, using the same\nproperties as before:\n\\begin{eqnarray*}\n&&(v_{\\alpha}^{{1\\over 2}(\\epsilon(y,\nP)-\\epsilon(x,C))}tr_{\\alpha}(S({\\buildrel \\alpha \\over A})\\buildrel\\alpha \\over \\mu \\buildrel\\alpha \\over u(x y)\n{\\buildrel \\alpha \\over A} \\buildrel\\alpha \\over u(y x)))\n(v_{\\beta}^{{1\\over 2}(\\epsilon(y, P)-\\epsilon(x,C))}tr_{\\beta}(S({\\buildrel\n\\beta \\over A})\\buildrel\\beta \\over \\mu \\buildrel\\beta \\over u(x y) {\\buildrel \\beta \\over A} \\buildrel\\beta \\over u(y x)))=\\\\\n&&=\\sum_{{\\buildrel {p,m,l,i} \\over {q,n,j,k}}}(v_{\\alpha}v_{\\beta})^{{1\\over\n2}(\\epsilon(y, P)-\\epsilon(x,C))}tr_{\\alpha \\beta}((S(\\theta^{-1 (2)}_p)\nS(\\theta^{(3)}_m) S(c_l) S( \\theta^{-1 (2)}_i) S(A) \\mu \\theta^{-1 (1)}_i\n\\theta^{(1)}_m \\theta^{-1 (11)}_p)_1 \\times\\\\\n&&\\times(S(\\theta^{-1 (3)}_p) S(A) \\mu \\theta^{-1 (3)}_i d_l \\theta^{(2)}_m\n\\theta^{-1 (12)}_p)_2 (\\buildrel\\alpha \\over u(x y)_1 \\buildrel\\beta \\over u(x y)_2)(S(\\theta^{-1 (11)}_q)\nS(\\theta^{(1)}_n) S( \\theta^{-1 (1)}_j)\\times\\\\\n&&\\times A \\theta^{-1 (2)}_j b_k \\theta^{(3)}_n \\theta^{-1 (2)}_q)_1\n(S(\\theta^{-1 (12)}_q) S(\\theta^{(2)}_n) S(a_k) S(\\theta^{-1 (3)}_j) A\n\\theta^{-1 (3)}_q)_2 (\\buildrel\\alpha \\over u(y x)_1 \\buildrel\\beta \\over u(y x)_2))=\\\\\n&&= (v_{\\alpha}v_{\\beta})^{{1\\over 2}(\\epsilon(y, P)-\\epsilon(x,C))}tr_{\\alpha\n\\beta}((S \\otimes S)(G_{21}R_{12})(\\mu \\otimes \\mu) (\\buildrel\\alpha \\over u(x y)_1 \\buildrel\\beta \\over u(x y)_2)\n(G_{21}R_{21}^{-1})\\times\\\\\n&&\\times (\\buildrel\\alpha \\over u(y x)_1 \\buildrel\\beta \\over u(y x)_2) )=\\\\\n&&= \\sum_{\\gamma}N^{\\alpha \\beta}_{\\gamma}v_{\\gamma}^{{1\\over 2}(\\epsilon(y,\nP)-\\epsilon(x,C))}tr_{\\gamma}(S({\\buildrel \\gamma \\over A})\\buildrel\\gamma \\over \\mu {\\buildrel\n\\gamma \\over u}(x y) {\\buildrel \\gamma \\over A} {\\buildrel \\gamma \\over u}(y\nx))\n\\end{eqnarray*}\nwhich ends the proof of the theorem.\n$\\Box$\n\n\\medskip\n\n\n\\begin{proposition}[commutation properties]\nIt can also be shown that\n$[\\buildrel\\alpha \\over W_C, \\buildrel\\beta \\over W_{C'}]=0 $\nfor all simple loops $C, C'$ without transverse intersections.\n\\end{proposition}\n\n\nAlthough the structure of the algebra $\\Lambda$ depends on the ciliation, it\nhas been\n shown in \\cite{AGS} that the algebra $\\Lambda^{inv}$ does not depend on it up\nto isomorphism.\nThis is\ncompletely consistent with the approach of V.V.Fock and A.A.Rosly: in their\nwork the graph needs to be\nendowed with a structure of ciliated fat graph in order to put on the space of\ngraph connections\n${\\cal A}^l$ a structure of Poisson algebra compatible with the action of the\ngauge group $G^{l}.$\nHowever, as a Poisson algebra ${\\cal A}^l\/G^{l}$ is canonically isomorphic to\nthe space ${\\cal M}^G$\nof flat connections modulo the gauge group, the Poisson structure of the\nlatter being independent of\nany choice of r-matrix \\cite{FR}.\n\n\\begin{definition}[zero-curvature projector]\nWe introduced a Boltzmann weight attached to any simple loop $C$ and defined\nby:\n\\begin{equation}\n\\delta_{C}=\\sum_{\\alpha\\in Phys(A)}[d_{\\alpha}] \\buildrel\\alpha \\over W_{C}.\n\\end{equation}\n\\end{definition}\n\\begin{proposition}\nThis element satisfies the flatness relation :\n\\begin{eqnarray}\n\\delta_{C}\\buildrel\\alpha \\over u_{C}{}^i_j&=&{\\buildrel \\alpha \\over\nB}^i_j\\delta_{C}.\\label{flatness}\n\\end{eqnarray}\nmoreover we have\n\\begin{eqnarray}\n(\\frac{(\\delta_{C})}{\\sum_{\\alpha \\in Phys({\\cal A})}\n[d_{\\alpha}]^{2}})^{2}&=&(\\frac{(\\delta_{C})}{\\sum_{\\alpha \\in Phys({\\cal A})}\n[d_{\\alpha}]^{2}}).\\label{flatness}\n\\end{eqnarray}\n\\end{proposition}\n\n{\\sl Proof:}\\,\nUsing the same properties as in the computation of fusion relations, we obtain\n:\n\\begin{eqnarray*}\n&&\\sum_{\\alpha \\in Phys({\\cal A})}[d_{\\alpha}](v_{\\alpha}^{{1\\over\n2}(\\epsilon(y, P)-\\epsilon(x,C))}tr_{\\alpha}(S({\\buildrel \\alpha \\over A})\\buildrel\\alpha \\over \\mu\n\\buildrel\\alpha \\over u(x y) {\\buildrel \\alpha \\over A} \\buildrel\\alpha \\over u(y x)))\\times\\\\\n&&\\times(v_{\\beta}^{{1\\over 2}(\\epsilon(y, P)-\\epsilon(x,C))}tr_{\\beta}(\nS({\\buildrel \\beta \\over A})\\buildrel\\beta \\over \\mu \\buildrel\\beta \\over u(x y) {\\buildrel \\beta \\over A} \\buildrel\\beta \\over u(y\nx)))=\\\\\n&&=\\sum_{{\\buildrel {\\alpha \\in Phys({\\cal A})} \\over {p,m,l,i,q,n,j,k}}}\n[d_{\\alpha}](v_{\\alpha}v_{\\beta})^{{1\\over 2}(\\epsilon(y,\nP)-\\epsilon(x,C))}tr_{\\alpha \\beta}((S(\\theta^{-1 (2)}_p) S(\\theta^{(3)}_m)\nS(c_l) S( \\theta^{-1 (2)}_i) S(A) \\mu \\theta^{-1 (1)}_i \\theta^{(1)}_m \\times\\\\\n&&\\theta^{-1 (11)}_p)_1 (\\theta^{-1 (3)}_i d_l \\theta^{(2)}_m \\theta^{-1\n(12)}_p)_2 (\\buildrel\\alpha \\over u(x y)_1 \\buildrel\\beta \\over u(x y)_2) (S(\\theta^{-1 (11)}_q) S(\\theta^{(1)}_n) S(\n\\theta^{-1 (1)}_j) A \\theta^{-1 (2)}_j b_k \\theta^{(3)}_n \\theta^{-1 (2)}_q)_1\n\\times\\\\\n&&\\times(S(\\theta^{-1 (12)}_q) S(\\theta^{(2)}_n) S(a_k) S(\\theta^{-1 (3)}_j) A\n\\theta^{-1 (3)}_q)_2 (\\buildrel\\alpha \\over u(y x)_1 \\buildrel\\beta \\over u(y x)_2)S(\\theta^{-1 (3)}_p)_2)=\\\\\n&&=\\sum_{\\alpha,\\gamma \\in Phys({\\cal A})}[d_{\\alpha}]\n\\sum_{p,m}\ntr_{\\alpha \\beta}(( S(\\theta^{(3)}_m) S(A) \\mu \\theta^{(2)}_m \\theta^{-1\n(12)}_p)_1 \\times\\\\\n&&\\times( \\theta^{(1)}_m \\theta^{-1 (11)}_p)_2 \\phi^{\\beta \\alpha}_{\\gamma}\n\\; \\buildrel\\gamma \\over u(x y) A \\buildrel\\gamma \\over u(y x) v_{\\gamma}^{{1\\over 2}(\\epsilon(y, P)-\\epsilon(x,C))}\n\\;\\psi^{\\gamma}_{\\beta \\alpha} f^{-1}_{21} \\; S(\\theta^{-1 (2)}_p)_1\nS(\\theta^{-1 (3)}_p)_2)\n\\end{eqnarray*}\nthe last equality uses again the quasitriangularity properties.\nThen we obtain, for any matrix $V$ in $End(\\alpha)$:\n\n\\begin{eqnarray*}\n&&\\delta_C \\; tr_{\\alpha}( S(A) \\buildrel\\alpha \\over \\mu V \\buildrel\\alpha \\over u_C )=\\\\\n&&=\\sum_{\\gamma \\in Phys({\\cal A})}\\sum_{p,m}\ntr_{\\gamma}( \\; (\\sum_{\\alpha \\in Phys({\\cal A})}[d_{\\alpha}]\n\\psi^{\\gamma}_{\\beta \\alpha} f^{-1}_{21} (S(\\theta^{-1 (2)}_p)\nS(\\theta^{(3)}_m) S(A) \\mu \\theta^{(2)}_m \\theta^{-1 (12)}_p)_1 \\times\\\\\n&&\\times( S(\\theta^{-1 (3)}_p) S(A) \\mu \\; V \\;\\theta^{(1)}_m \\theta^{-1\n(11)}_p)_2 \\phi^{\\beta \\alpha}_{\\gamma} ) \\; (\\buildrel\\gamma \\over u(x y) A \\buildrel\\gamma \\over u(y x)\nv_{\\gamma}^{{1\\over 2}(\\epsilon(y, P)-\\epsilon(x,C))}) \\; )\n\\end{eqnarray*}\nDue to the intertwining properties and the normalizations mentioned before of\nthe $\\phi,\\psi$s\nwe can conclude (see \\cite{AGS}) that it does exist some complex coefficients\n$A(\\alpha \\beta\n\\gamma)$ such that: \\begin{eqnarray}\n&&id_K \\psi^{0}_{\\beta \\bar{\\beta}} = \\sum_{\\alpha}A(\\alpha \\beta\n\\gamma) \\psi^{\\gamma}_{ \\beta \\alpha\n}\\psi^{\\alpha}_{\\bar{\\beta} \\gamma }\\\\ &&\\mbox{ and }\\nonumber\\\\\n&&\\phi^{\\gamma}_{\\beta \\alpha}=A(\\alpha \\beta\n\\gamma)\\frac{[d_{\\gamma}]}{[d_{\\alpha}]}(\\psi^{\\alpha}_{\\bar{\\beta}\\gamma}{\\buildrel {\\beta\n\\bar{\\beta} \\gamma} \\over \\Theta}\\otimes id_{\\beta}) (\\phi^{\\beta\n\\bar{\\beta}}_{0}\\otimes\nid_{\\gamma}) \\end{eqnarray}\nwe obtain \\begin{eqnarray*}\n&&\\sum_{p,m}\n\\sum_{\\alpha \\in Phys({\\cal A})}[d_{\\alpha}]\n\\psi^{\\gamma}_{\\beta \\alpha} f^{-1}_{21} (S(\\theta^{-1 (2)}_p)\nS(\\theta^{(3)}_m) S(A)\n\\mu \\theta^{(2)}_m \\theta^{-1 (12)}_p)_1 \\times\\\\\n&&\\times( S(\\theta^{-1 (3)}_p) S(A) \\mu \\; V \\;\\theta^{(1)}_m \\theta^{-1\n(11)}_p)_2\n\\phi^{\\beta \\alpha}_{\\gamma} =[d_{\\gamma}]id_{\\gamma} tr_{\\beta}(S(A)\\mu \\; V\n\\; B)\n\\end{eqnarray*}\nthen for any matrix $V$ we have $\\delta_C \\; tr_{\\alpha}({\\buildrel \\alpha\n\\over S(A)}\n\\buildrel\\alpha \\over \\mu {\\buildrel \\alpha \\over V }\\buildrel\\alpha \\over u_C )=\\delta_C \\; tr_{\\alpha}( {\\buildrel\n\\alpha \\over S(A)} \\buildrel\\alpha \\over \\mu\n{\\buildrel \\alpha \\over V }{\\buildrel \\alpha \\over B})$, and the linear\nindependance of the\ngenerators of our algebra ensures the final result.\\\\ The last formula of the\nproposition is a\ntrivial consequence of the last result. $\\Box$\n\n\nWe were led to define an element that we called $a_{YM}=\\prod_{f\\in {\\cal\nF}}\\delta_{\\partial\nf}.$ This element is the non commutative\nanalogue of the projector on the space of flat connections.\n\nIn \\cite{AGS}\\cite{BR1} it was proved that $\\delta_{\\partial f}$ is a central\nelement of\n $\\Lambda^{inv}$ and the algebra $\\Lambda_{CS}=\\Lambda^{inv}a_{YM}$ was shown\nto be independant, up\nto isomorphism, of the triangulation. The proof is based on the lemma of\ndecomposition rules of the\nholonomies shown before and on the quite obvious property: let $C_1$ and $C_2$\nbe two simple\ncontractile loops which interiors are disjoint and with a segment $[x y]$ of\ntheir boundary in\ncommon: \\begin{equation}\n\\int dh(u(xy)) \\buildrel\\alpha \\over W_{C_1} \\buildrel\\beta \\over W_{C_2} = \\delta_{\\alpha, \\beta}\\buildrel\\alpha \\over W(C_1 \\# C_2)\n\\end{equation}\n\n As a result it was advocated that $\\Lambda_{CS}$ is the algebra of\nobservables of the\nChern Simons theory on the manifold $\\Sigma\\times [0,1].$\nThis is supported by the topological invariance of $\\Lambda_{CS}$ (i.e this\nalgebra depends only on\nthe topological structure of the surface $\\Sigma$) and the flatness of the\nconnection.\n\nOur aim is now to construct in the algebra $\\Lambda_{CS}$ the observables\n associated to any link in $\\Sigma\\times [0,1].$\n\n\n\\subsection{Links, chord diagrams and quantum observables}\nIn the following subsection and in the chapter $3$ the computations will be\nmade in the case of\n $q$ generic\nto simplify the notations but the generalization to $q$ root of unity\ncan be made exactly in the same way.\\\\\nWe will consider a compact connected surface $\\Sigma$ with boundary $\\partial\n\\Sigma$.\nThe boundary is a set of disjoint simple closed curves\n which are designed to be\n\"In\" or \"Out\". Let us draw some oriented curves on the surface $\\Sigma$\ndefining a link $L$ ,\n assuming that their boundary is contained in $\\partial \\Sigma$ and\nwith simple, transverse intersections,\nwith the specification of over- or undercrossing at each intersection. We will\nalso consider that\nrepresentations of the quantum group are attached to connected components of\nthe link.\\\\\n\nThe data ( surface with boundary + colored link ) will be called \"striped\nsurface\", the data of \"In\" (resp. \"Out\") boundary of $\\Sigma$ and $L$ with\ncorresponding colors will be called the \"In state\" (resp. the \"Out state\") of\nthe striped surface.\\\\\n\n To describe such objects we will choose a Morse function which gives a time\ndirection and the set\nof \"equitime planes\" $({\\cal P}_t )_{t_i \\leq t \\leq t_f }$ cutting the\nsurface.\nAn equitime plane $P_t$ divides the surface in two parts called respectively\n\"future\" and \"past\". On any simple curve drawn on an equitime plane the time\ndirection give us\nan orientation of the curve, if we impose moreover a departure point $x$ on\nthis curve we are able\nto decide if a point $z$ is on the left (resp. on the right) of another point\n$z',$ if $x, z, z'$\n(resp. $x, z', z$) appear in the order given by this orientation. We will\nconsider the surface in a\ncanonical position defined by the following conditions. The intersection\nbetween the surface and the\nplane for $tt_f$ is empty. The \"In\" (resp. \"out\") boundary is\ncontained in the $t=t_i$\n(resp. $t=t_f$) plane. The intersection between the $({\\cal P}_t )\n _{t_i \\leq t \\leq t_f }$ and the surface is a set of disjoint simple closed\ncurves\n $(C^t_i)_{i=1,..., n(t)} $\n(not necessary disjoint at the singular times), where $n(t)$ is the number of\nconnected components\n of $\\Sigma \\cap {\\cal P}_t.$\n We will call \"$\\varphi^3-$diagram\" of the surface\na graph drawn on it which intersections with $({\\cal P}_t )\n _{t_i \\leq t \\leq t_f }$ determine\n departure points on each closed curve in these sets.\nWe impose that the $\\varphi^3-$diagram never turn around any handle of the\nsurface.\\\\\nOur aim is now to define a ciliated fat graph which will encode\nthe topology of the striped surface,i.e. this decomposition involves only\ncontractile plaquettes,\nit is sufficiently fine to allow us to put the link on the graph in a generic\nposition and allow us\nto distinguish two situations related by a Dehn twist of the surface. We then\ndecompose the surface\nin blocks, their number being chosen with respect to the singularities of the\nMorse function (\nconsidered as a function over the points of the surface and of the link). The\ninformation contained\nin the Morse function is not sufficient to deal with the problem of possible\nnon trivial cycles of\n$L$ around handles of $\\Sigma$. We will rule out this problem by adding\nfictively two disjoint\n$\\varphi^3-$diagrams of $\\Sigma$ to the link $L,$ the intersections between the\nlink and the\n$\\varphi^3-$diagrams will detect the rotation of the link around an handle of\n$\\Sigma$, we will\nthen refine the decomposition with respect to these datas. We will assume that\nthe singularities\n of the Morse function $f$, considered now as a function of the points of\nthe surface, points of the link and points of the $\\varphi^3-$diagrams,\ncorrespond to different times.\nWe will denote by $t_0=t_i,t_1,\\cdots,t_{n-1},t_n=t_f$ the different instants\ncorresponding\nto the singularities of the Morse function. We will consider a decomposition of\nthe surface\nand of the link in \"elementary blocks\" ${\\cal B}_0,\\cdots,{\\cal B}_n$\ncorresponding to the\nsubdivision\n$[t_i,t_f] = [\\tau_0,\\tau_1] \\cup [\\tau_1,\\tau_2] \\cup\n\\cdots \\cup [\\tau_n,\\tau_{n+1}]$ where\n$\\tau_0=t_0, \\tau_1={1\\over2}(t_0+t_1), \\cdots,\n\\tau_n={1\\over2}(t_{n-1}+t_{n}), \\tau_{n+1}=t_n$\nare called \"cutting times\".\nAn example of a striped surface with the block decomposition\ndescribed before is shown in the following figure:\n\\par\n\\centerline{\\psfig{figure=stripexa.ps}}\n\\par\n We will consider a triangulation ${\\cal T}$ and a ciliation induced by this\nblock decomposition. The set ${\\cal V}$ of vertices of ${\\cal T}$ contains all\nelements of the sets $L \\cap {\\cal P}_{\\tau_k}$ and the singularities of the\nMorse function considered as a function of the link and of the surface.\nThe edges of the triangulation are either a segment of the link or of the\n$\\varphi^3-$diagram\nbetween two consecutiv vertices, or a segment drawn on the surface at the same\ntime $\\tau_i$ between two emerging strings.\nThe plaquettes of the triangulation are the connected regions of $\\Sigma$\nsurrounded by the edges described before.\nLet us denote by $(x^k_l)$ the intersections of the link with the cutting\nplanes\n$({\\cal P}_{\\tau_k}).$ The ciliation is chosen to be:\nat each vertex $x^k_l$, directed to the past and to the left, just \"before\" the\nequitime line, and\nat each crossing of the oriented link, between the two outgoing strands.\nWe will choose a practical indexation satisfying the following properties:\n$x^k_l \\in C^{\\tau_k}_i$ for $l \\in \\{ 1+\\sum_{jq.$)\n\n\\begin{definition}\nLet $P$ be a connected piece of one of the $\\S{i}$. Let us choose for\nsimplicity $P=[\\buildrel i\\over z_{n+1},\n\\buildrel i\\over y_n, \\buildrel i\\over z_n, \\cdots, \\buildrel i\\over y_1, \\buildrel i\\over z_{1}]$, we will denote the holonomy associated\nto\nit, by\n\\begin{equation}\n{\\cal U}_P=\\omega(P)U_{[\\buildrel i\\over z_{n+1}\\buildrel i\\over y_{n}]}\\cR{\\buildrel i\\over y_{n}}(<_l)U_{[\\buildrel i\\over y_n \\buildrel i\\over z_n]}\n\\cR{\\buildrel i\\over z_n}(<_l)\n\\cdots \\cR{\\buildrel i\\over y_1 }(<_l) U_{[\\buildrel i\\over y_1 \\buildrel i\\over z_1]},\n\\end{equation}\nwhere\n$\\omega(P)=v_{\\alpha_i}^{-{1\\over 2}\n\\sum_{x\\in P\\setminus\\{\\buildrel i\\over z_{n+1}, \\buildrel i\\over z_1\\}}\\epsilon(\\phi(x),\\S{i})}.$\nWe will denote by $\\cUS{i}$ the holonomy associated to the entire circle\n$\\S{i}$.\nWe will also define the permutation operator:\n$\\sigma_P = \\prod_{x=\\buildrel i\\over y_n}^{\\buildrel i\\over y_1} P_{\\buildrel i\\over z_{n+1}, x}$ (where the order is given\nby the order of vertices along $P$) and $\\sig{i}$ will denote $\\sigma_{\\S{i}}.$\n\\end{definition}\n\n\\begin{definition}[Generalized Holonomies and Wilson loops]\nTo each link in $\\Sigma\\times[0,1]$ we associate an element $W_L$ by the\nfollowing procedure:\nlet us denote by ${\\cal W}_{L}$ the element\n\\begin{equation}\n{\\cal W}_{L}=\n\\mu_{{\\cal S}}\\prod_{i=1}^p\\sig{i} \\prod_{i=1}^{p}\n\\cUS{i};\n\\end{equation}\nwhere $\\mu_{{\\cal S}}=\\bigotimes_{x \\in Z \\setminus Z^{\\partial \\Sigma}}\\mu_{x^+}$ :\n\n\n The element associated to the link $L$ is defined by\n\\begin{equation}\nW_{L}=tr_{\\bigotimes_{x \\in Z \\setminus Z^{\\partial \\Sigma}}V_{x^+}}{\\cal\nW}_{L}\n\\end{equation}\n where $tr_{V_{+}}$ means the partial trace over the space $V_{+}$ after the\nnatural\nidentification $V_{+}=V_{-}.$\n\\end{definition}\n\n\n\nThis element satisfies important properties described by the following theorem\n\\cite{BR2}:\n\n\\begin{theorem}\nLet $L$ be a link satisfying the set of assumptions, then $W_L$ does not depend\non the labelling of the components nor does it depend on the choice of\ndeparture points of the\n components. As a result W is a function on the space of links with values in\n$\\Lambda\\otimes\n\\bigotimes_{P \\in {\\cal C}_2}End(V_{d(P)^-},V_{e(P)^+}).$ Moreover this mapping\nis invariant under\nthe coaction of the gauge group at a vertex interior to the surface. If $L$ and\n$L'$ are two links ,\nwe have the morphism property $W_{L*L'}=W_{L}W_{L'}.$ \\end{theorem}\n\n\nOur principal aim is the computation of the correlation function defined in an\nobvious way:\n\n\\begin{proposition}[Correlation functions and Ribbons invariants]\nThe correlation function of the link $L$ considered as immersed in\n$\\Sigma\\times [0,1]$ is simply defined by:\n\\begin{equation}\n< W_L >_{q-YM(\\Sigma)}= \\int \\prod_{l \\in {\\cal L}^{int}}dh(U_l) \\;\\;W_L\n\\;\\;\\prod_{F\\in {\\cal F}} \\delta_{\\partial F}\n\\end{equation}\n\nThe observable associated to $L$ will be denoted by\n${\\widehat W}_L=W_L\\prod_{F\\in {\\cal F}}\\delta_{\\partial F}.$ This element of\n$\\Lambda_{CS}$ depends\nonly on the regular isotopy class of the link $L,$ i.e it satisfies the\nReidemeister moves of type\n0,2,3. This fact was established in \\cite{BR2}.\\\\\n\n\\par\n\\centerline{\\psfig{figure=reidem.ps}}\n\\par\n\nMoreover, let $L$ be as usual a link in $\\Sigma\\times [0, 1]$ and $P$ the set\nof projected\ncurves on $\\Sigma$ and let $L^{\\propto\\pm}$ be another link\n whose projection $P^{\\propto\\pm}$ differs from $P$ by a move of type I\n\\par\n\\centerline{\\psfig{figure=move1.ps}}\n\\par\n applied to a curve colored by $\\alpha$,\nwe have the following relation:\n\\begin{equation}\n{\\widehat W}_{L^{\\propto\\pm}}=v_{\\alpha}^{\\pm 1} {\\widehat W}_{L}\n\\end{equation}\nThe expectation value of a Wilson loop on a Riemann surface can be considered\nas an\ninvariant associated to a ribbon glued on the surface with the blackboard\nframing.\n\\end{proposition}\n\\bigskip\n\n\\section{Computation of the correlation functions}\nOur first aim is the computation of the invariants associated to links drawn on\na closed\nRiemann surface embedded in $S^3.$ To realize this program we want to decompose\nthe computation by\nintroducing surfaces with boundaries and links drawn on them, already called\n\"striped surfaces\", and\nby describing the gluing operation of the latter.\n\nThis decomposition allows us to reduce the striped surface to the gluing of the\nfollowing objects,\ncalled \"elementary blocks\" : \\begin{enumerate}\n\\item {\\bf the cups}\n\\item {\\bf the caps}\n\\item {\\bf the (n,m)(n+m) trinions}\n\\item {\\bf the (n+m)(n,m) trinions}\n\\item {\\bf the free propagation of n strands}\n\\item {\\bf the propagation of n strands with one overcrossing}\n\\item {\\bf the propagation of n strands with one undercrossing}\n \\item {\\bf the (n-2)(n) creation}\n\\item {\\bf the (n)(n-2) annihilation}\n \\end{enumerate}\nThese objects are described in the following figure\\\\\n\n\\par\n\\centerline{\\psfig{figure=blocselement.ps}}\n\\par\n\n\n\n\n\\medskip\n\nThe correlation functions can be put in a more convenient form to reduce the\ncomputation to the gluing of elements associated to elementary blocks.\n\n\\begin{lemma}\nLet us consider an element $l$ of ${\\cal L}$ and an element $P$ of ${\\cal F}$\nthen\n\\begin{eqnarray*}\nP \\in Past(l) \\Rightarrow \\delta_P \\Ua{l} = \\Ua{l} \\delta_P.\n\\end{eqnarray*}\n\\end{lemma}\n\n{\\sl Proof:}\\,\n\nthis result is a trivial consequence of the choice of ciliation and of the\ncommutation properties\n developed in \\cite{BR1}\n\n$\\Box$\n\n\\medskip\n\n From now the order induced by the orientation of the link will not be\nconvenient anymore, prefering\ntime ordering we will introduce the vector spaces $V_{x^a}$ and $V_{x^b}$\n(\"after\" and \"before\")\nrather than $V_{x^+}$ and $V_{x^-}.$ Let $\\alpha$ denote the representation\nassociated to the circle\nwhere x is taken, then $V_{x^+}=V_{x^-}=\\V{\\alpha}.$ If $x$ is in ${\\cal\nV}_+$, then\n$V_{x^a}=V_{x^b}=\\V{\\alpha}$ and we will introduce the canonical identification\nmaps\n$id_{(V_{x^a},V_{x^+})}$ and $id_{(V_{x^b},V_{x^-})}.$ If $x$ is in ${\\cal\nV}_-$, then\n$V_{x^a}=V_{x^b}=\\V{\\bar{\\alpha}}$ and we will introduce the canonical maps\n${\\phi^{\\bar{\\alpha}\\alpha}_{0}}_{(V_{x^b},V_{x^+})}$ and\n${\\psi^{0}_{\\bar{\\alpha}\\alpha}}_{(V_{x^a},V_{x^-})}.$\nWe can define a new holonomy by ${U}^{\\#}_{[ x, y]}$ :\n\\begin{eqnarray*} {U}^{\\#}_{[x, y]}&=& id_{(V_{x^a},V_{x^+})} {\\buildrel{\\alpha} \\over U}\\!_{[ x, y ]}\nid_{(V_{y^b},V_{y^-})}, \\mbox{ if } x \\in {\\cal V}_+, y \\in {\\cal V}_+\\\\ &=&\n{\\psi_{\\bar{\\alpha}\\alpha}^{0}}_{(V_{x^b},V_{x^+})} {{\\buildrel{\\alpha} \\over U}\\!}_{[ x, y ]}\n{\\phi_{0}^{\\bar{\\alpha}\\alpha}}_{(V_{y^a},V_{y^-})}= {{\\buildrel{\\bar\\alpha} \\over U}\\!}_{[ y, x ]},\n\\mbox{ if } x \\in {\\cal\nV}_-, y \\in {\\cal V}_-\\\\ &=&\n{\\psi_{\\bar{\\alpha}\\alpha}^{0}}_{(V_{x^b},V_{x^+})} {{\\buildrel{\\alpha} \\over U}\\!}_{[ x, y ]}\nid_{(V_{y^b},V_{y^-})}, \\mbox{ if } x \\in {\\cal V}_-, y \\in {\\cal V}_+\\\\ &=&\nid_{(V_{x^a},V_{x^+})}\n{{\\buildrel{\\alpha} \\over U}\\!}_{[ x, y ]} {\\phi_{0}^{\\bar{\\alpha}\\alpha}}_{(V_{y^a},V_{y^-})}, \\mbox{\nif } x \\in {\\cal\nV}_+, y \\in {\\cal V}_- \\end{eqnarray*}\n Despite its apparent complexity, this definition has a very\nsimple meaning. It describes the usual fact that a strand in the direction of\nthe past coloured by a\nrepresentation $\\alpha$ can be described by a strand in the direction of the\nfuture coloured by a\nrepresentation $\\bar{\\alpha}.$\n\nWe will denote in the following:\n\\begin{eqnarray*}\nA_k = \\prod_{j=1}^{Card({\\cal L}_{\\tau_k \\le t \\le \\tau_{k+1}})} ( (\\prod_{P\n\\in Present(l^k_j)}\n\\delta_{\\partial P}) {U}^{\\#}_{l^k_j} ),\n\\end{eqnarray*}\n\nthe elements ( $(\\prod_{P \\in Present(l)}\\delta_P U^{\\#}_l)$ if $l$ does not\nbelong to a crossing\nand $(\\prod_{P \\in Present(l)}\\delta_P U^{\\#}_l U^{\\#}_{l'})$ if $l$ and $l'$\ncross themselves )\nwill be called \"square plaquettes\" elements in the following.\\\\\n\n\nWe will also use the following permutation operator:\n\\begin{eqnarray*}\n\\sigma^{\\tau_k} = \\prod_{j=1}^{Card({\\cal V}_{\\tau_k})} (\\prod_{i=Card({\\cal\nV}_{\\tau_{k+1}})}^{1} P_{(x^k_j)^a,(x^{k+1}_i)^a} {})\n\\end{eqnarray*}\n\n\\begin{lemma}[chronologically ordered observables]\nUsing these definitions, the element associated to the striped surface can be\nput\nin a form which respects the ordering induced by the time order:\n\\begin{equation}\n{\\widehat W}_L=v_{\\alpha}^{{1\\over2}\\sum_{x \\in {\\cal V}^{int}}\n\\epsilon(x^b,x^a)} tr_{\\bigotimes_{x \\in {\\cal\nV}^{int}}V_{x^a}}((\\prod_{k=1}^{n}\\sigma^{\\tau_k})(\\prod_{k=0}^{n}A_k))\n\\end{equation}\n\\end{lemma}\n{\\sl Proof:}\\,\nWe begin with the ordering of the holonomies attached to the link.\nUsing the commutation relations and the properties of the $R$ matrix we obtain:\n\\begin{eqnarray*}\nW_L&=& tr_{\\bigotimes_{x \\in {\\cal\nV}^{int}}V_{x^+}}((\\prod_{k=1}^{n}\\prod_{j=1}^{Card({\\cal V}_{\\tau_k})}\n\\prod_{i=Card({\\cal V}_{\\tau_{k+1}})}^{1} P_{(x^k_j)^+,(x^{k+1}_i)^+} )\\times\\\\\n&\\times&(\\otimes_{x\\in {\\cal V}^{int}}\\mu_{x^+})\n(\\prod_{k=0}^{n}\\prod_{j=1}^{Card({\\cal L}_{\\tau_k \\le t \\le \\tau_{k+1}})}\n{U}_{l^k_j}))\nv_{\\alpha}^{{1\\over2}(\\sum_{x \\in {\\cal V}^{int}_+} \\epsilon(x^-,x^+) +\n\\sum_{x \\in {\\cal V}^{int}_-} \\epsilon(x^+,x^-))}\n\\end{eqnarray*}\nnow the commutation lemma gives easily:\n\\begin{eqnarray*}\n(\\prod_{P \\in {\\cal F}}\\delta_P)(\\prod_{k=0}^{n}\n\\prod_{j=1}^{Card({\\cal L}_{\\tau_k \\le t \\le \\tau_{k+1}})}\n{U}_{l^k_j})=\\prod_{k=0}^{n}\\prod_{j=1}^{Card({\\cal L}_{\\tau_k \\le t \\le\n\\tau_{k+1}})} ( (\\prod_{P\n\\in Present(l^k_j)} \\delta_P) {U}_{l^k_j} ), \\end{eqnarray*}\nand with the definition of ${U}^{\\#}_{l^k_j}$\nwe then obtain:\n\\begin{eqnarray*}\n{\\widehat W}_L=v_{\\alpha}^{{1\\over2}\\sum_{x \\in {\\cal V}^{int}}\n\\epsilon(x^b,x^a)}\n tr_{\\bigotimes_{x \\in {\\cal V}^{int}}V_{x^a}}((\\prod_{k=1}^{n}\n\\prod_{j=1}^{Card({\\cal\nV}_{\\tau_k})} \\prod_{i=Card({\\cal V}_{\\tau_{k+1}})}^{1}\nP_{(x^k_j)^a,(x^{k+1}_i)^a} )\n(\\prod_{k=0}^{n}A_k))\n\\end{eqnarray*}\n$\\Box$\n\n\\bigskip\n\nThis lemma leads us to a new definition of elements associated to \"striped\nsurfaces\" which\nis based on gluing chronologically ordered elementary blocks.\n\n\\begin{proposition}[Correlation functions and gluing operation]\nLet us consider a \"striped surface\" $\\Sigma+L$.\nLet us define an element corresponding to $\\Sigma+L$\n( which will be denoted by ${\\cal A}_{\\Sigma+L}$) by the following rules: \\\\\n\\begin{itemize}\n\\item if $\\Sigma+L$ is an elementary block ${\\cal B}_1$, the element of the\ngauge algebra\n associated to it is:\n\\begin{equation}\n {\\cal A}_{{\\cal B}_1}=\\int \\prod_{l \\in {\\cal L}_{\\left] t,t' \\right[ }}\ndh(U_l)\n\\prod_{j=1}^{Card({\\cal L}_1)} ( (\\prod_{P \\in Present(l^1_j)} \\delta_P)\n{U}^{\\#}_{l^1_j} )\nv_{\\alpha}^{{1\\over2}\\sum_{x \\in {\\cal V}^{t'}} \\epsilon(x^b,x^a)} \\end{equation}\n\\item if $\\Sigma+L$ is a disjoint union of $N$ elementary blocks placed between\n$t$ and $t'$ then\nthe element of the algebra associated to $\\Sigma+L$ is obviously the product\nof the elements\nassociated to each elementary block, the order between them being irrelevant\nbecause they are\ncommuting.\\\\\n\n\\item if there exists a time $t''$ between $t$ and $t'$ such that $\\Sigma+L$ is\nobtained by gluing\ntwo \"striped surfaces\" $\\Sigma_1+L_1$ and $\\Sigma_2+L_2$ placed respectively\nbetween $t$ and $t''$,\nand between $t''$ and $t'.$ The element associated to $\\Sigma+L$ will be\ndefined by: \\begin{eqnarray}\n{\\cal A}_{\\Sigma+L}&=&{\\cal A}_{\\Sigma_1+L_1} \\circ {\\cal A}_{\\Sigma_2+L_2}\\\\\n&=& \\int \\prod_{l \\in {\\cal L}_{t''}} dh(U_l) tr_{\\bigotimes_{x \\in {\\cal\nV}^{t''}}V_{x^a}}(\\sigma^{t''}{\\cal A}_{\\Sigma_1+L_1} {\\cal\nA}_{\\Sigma_2+L_2})\\nonumber\n\\end{eqnarray}\n( the canonical choice of ciliation defined for any striped surface is\nobviously compatible with the gluing operation )\\\\\n\\end{itemize}\nthese properties give us a new way to compute the invariants associated to\nlinks on a closed\nsurface:\n\\begin{equation}\n< W_L >_{q-YM(\\Sigma)}= {\\cal A}_{\\Sigma+L}.\n\\end{equation}\n\\end{proposition}\n\n\n\\medskip\n\n From now the computation of the correlation functions\nis reduced to the computation of elements\nassociated to elementary blocks. After some definitions we\nwill give the result of the explicit computation of these\nelements.\n\n\\begin{definition}[In and Out states]\n\nLet us consider a striped surface $\\Sigma+L$.\nA connected component of its \"In state\" is a simple loop ${\\cal\nC}=[x_{n+1}=x_1, x_n, \\cdots, x_1]$\n oriented in the inverse clockwise sense with $n+1$ strands going through it at\neach $x_i$\nin the direction of the past with a representation $\\alpha_i.$\\\\\n ( a strand in the\ndirection of the future with a representation $\\alpha$ is reversed to the past\nby changing its representation in $\\bar{\\alpha}$). The ciliation at each $x_i$\nis chosen as in the general construction of striped surfaces.\nThen, choosing $n$ other representations $(\\beta_i)_{i=1,...,n}$ we define\n${\\cal O} \\in \\Lambda \\otimes End(\\otimes_{x \\in {\\cal C}} V_{x^b},{\\bf C})$\nand\n${\\cal I} \\in \\Lambda \\otimes End({\\bf C},\\otimes_{x \\in {\\cal C}} V_{x^a})$:\\\\\nif there is at least one emerging strand through ${\\cal C}$,\n\\begin{eqnarray*}\n\\Outstate{\\beta_n\n}{\\alpha_n}{x_n}{\\beta_{n-1}}{\\alpha_{n-1}}{x_{n-1}}{\\beta_1}{\\alpha_1}{x_1}&=&\nv^{-1}_{\\beta_n}tr_{\\V{\\beta_n}}\n(\\psi^{\\beta_n}_{\\beta_1\n\\alpha_1}\\guf{\\beta_1}_{[x_1,x_2]}\\psi^{\\beta_1}_{\\beta_2 \\alpha_2}\n\\cdots \\psi^{\\beta_{n-1}}_{\\beta_n\\alpha_{n}}\\guf{\\beta_n}_{[x_{n},x_1]}\n\\Rmff{\\beta_n}{\\alpha_1}\\muf{\\beta_n})\\\\\n\\Instate{\\beta_n}{\\alpha_n}{x_n}{\\beta_{n-1}}{\\alpha_{n-1}}{x_{n-1}}{\\beta_1}{\\alpha_1}{x_1}&=&\nv_{\\beta_n}tr_{\\V{\\beta_n}}(\\muf{\\beta_n}\\Rff{\\beta_n}{\\alpha_1}\n\\guf{\\beta_n}_{[x_1,x_{n}]}\\phi^{\\beta_n \\alpha_{n}}_{\\beta_{n-1}} \\cdots\n\\phi^{\\beta_2\n\\alpha_2}_{\\beta_1}\\guf{\\beta_1}_{[x_2,x_{1}]}\\phi^{\\beta_1\n\\alpha_1}_{\\beta_n}) \\end{eqnarray*}\nHere and in the following we will often forget the multiplicities $m_i$ for\nreadibility.\\\\ If the\nconnected component of this \"In state\" has no emerging strand we will define\n${\\cal O}$ and ${\\cal\nI}$ to be: \\begin{equation} {\\cal O}(\\beta_0)=W^{\\beta_0}_{C^{-1}}\\mbox{ and }{\\cal\nI}(\\beta_0)=W^{\\beta_0}_{C} \\end{equation}\n\\end{definition}\n\n\\bigskip\n\nThe properties of the latter objects are described in the following lemma.\n\\begin{lemma}\nThe properties of the In and Out states are generalizations of those of Wilson\nloops.\\\\\nCyclicity\n\\begin{eqnarray}\n\\Outstate{\\beta_n}{\\alpha_n}{x_n}{\\beta_{n-1}}{\\alpha_{n-1}}{x_{n-1}}{\\beta_1}{\\alpha_1}{x_1}&=&\\Outstate{\\beta_{n-1}}{\\alpha_{n-1}}{x_{n-1}}{\\beta_1}{\\alpha_1}{x_1}{\\beta_n}{\\alpha_n}{x_n}\\nonumber\\\\\n\\Instate{\\beta_n}{\\alpha_n}{x_n}{\\beta_{n-1}}{\\alpha_{n-1}}{x_{n-1}}{\\beta_1}{\\alpha_1}{x_1}&=&\\Instate{\\beta_{n-1}}{\\alpha_{n-1}}{x_{n-1}}{\\beta_1}{\\alpha_1}{x_1}{\\beta_n}{\\alpha_n}{x_n}\\nonumber\\\\\n\\end{eqnarray}\nGauge transformation\n\\begin{eqnarray}\n\\Omega(\\Outstate{\\beta_n}{\\alpha_n}{x_n}{\\beta_{n-1}}{\\alpha_{n-1}}{x_{n-1}}{\\beta_1}{\\alpha_1}{x_1})&=&\\Outstate{\\beta_n}{\\alpha_n}{x_n}{\\beta_{n-1}}{\\alpha_{n-1}}{x_{n-1}}{\\beta_1}{\\alpha_1}{x_1} \\; \\prod_{i=1}^{n} S(\\gf{\\alpha_i}_{x_i})\\nonumber\\\\\n\\Omega(\\Instate{\\beta_n}{\\alpha_n}{x_n}{\\beta_{n-1}}{\\alpha_{n-1}}{x_{n-1}}{\\beta_1}{\\alpha_1}{x_1})&=& \\prod_{i=1}^{n} \\gf{\\alpha_i}_{x_i}\\; \\Instate{\\beta_n}{\\alpha_n}{x_n}{\\beta_{n-1}}{\\alpha_{n-1}}{x_{n-1}}{\\beta_1}{\\alpha_1}{x_1}\\nonumber\\\\\n\\end{eqnarray}\nScalar product\n\\begin{eqnarray}\n\\int \\prod_{l \\in {\\cal C}}dh(U_l)&&tr_{\\otimes_{i}V_{x_i^a}}(\\sigma_{{\\cal C}}\n\\Outstate{\\beta_n}{\\alpha_n}{x_n}{\\beta_{n-1}}{\\alpha_{n-1}}{x_{n-1}}{\\beta_1}{\\alpha_1}{x_1}\\times\\\\\n&&\\times\\Instate{{\\beta'}_n}{\\alpha_n}{x_n}{{\\beta'}_{n-1}}{\\alpha_{n-1}}{x_{n-1}}{{\\beta'}_1}\n{\\alpha_1}{x_1})\\;\\;=\\;\\;\n\\prod_{i=1}^{n}\\delta_{\\beta_i,{\\beta'}_i}\\nonumber \\end{eqnarray}\n\\end{lemma}\n\n{\\sl Proof:}\\,\n\nThe cyclicity property is not completely obvious. We give here a detailed proof\nof this fact:\n\\begin{eqnarray*}\n&&\\Outstate{\\beta_n}{\\alpha_n}{x_n}{\\beta_{n-1}}{\\alpha_{n-1}}{x_{n-1}}{\\beta_1}{\\alpha_1}{x_1}=\\\\\n&&=v^{-1}_{\\beta_n}tr_{V_{\\beta_n}}\n(\\psi^{\\beta_n}_{\\beta_1\n\\alpha_1}\\guf{\\beta_1}_{[x_1,x_2]}\\psi^{\\beta_1}_{\\beta_2\n\\alpha_2}\\guf{\\beta_2}_{[x_2,x_3]} \\cdots\n\\psi^{\\beta_{n-1}}_{\\beta_n\\alpha_{n}}\\guf{\\beta_n}_{[x_{n},x_1]}\n\\Rmff{\\beta_n}{\\alpha_1}\\muf{\\beta_n})\\\\\n&&=v^{-1}_{\\beta_n}tr_{V_{\\beta_1}}\n(\\guf{\\beta_1}_{[x_1,x_2]}\\psi^{\\beta_1}_{\\beta_2 \\alpha_2}\n\\guf{\\beta_2}_{[x_2,x_3]} \\cdots\n\\psi^{\\beta_{n-1}}_{\\beta_n\\alpha_{n}}\\guf{\\beta_n}_{[x_{n},x_1]}\n\\psi^{\\beta_n}_{\\beta_1 \\alpha_1}\n\\Rmff{\\beta_1}{\\alpha_1}\\muf{\\beta_1}v_{\\alpha_2})\\\\\n&&=v^{-1}_{\\beta_n}tr_{V_{\\beta_1}\\otimes {V'}_{\\beta_1}}\n(P_{V_{\\beta_1},{V'}_{\\beta_1}}\\guf{\\beta_1}_{[x_1,x_2]}\\psi^{{\\beta'}_1}_{\\beta_2 \\alpha_2}\n\\guf{\\beta_2}_{[x_2,x_3]} \\cdots\n\\psi^{\\beta_{n-1}}_{\\beta_n\\alpha_{n}}\\guf{\\beta_n}_{[x_{n},x_1]}\n\\psi^{\\beta_n}_{{\\beta'}_1 \\alpha_1}\n\\Rmff{{\\beta'}_1}{\\alpha_1}\\muf{{\\beta'}_1}v_{\\alpha_2})\\\\\n&&=\\sum_{(i),(j)}v^{-1}_{\\beta_n}tr_{V_{\\beta_1}}\n(\\psi^{\\beta_1}_{\\beta_2 \\alpha_2} b_{(i)}^{\\beta_2} \\guf{\\beta_2}_{[x_2,x_3]}\n\\cdots\n\\psi^{\\beta_{n-1}}_{\\beta_n\\alpha_{n}}\\guf{\\beta_n}_{[x_{n},x_1]}\nS(a_{(j)}^{\\beta_n})\n\\psi^{\\beta_n}_{\\beta_1 \\alpha_1}\n\\Rmff{\\beta_1}{\\alpha_1}\\muf{\\beta_1}v_{\\alpha_2}\nb_{(j)}^{\\beta_1}\\guf{\\beta_1}_{[x_1,x_2]} S^{2}(a_{(i)}^{\\beta_1}))\\\\\n&&=\\sum_{(i)}v^{-1}_{\\beta_n}tr_{V_{\\beta_1}}\n(S^{2}(a_{(i)}^{\\beta_1})b_{(i)}^{\\beta_1}\n\\psi^{\\beta_1}_{\\beta_2 \\alpha_2} \\guf{\\beta_2}_{[x_2,x_3]} \\cdots\n\\psi^{\\beta_{n-1}}_{\\beta_n\\alpha_{n}}\\guf{\\beta_n}_{[x_{n},x_1]}\\psi^{\\beta_n}_{\\beta_1 \\alpha_1}\n\\guf{\\beta_1}_{[x_1,x_2]} \\Rmff{\\beta_1}{\\alpha_2}v_{\\beta_n})\\\\\n&&=\\sum_{(i)}v_{\\beta_1}^{-1}tr_{V_{\\beta_1}} (\\psi^{\\beta_1}_{\\beta_2\n\\alpha_2}\n\\guf{\\beta_2}_{[x_2,x_3]} \\cdots\n\\psi^{\\beta_{n-1}}_{\\beta_n\\alpha_{n}}\\guf{\\beta_n}_{[x_{n},x_1]}\\psi^{\\beta_n}_{\\beta_1 \\alpha_1}\n\\guf{\\beta_1}_{[x_1,x_2]} \\Rmff{\\beta_1}{\\alpha_2}\\mu_{\\beta_1})\\\\\n&&=\\Outstate{\\beta_1}{\\alpha_1}{x_1}{\\beta_{n}}{\\alpha_{n}}{x_{n}}{\\beta_2}{\\alpha_2}{x_2} \\end{eqnarray*} The\ngauge transformation is very simple to derive using the decomposition rules of\nthe elements of the\ngroup and we can proove the scalar product property using simply the\nintegration formula and the\nunitarity relations of Clebsch-Gordan maps. \\begin{eqnarray*} &&\\int \\prod_{l \\in {\\cal\nC}}dh(U_l)tr_{\\otimes_{i}V_{x_i^a}}(\\sigma_{{\\cal C}} \\Outstate{\\beta_n\nm_n}{\\alpha_n}{x_n}{\\beta_{n-1} m_{n-1}}{\\alpha_{n-1}}{x_{n-1}}{\\beta_1\nm_1}{\\alpha_1}{x_1}\\times\\\\\n&&\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\Instate{{\\beta'}_n\n{m'}_n}{\\alpha_n}{x_n}{{\\beta'}_{n-1}\n{m'}_{n-1}}{\\alpha_{n-1}}{x_{n-1}}{{\\beta'}_1 {m'}_1}{\\alpha_1}{x_1})=\\\\\n&&=\\int \\prod_{l \\in {\\cal\nC}}dh(U_l) v^{-1}_{\\beta_n}v_{{\\beta'}_n}tr_{V_{\\beta_n} \\otimes\nV_{{\\beta'}_n}} (\\psi^{\\beta_n\nm_n}_{\\beta_1 \\alpha_1}\\guf{\\beta_1}_{[x_1,x_2]}\\psi^{\\beta_1 m_1}_{\\beta_2\n\\alpha_2} \\cdots\n\\psi^{\\beta_{n-1}\nm_{n-1}}_{\\beta_n\\alpha_{n}}\\muf{{\\beta'}_n}\\Rff{{\\beta'}_n}{\\alpha_1}\n\\guf{\\beta_n}_{[x_{n},x_1]}\\times\\\\ &&\\times\\guf{{\\beta'}_n}_{[x_1,x_{n}]}\n \\Rff{\\beta_n}{\\alpha_1}^{-1}\\muf{\\beta_n}\n\\phi^{{\\beta'}_n \\alpha_{n}}_{{\\beta'}_{n-1} {m'}_{n-1}} \\cdots\n\\phi^{{\\beta'}_2 \\alpha_2}_{{\\beta'}_1\n{m'}_1}\\guf{{\\beta'}_1}_{[x_2,x_{1}]}\\phi^{{\\beta'}_1\n\\alpha_1}_{{\\beta'}_n {m'}_n})=\\\\ &&=\\int \\prod_{l \\in {\\cal\nC}\\setminus{[x_1,x_n]}}dh(U_l)\n\\frac{\\delta_{\\beta_n,{\\beta'}_n}\n\\delta_{m_n,{m'}_n}}{[d_{\\beta_n}]}tr_{V_{\\beta_n}}\n(\\muf{\\beta_n}\\psi^{\\beta_n m_n}_{\\beta_1\n\\alpha_1}\\guf{\\beta_1}_{[x_1,x_2]}\\psi^{\\beta_1\nm_1}_{\\beta_2 \\alpha_2} \\cdots \\psi^{\\beta_{n-1} m_{n-1}}_{\\beta_n\\alpha_{n}}\n\\phi^{{\\beta}_n\n\\alpha_{n}}_{{\\beta'}_{n-1} {m'}_{n-1}} \\cdots \\\\ &&\\cdots \\phi^{{\\beta'}_2\n\\alpha_2}_{{\\beta'}_1\n{m'}_1}\\guf{{\\beta'}_1}_{[x_2,x_{1}]}\\phi^{{\\beta'}_1 \\alpha_1}_{{\\beta'}_n\n{m'}_n})=\\\\\n&&=\\frac{\\delta_{\\beta_n,{\\beta'}_n}\\delta_{m_n,{m'}_n}}{[d_{\\beta_n}]}tr_{V_{\\beta_n}}\n(\\muf{\\beta_n})\\prod_{i=1}^{n-1}(\\delta_{\\beta_i,{\\beta'}_i}\\delta_{m_i,{m'}_i})=\\prod_{i=1}^{n}(\\delta_{\\beta_i,{\\beta'}_i}\\delta_{m_i,{m'}_i}).\\\\\n\\end{eqnarray*} This ends the proof of the lemma.\\\\\n\n$\\Box$\n\n\n\nAll elements associated to elementary blocks can be computed in terms of \"In\"\nand \"Out\" states of the latter form.\n\\begin{proposition}\nLet us give here the expression of the elements associated to the elementary\nblocks\nenumerated before:\\\\\n\\begin{eqnarray}\n&&{\\bf {\\cal A}^{elem}_{cup}}= \\sum_{\\beta_0} [d_{\\beta_0}] {\\cal O}(\\beta_0)\\\\\n&&{\\bf {\\cal A}^{elem}_{cap}}= \\sum_{\\beta_0} [d_{\\beta_0}] {\\cal I}(\\beta_0)\\\\\n&&{\\bf {\\cal A}^{elem}_{(n,m)(n+m) tri.}}= \\sum_{\\beta_1,\\cdots, \\beta_{n+m}}\n[d_{\\beta_{n+m}}]^{-1}\n\\Intri{{\\beta'}_n}{{\\alpha'}_n}{{x'}_n}{{\\beta'}_1}{{\\alpha'}_1}{{x'}_1}\\Intri{{\\beta''}_n}{{\\alpha''}_n}{{x''}_n}{{\\beta''}_1}{{\\alpha''}_1}{{x''}_1}\n\\times\\\\\n&&\\times\\Outtri{{\\beta}_n}{{\\alpha}_n}{{x}_n}{{\\beta}_1}{{\\alpha}_1}{{x}_1}\n\\delta_{{\\beta'}_n, {\\beta''}_m, {\\beta}_{n+m}, \\beta_m}\n\\prod_{k=1}^{n}\\delta_{\\alpha_{m+k}, {\\alpha'}_{k}}\n\\prod_{k=1}^{m}\\delta_{\\alpha_{k}, {\\alpha''}_{k}}\n\\prod_{k=1}^{n-1}\\delta_{\\alpha_{m+k}, {\\alpha'}_{k}}\n\\prod_{k=1}^{m-1}\\delta_{\\alpha_{k}, {\\alpha''}_{k}}\\nonumber\\\\\n&&{\\bf {\\cal A}^{elem}_{(n+m)(n,m) tri.}}= \\sum_{\\beta_1,\\cdots, \\beta_{n+m}}\n[d_{\\beta_{n+m}}]^{-1}\n\\Intri{{\\beta}_n}{{\\alpha}_n}{{x}_n}{{\\beta}_1}{{\\alpha}_1}{{x}_1}\n\\Outtri{{\\beta'}_n}{{\\alpha'}_n}{{x'}_n}{{\\beta'}_1}{{\\alpha'}_1}{{x'}_1}\\times\\\\\n&&\\times\n\\Outtri{{\\beta''}_n}{{\\alpha''}_n}{{x''}_n}{{\\beta''}_1}{{\\alpha''}_1}{{x''}_1}\n\\delta_{{\\beta'}_n, {\\beta''}_m, {\\beta}_{n+m}, \\beta_m}\n\\prod_{k=1}^{n}\\delta_{\\alpha_{m+k}, {\\alpha'}_{k}}\n\\prod_{k=1}^{m}\\delta_{\\alpha_{k}, {\\alpha''}_{k}}\n\\prod_{k=1}^{n-1}\\delta_{\\alpha_{m+k}, {\\alpha'}_{k}}\n\\prod_{k=1}^{m-1}\\delta_{\\alpha_{k}, {\\alpha''}_{k}}\\nonumber\\\\\n&&{\\bf {\\cal A}^{elem}_{free}}=\\sum_{\\beta_1,\\cdots,\\beta_n}\n\\Intri{\\beta_n}{\\alpha_{n}}{x_{n}}{\\beta_1}{\\alpha_1}{x_1}\n\\Outtri{{\\beta'}_n}{{\\alpha'}_{n}}{{x'}_{n}}{{\\beta'}_1}{{\\alpha'}_1}{{x'}_1}\n\\prod_{i=1}^{n}\\delta_{\\beta_i,{\\beta'}_i}\\prod_{i=1}^{n}\\delta_{\\alpha_i,{\\alpha'}_i}\\\\\n&&{\\bf {\\cal A}^{elem}_{creation}}=\\sum_{{\\beta'}_1,\\cdots,{\\beta'}_n}\n\\Incross{\\beta_n}{\\alpha_{k+2}}{x_{k+2}}{\\beta_k}{\\alpha_{k-1}}{x_{k-1}}{\\alpha_1}{x_1}\n\\Outtri{{\\beta'}_n}{{\\alpha'}_{n}}{{x'}_{n}}{{\\beta'}_1}{{\\alpha'}_1}{{x'}_1}\\times\\\\\n&&\\times\\prod_{i=k+2}^{n}(\\delta_{\\beta_i,{\\beta'}_i}\\delta_{\\alpha_i,{\\alpha'}_i})\n\\delta_{{\\beta'}_{k+1},\\beta_k,{\\beta'}_{k-1}}\\delta_{{\\alpha'}_{k+1},{\\bar{\\alpha'}}_{k}}\n(v_{{\\beta'}_k} v_{{\\beta}_k})^{{1\\over 2}}\n(\\frac{[d_{{\\beta'}_k}]}{[d_{{\\beta}_k}]})^{{1\\over 2}}\nN^{{\\beta'}_k,{m'}_k}_{{\\beta'}_{k-1}\\alpha_k} \\prod_{i=1}^{k-1}\n\\delta_{\\alpha_i,{\\alpha'}_i}\n\\prod_{i=1}^{k-2} \\delta_{\\beta_i,{\\beta'}_i}\\nonumber\\\\ &&{\\bf {\\cal\nA}^{elem}_{annihil.}}\n=\\sum_{{\\beta'}_1,\\cdots,{\\beta'}_n}\n\\Intri{{\\beta'}_n}{{\\alpha'}_{n}}{{x'}_{n}}{{\\beta'}_1}{{\\alpha'}_1}{{x'}_1}\n\\Outcross{\\beta_n}{\\alpha_{k+2}}{x_{k+2}}{\\beta_k}{\\alpha_{k-1}}{x_{k-1}}{\\alpha_1}{x_1} \\times\\\\\n&&\\times\\prod_{i=k+2}^{n}(\\delta_{\\beta_i,{\\beta'}_i}\\delta_{\\alpha_i,{\\alpha'}_i})\n\\delta_{{\\beta'}_{k+1},\\beta_k,{\\beta'}_{k-1}}\\delta_{{\\alpha'}_{k+1},{\\bar{\\alpha'}}_{k}}\n(v_{{\\beta'}_k} v_{{\\beta}_k})^{-{1\\over 2}}\n(\\frac{[d_{{\\beta'}_k}]}{[d_{{\\beta}_k}]})^{{1\\over 2}}\n N^{{\\beta'}_k,{m'}_k}_{{\\beta'}_{k-1}\\alpha_k} \\prod_{i=1}^{k-1}\n\\delta_{\\alpha_i,{\\alpha'}_i} \\prod_{i=1}^{k-2}\n\\delta_{\\beta_i,{\\beta'}_i}\\nonumber\\\\ &&{\\bf {\\cal\nA}^{elem}_{overcross.}}=\\sum_{\\beta_1,\\cdots,\\beta_n,{\\beta'}_k}\n\\Incross{\\beta_n}{\\alpha_{k+1}}{x_{k+1}}{\\beta_k}{\\alpha_k}{x_k}{\\alpha_1}{x_1}\n\\Outcross{{\\beta'}_n}{{\\alpha'}_{k+1}}{{x'}_{k+1}}{{\\beta'}_k}{{\\alpha'}_k}{{x'}_k}{{\\alpha'}_1}{{x'}_1}\\times\\nonumber\\\\\n&&\\times\\prod_{i\\not=\nk}\\delta_{\\beta_i,{\\beta'}_i}\\prod_{i\\not=k,k+1}\\delta_{\\alpha_i,{\\alpha'}_i}\n\\frac\n{tr_q(\\psi_{\\beta_{k-1}\\alpha_{k-1}}^{\\beta_{k-2}}\\psi_{\\beta_{k}\\alpha_{k}}^{\\beta_{k-1}}\\Rtff{\\alpha_k}{\\alpha_{k-1}}\n\\phi^{\\beta_{k}\\alpha_{k-1}}_{{\\beta'}_{k-1}}\\phi^{{\\beta'}_{k-1}\\alpha_{k}}_{\\beta_{k-2}})}{[d_{\\beta_{k-2}}]v_{{\\beta}_k}^{{1\\over\n2}}v_{{\\beta'}_k}^{-{1\\over 2}}}\n\\delta_{\\alpha_{k+1},{\\alpha'}_k}\\delta_{\\alpha_{k},{\\alpha'}_{k+1}}\\\\ &&{\\bf\n{\\cal\nA}^{elem}_{undercross.}}=\\sum_{\\beta_1,\\cdots,\\beta_n,{\\beta'}_k}\n\\Incross{\\beta_n}{\\alpha_{k+1}}{x_{k+1}}{\\beta_k}{\\alpha_k}{x_k}{\\alpha_1}{x_1}\n\\Outcross{{\\beta'}_n}{{\\alpha'}_{k+1}}{{x'}_{k+1}}{{\\beta'}_k}{{\\alpha'}_k}{{x'}_k}{{\\alpha'}_1}{{x'}_1}\\times\\nonumber\\\\\n&&\\times\\prod_{i\\not=k}\\delta_{\\beta_i,{\\beta'}_i}\\prod_{i\\not=k,k+1}\\delta_{\\alpha_i,{\\alpha'}_i}\n\\frac\n{tr_q(\\psi_{\\beta_{k-1}\\alpha_{k-1}}^{\\beta_{k-2}}\\psi_{\\beta_{k}\\alpha_{k}}^{\\beta_{k-1}}\\Rtmff{\\alpha_k}{\\alpha_{k-1}}\n\\phi^{\\beta_{k}\\alpha_{k-1}}_{{\\beta'}_{k-1}}\\phi^{{\\beta'}_{k-1}\\alpha_{k}}_{\\beta_{k-2}})}{[d_{\\beta_{k-2}}]v_{{\\beta}_k}^{{1\\over\n2}}v_{{\\beta'}_k}^{-{1\\over 2}}}\n\\delta_{\\alpha_{k+1},{\\alpha'}_k}\\delta_{\\alpha_{k},{\\alpha'}_{k+1}}\n\\end{eqnarray} \\end{proposition}\n\n\\medskip\n\n{\\sl Proof:}\\, \\\\\nThe result for ${\\cal A}^{elem}_{cup}$ and ${\\cal A}^{elem}_{cap}$ is clearly\ngiven by the Boltzmann\nweight . The computation of the other elements need a careful description. The\nidea is very simple.\nWe first absorb each link segment in the attached boltzmann weight to put each\nelements associated\nto \"square plaquettes\" in a same practical form where all edges of the boundary\nappear one and only\none time and always in the same order. This form allows us to reduce the gluing\nof \"plaquettes\" to\none commutation plus one integration only. \\\\ In the case of an empty square\nplaquette the\ncorresponding Boltzmann weight is already in the reduced form.\\\\\n For example the square plaquette element involved in the computation of a free\npropagation is\ngiven by:\n\\begin{eqnarray}\n&&\\delta_{[{x'}_n,{x}_{n},{x}_{n-1},{x'}_{n-1}]}\\guf{\\alpha_{n-1}}_{[x_{n-1},{x'}_{n-1}]}=\n\\sum_{\\beta_{n-1}{\\beta'}_{n-2}}[d_{\\beta_{n-1}}]\\lambda_{\\beta_{n-1}\n\\alpha_{n-1}\\beta_{n-2}}^{-1}\\times\\\\ &&\\times\ntr_{V_{\\beta_{n-1}}}(\\muf{\\beta_{n-1}}\\guf{\\beta_{n-1}}_{[{x'}_{n},{x}_{n}]}\n\\guf{\\beta_{n-1}}_{[x_{n},{x}_{n-1}]}\\phi^{\\beta_{n-1}\n\\alpha_{n-1}}_{{\\beta'}_{n-2}}\\guf{{\\beta'}_{n-2}}_{[x_{n-1},{x'}_{n-1}]}\\psi_{\\beta_{n-1}\n\\alpha_{n-1}}^{{\\beta'}_{n-2}}\n\\guf{\\beta_{n-1}}_{[{x'}_{n-1},{x'}_{n}]})\\nonumber\n\\end{eqnarray} where the\nnotations are summarized on the following figure:\n\n\\par\n\\centerline{\\psfig{figure=squaplaq1.ps}}\n\\par\n\n\nIn the case of a creation element for example, with a similar computation and\nthe same notations, we have :\n\\begin{eqnarray}\n&&\\delta_{[{x'}_n,{x}_{n},{x}_{n-1},{x'}_{n-1}]}\\guf{\\alpha_{n-1}}_{[{x'}_{n},{x'}_{n-1}]}^{\\#}=\n\\nonumber\\\\\n&&=\\sum_{\\beta_n,{\\beta'}_n} ([d_{\\beta_n}][d_{{\\beta'}_n}])^{1 \\over 2}\nN^{{\\beta'}_n \\alpha_n}_{\\beta_n}tr_{V_{\\beta_n}}(\\muf{\\beta_n}\n\\guf{\\beta_n}_{[{x'}_n,x_n]}\\guf{\\beta_n}_{[x_n,x_{n-1}]}\\guf{\\beta_n}_{[x_{n-1},{x'}_{n-1}]}\n\\times\\nonumber\\\\\n&&\\times\\psi_{{\\beta'}_n \\alpha_n}^{\\beta_n}\n\\guf{{\\beta'}_n}_{[{x'}_n,{x'}_{n-1}]}\\psi_{{\\beta}_n\n\\bar{\\alpha_n}}^{{\\beta'}_n}\\Rmff{\\beta_n}{\\bar{\\alpha_n}}\\lambda_{\\beta_n\n\\bar{\\alpha_n}\n{\\beta'}_n}) \\end{eqnarray}\n\n\n When the square contains a crossing the reduction is less obvious and is given\nby the\nfollowing lemma:\n\\begin{lemma}[integration over a crossing]\nWith the notations of the figure:\\\\\n\n\\par\n\\centerline{\\psfig{figure=squaplaq2.ps}}\n\\par\nit was shown in our last work \\cite{BR2} that:\n\\begin{eqnarray}\n&&\\int\ndh(U_{[x_{n-1},y]})dh(U_{[y,{x'}_n]})dh(U_{[x_{n},y]})dh(U_{[y,{x'}_{n-1}]})\\times\\nonumber\\\\\n&&\\times\\delta_{[y x_n x_{n-1}]}\\delta_{[y x_{n-1} {x'}_{n-1}]}\\delta_{[y\n{x'}_{n-1} {x'}_{n}]}\n\\delta_{[y {x'}_n x_{n}]}\\guf{\\alpha_n}_{[x_n y {x'}_{n-1}]}\n\\guf{\\alpha_{n-1}}_{[x_{n-1} y\n{x'}_n]}=\\nonumber\\\\\n&&=\\sum_{\\beta_n}[d_{\\beta_n}](\\frac{v_{{\\beta'}_{n-1}}v_{{\\beta}_{n-2}}}{v_{{\\beta}_{n-1}}\nv_{{\\beta}_{n}}})^{1\n\\over 2}\\frac\n{tr_q(\\psi_{\\beta_{k-1}\\alpha_{k-1}}^{\\beta_{k-2}}\\psi_{\\beta_{k}\\alpha_{k}}^{\\beta_{k-1}}\n\\Rtff{\\alpha_k}{\\alpha_{k-1}}\n\\phi^{\\beta_{k}\\alpha_{k-1}}_{{\\beta'}_{k-1}}\n\\phi^{{\\beta'}_{k-1}\\alpha_{k}}_{\\beta_{k-2}})}{[d_{\\beta_{k-2}}]}\ntr_{V_{\\beta_n}}(\\muf{\\beta_n} \\guf{\\beta_n}_{[{x'}_{n}\n{x}_{n}]}\\times\\nonumber\\\\\n&&\\times\\phi^{\\beta_n \\alpha_n}_{\\beta_{n-1}}\\guf{\\beta_{n-1}}_{[{x}_{n}\n{x}_{n-1}]}\n\\phi^{\\beta_{n-1} \\alpha_{n-1}}_{\\beta_{n-2}}\\guf{\\beta_{n-2}}_{[{x}_{n-1}\n{x'}_{n-1}]}\n\\psi^{\\beta_{n-2}}_{{\\beta'}_{n-1} \\alpha_n}\\guf{{\\beta'}_{n-1}}_{[{x'}_{n-1}\n{x'}_n]}\\psi_{\\alpha_{n-1}\\beta_n\n}^{{\\beta'}_{n-1}}\\Rpff{\\alpha_{n-1}}{\\beta_n}) \\end{eqnarray} and the\nanalog relation for the undercrossing. \\end{lemma}\n\n\\medskip\n\nNow all square plaquettes elements are in the reduced form.\nThen the problem of computing the whole elementary block element is reduced to\nthe gluing of elements associated to each of the square plaquettes given in the\nreduced form, i.e. a commutation $+$ an integration. Now, let us give a careful\ncomputation\nin the case of ${\\cal A}_{free}^{elem}$ and the other ones, very similar to\nthis one,\nwill be led to the reader.\n\n\nThe notations are summarized on the figure:\n\n\\par\n\\centerline{\\psfig{figure=free.ps}}\n\\par\n\n\n({\\bf Remark:} In the following computation we forget again the multiplicities\nof\nrepresentations in all\ndecompositions, but we must take care of them...)\nLet us first describe the gluing of two square plaquettes:\n\n\\begin{eqnarray*}\n&&\\int dh(U_{[x_{n-1},{x'}_{n-1}]})\n\\delta_{[{x'}_n,{x}_{n},{x}_{n-1},{x'}_{n-1}]}\\guf{\\alpha_{n-1}}_{[x_{n-1},{x'}_{n-1}]}\n\\delta_{[{x'}_{n-1},{x}_{n-1},{x}_{n-2},{x'}_{n-2}]}\\guf{\\alpha_{n-2}}_{[x_{n-2},{x'}_{n-2}]}=\\\\\n&&=\\int dh(U_{[x_{n-1},{x'}_{n-1}]})\n\\sum_{\\beta_{n-1},\\beta_{n-2},{\\beta'}_{n-2},{\\beta'}_{n-3}}[d_{\\beta_{n-1}}][d_{\\beta_{n-2}}]\\lambda_{\\beta_{n-1} \\alpha_{n-1}\\beta_{n-2}}^{-1}\\lambda_{\\beta_{n-2} \\alpha_{n-2}\\beta_{n-3}}^{-1}\\times\\\\\n&&\\times\ntr_{V_{\\beta_{n-1}}}(\\muf{\\beta_{n-1}}\\guf{\\beta_{n-1}}_{[{x'}_{n},{x}_{n}]}\\guf{\\beta_{n-1}}_{[x_{n},{x}_{n-1}]}\\phi^{\\beta_{n-1} \\alpha_{n-1}}_{{\\beta'}_{n-2}}\\guf{{\\beta'}_{n-2}}_{[x_{n-1},{x'}_{n-1}]}\\psi_{\\beta_{n-1} \\alpha_{n-1}}^{{\\beta'}_{n-2}}\n\\guf{\\beta_{n-1}}_{[{x'}_{n-1},{x'}_{n}]})\\\\\n&&\\times\ntr_{V_{\\beta_{n-2}}}(\\muf{\\beta_{n-2}}\\guf{\\beta_{n-2}}_{[{x'}_{n-1},{x}_{n-1}]}\\guf{\\beta_{n-2}}_{[x_{n-1},{x}_{n-2}]}\\phi^{\\beta_{n-2} \\alpha_{n-2}}_{{\\beta'}_{n-3}}\\guf{{\\beta'}_{n-3}}_{[x_{n-2},{x'}_{n-2}]}\\psi_{\\beta_{n-2} \\alpha_{n-2}}^{{\\beta'}_{n-3}}\\guf{\\beta_{n-2}}_{[{x'}_{n-2},{x'}_{n-1}]})=\\\\\n&&=\\sum_{(i),(j)}\\int dh(U_{[x_{n-1},{x'}_{n-1}]})\n\\sum_{\\beta_{n-1},\\beta_{n-2},{\\beta'}_{n-2},{\\beta'}_{n-3}}[d_{\\beta_{n-1}}][d_{\\beta_{n-2}}]\\lambda_{\\beta_{n-1} \\alpha_{n-1}\\beta_{n-2}}^{-1}\\lambda_{\\beta_{n-2} \\alpha_{n-2}{\\beta'}_{n-3}}^{-1}\\times\\\\\n&&\\times tr_{V_{\\beta_{n-1}}\\otimes\nV_{\\beta_{n-2}}}(\\muf{\\beta_{n-1}}\\guf{\\beta_{n-1}}_{[{x'}_{n},{x}_{n}]}\\guf{\\beta_{n-1}}_{[x_{n},{x}_{n-1}]}\\phi^{\\beta_{n-1} \\alpha_{n-1}}_{{\\beta'}_{n-2}}\\muf{\\beta_{n-2}}a_{(i)}^{\\beta{n-2}}\\muf{\\beta_{n-2}}^{-1}\\times\\\\\n&&\\guf{{\\beta'}_{n-2}}_{[x_{n-1},{x'}_{n-1}]}\n\\muf{\\beta_{n-2}}\\guf{\\beta_{n-2}}_{[{x'}_{n-1},{x}_{n-1}]}\\guf{\\beta_{n-2}}_{[x_{n-1},{x}_{n-2}]}\n\\phi^{\\beta_{n-2}\n\\alpha_{n-2}}_{{\\beta'}_{n-3}}\\guf{{\\beta'}_{n-3}}_{[x_{n-2},{x'}_{n-2}]}\\psi_{\\beta_{n-2} \\alpha_{n-2}}^{{\\beta'}_{n-3}}\\times\\\\\n&&\\guf{\\beta_{n-2}}_{[{x'}_{n-2},{x'}_{n-1}]}S(a_{(j)}^{\\beta_{n-2}})\n\\psi_{\\beta_{n-1} \\alpha_{n-1}}^{{\\beta'}_{n-2}}\nb_{(i)}^{\\beta_{n-1}}b_{(j)}^{\\beta_{n-1}}\\guf{\\beta_{n-1}}_{[{x'}_{n-1},{x'}_{n}]})=\\\\\n&&=\\sum_{(i),(j)}\\sum_{\\beta_{n-1},\\beta_{n-2},{\\beta'}_{n-2},{\\beta'}_{n-3}}[d_{\\beta_{n-1}}][d_{\\beta_{n-2}}]\\lambda_{\\beta_{n-1} \\alpha_{n-1}\\beta_{n-2}}^{-1}\\lambda_{\\beta_{n-2} \\alpha_{n-2}{\\beta'}_{n-3}}^{-1}\\frac{\\delta_{\\beta_{n-2},{\\beta'}_{n-2}}}{[d_{\\beta_{n-2}}]}\\times\\\\\n&&\\times\ntr_{V_{\\beta_{n-1}}}(\\muf{\\beta_{n-1}}\\guf{\\beta_{n-1}}_{[{x'}_{n},{x}_{n}]}\\guf{\\beta_{n-1}}_{[x_{n},{x}_{n-1}]}\\phi^{\\beta_{n-1} \\alpha_{n-1}}_{{\\beta}_{n-2}}\\guf{\\beta_{n-2}}_{[x_{n-1},{x}_{n-2}]}\n\\phi^{\\beta_{n-2}\n\\alpha_{n-2}}_{{\\beta'}_{n-3}}\\guf{{\\beta'}_{n-3}}_{[x_{n-2},{x'}_{n-2}]}\\times\\\\\n&&\\times\\psi_{\\beta_{n-2}\n\\alpha_{n-2}}^{{\\beta'}_{n-3}}\\guf{\\beta_{n-2}}_{[{x'}_{n-2},{x'}_{n-1}]}S(a_{(j)}^{\\beta_{n-2}})\nS^2(a_{(i)}^{\\beta{n-2}})\n\\psi_{\\beta_{n-1} \\alpha_{n-1}}^{{\\beta}_{n-2}}\nb_{(i)}^{\\beta_{n-1}}b_{(j)}^{\\beta_{n-1}}\\guf{\\beta_{n-1}}_{[{x'}_{n-1},{x'}_{n}]})=\\\\\n&&=\\sum_{\\beta_{n-1},\\beta_{n-2},{\\beta'}_{n-3}}[d_{\\beta_{n-1}}]\\lambda_{\\beta_{n -1}\\alpha_{n-1}\\beta_{n-2}}^{-1}\\lambda_{\\beta_{n-2} \\alpha_{n-2}{\\beta'}_{n-3}}^{-1} tr_{V_{\\beta_{n-1}}}(\\muf{\\beta_{n-1}}\\guf{\\beta_{n-1}}_{[{x'}_{n},{x}_{n}]}\\guf{\\beta_{n-1}}_{[x_{n},{x}_{n-1}]}\\times\\\\\n&&\\phi^{\\beta_{n-1}\n\\alpha_{n-1}}_{{\\beta}_{n-2}}\\guf{\\beta_{n-2}}_{[x_{n-1},{x}_{n-2}]}\n\\phi^{\\beta_{n-2}\n\\alpha_{n-2}}_{{\\beta'}_{n-3}}\\guf{{\\beta'}_{n-3}}_{[x_{n-2},{x'}_{n-2}]}\\psi_{\\beta_{n-2} \\alpha_{n-2}}^{{\\beta'}_{n-3}}\\guf{\\beta_{n-2}}_{[{x'}_{n-2},{x'}_{n-1}]}\\times\\\\\n&&\\times\\psi_{\\beta_{n-1}\n\\alpha_{n-1}}^{{\\beta}_{n-2}}\\guf{\\beta_{n-1}}_{[{x'}_{n-1},{x'}_{n}]})\\\\\n\\end{eqnarray*}\nIn the same way we can glue $n-1$ Boltzmann weights and links. As a result we\nhave obviously:\n\\begin{eqnarray*}\n&&\\int \\prod_{i=1}^{n-1} dh(U_{[x_{i},{x'}_{i}]})\n\\prod_{i=1}^{n-1}(\\delta_{[{x'}_{i+1},{x}_{i+1},{x}_{i},{x'}_{i}]}\\guf{\\alpha_{i}}_{[x_{i},{x'}_{i}]})=\\\\\n&&=\\sum_{\\beta_{n-1},\\cdots,\\beta_{2},{\\beta}_{1}}[d_{\\beta_{n-1}}]\\prod_{i=2}^{n-1}\\lambda_{\\beta_i \\alpha_{i}\\beta_{i-1}}^{-1}\\lambda_{\\beta_{1} \\alpha_{1}{\\beta'}_{n}}^{-1} tr_{V_{\\beta_{n-1}}}(\\muf{\\beta_{n-1}}\\guf{\\beta_{n-1}}_{[{x'}_{n},{x}_{n}]}\\guf{\\beta_{n-1}}_{[x_{n},{x}_{n-1}]}\\phi^{\\beta_{n-1} \\alpha_{n-1}}_{{\\beta}_{n-2}}\\cdots\\\\\n&&\\cdots \\phi^{\\beta_{1}\n\\alpha_{1}}_{{\\beta'}_{n}}\\guf{{\\beta'}_{n}}_{[x_{1},{x'}_{1}]}\\psi_{{\\beta}_{1} \\alpha_{1}}^{{\\beta'}_{n}}\\cdots\n\\psi_{\\beta_{n-1}\n\\alpha_{n-1}}^{{\\beta}_{n-2}}\\guf{\\beta_{n-1}}_{[{x'}_{n-1},{x'}_{n}]})\\\\\n\\end{eqnarray*}\nThe computation of ${\\cal A}^{elem}_{free}$ can be achieved by the gluing of\nthe last\nBoltzmann weight to obtain the cylinder with $n$ strands.This Boltzmann weight\nmust be changed to the\nsquare plaquette element corresponding to crossings,... in the computations of\nthe other blocks\nelements. \\begin{eqnarray*} &&\\int \\prod_{i=1}^{n} dh(U_{[x_{i},{x'}_{i}]})\n\\prod_{i=1}^{n}(\\delta_{[{x'}_{i+1},{x}_{i+1},{x}_{i},{x'}_{i}]}\\guf{\\alpha_{i}}_{[x_{i},{x'}_{i}]})=\\\\\n&&=\\int dh(U_{[x_{n},{x'}_{n}]})dh(U_{[x_{1},{x'}_{1}]})\\times\\\\\n&&(\\sum_{\\beta_n,{\\beta'}_{n-1}}[d_{\\beta_n}]\\lambda_{\\beta_n \\alpha_n\n{\\beta'}_{n-1}}^{-1}tr_{V_{\\beta_n}}(\\muf{\\beta_n}\\guf{\\beta_n}_{[x_1,x_n]}\n\\phi^{\\beta_n\n\\alpha_n}_{{\\beta'}_{n-1}}\\guf{{\\beta'}_{n-1}}_{[x_n,{x'}_n]}\\psi_{\\beta_n\n\\alpha_n}^{{\\beta'}_{n-1}}\\guf{\\beta_n}_{[{x'}_n,{x'}_1]}\\guf{\\beta_n}_{[{x'}_1,{x}_1]})) \\times\\\\\n&&\\times\n(\\sum_{\\beta_{n-1},\\cdots,\\beta_{2},{\\beta}_{1}}[d_{\\beta_{n-1}}]\\prod_{i=2}^{n-1}\\lambda_{\\beta_i\n\\alpha_{i}\\beta_{i-1}}^{-1}\\lambda_{\\beta_{1} \\alpha_{1}{\\beta'}_{n}}^{-1}\ntr_{V_{\\beta_{n-1}}}(\\muf{\\beta_{n-1}}\\guf{\\beta_{n-1}}_{[{x'}_{n},{x}_{n}]}\\guf{\\beta_{n-1}}_{[x_{n},{x}_{n-1}]}\n\\phi^{\\beta_{n-1} \\alpha_{n-1}}_{{\\beta}_{n-2}}\\cdots\\\\ &&\\cdots\n\\phi^{\\beta_{1}\n\\alpha_{1}}_{{\\beta'}_{n}}\\guf{{\\beta'}_{n}}_{[x_{1},{x'}_{1}]}\\psi_{{\\beta}_{1}\n\\alpha_{1}}^{{\\beta'}_{n}}\\cdots \\psi_{\\beta_{n-1}\n\\alpha_{n-1}}^{{\\beta}_{n-2}}\\guf{\\beta_{n-1}}_{[{x'}_{n-1},{x'}_{n}]}))=\\\\\n&&=\\int\ndh(U_{[x_{n},{x'}_{n}]})dh(U_{[x_{1},{x'}_{1}]})\n\\sum_{{\\beta'}_{n-1},\\beta_n,\\cdots,\\beta_{2},{\\beta'}_{1}}[d_{\\beta_{n-1}}][d_{\\beta_n}]\n\\prod_{i=2}^{n-1}\\lambda_{\\beta_i \\alpha_{i}\\beta_{i-1}}^{-1}\\lambda_{\\beta_{1}\n\\alpha_{1}{\\beta'}_{n}}^{-1} \\lambda_{\\beta_n \\alpha_n\n{\\beta'}_{n-1}}^{-1}\\times\\\\\n&&tr_{V_{\\beta_n} \\otimes\nV_{\\beta_{n-1}}}(\\muf{\\beta_n}\\guf{\\beta_n}_{[x_1,x_n]} \\phi^{\\beta_n\n\\alpha_n}_{{\\beta'}_{n-1}} \\muf{\\beta_{n-1}} a_{(j)}^{\\beta_{n-1}}\n\\muf{\\beta_{n-1}}^{-1}\n\\guf{{\\beta'}_{n-1}}_{[x_n,{x'}_n]}\\muf{\\beta_{n-1}}\\guf{\\beta_{n-1}}_{[{x'}_{n},{x}_{n}]}\n\\guf{\\beta_{n-1}}_{[x_{n},{x}_{n-1}]}\\times\\\\ &&\\times\\phi^{\\beta_{n-1}\n\\alpha_{n-1}}_{{\\beta}_{n-2}}\\cdots\\guf{\\beta_{1}}_{[x_{1},{x}_{n}]}b_{(i)}^{\\beta_1}\\phi^{\\beta_{1}\n\\alpha_{1}}_{{\\beta'}_{n}} \\psi_{\\beta_n\n\\alpha_n}^{{\\beta'}_{n-1}}b_{(j)}^{\\beta_n}\\guf{{\\beta}_{n}}_{[{x'}_n,{x'}_1]}\n\\guf{\\beta_n}_{[{x'}_1,{x}_1]}\\guf{{\\beta'}_{n}}_{[x_{1},{x'}_{1}]}S(a_{(i)}^{{\\beta'}_{n}})\n\\psi_{{\\beta}_{1} \\alpha_{1}}^{{\\beta'}_{n}}\\cdots\\\\ &&\\cdots\\psi_{\\beta_{n-1}\n\\alpha_{n-1}}^{{\\beta}_{n-2}}\\guf{\\beta_{n-1}}_{[{x'}_{n-1},{x'}_{n}]}))=\\\\\n&&=\\sum_{\\beta_n,\\cdots,{\\beta}_{1}}[d_{\\beta_n}]\n\\prod_{i=1}^{n}v_{\\alpha_{i}}^{-{1 \\over\n2}}tr_{V_{\\beta_n}}(\\muf{\\beta_n}\\guf{\\beta_n}_{[x_1,x_n]} \\phi^{\\beta_n\n\\alpha_n}_{{\\beta}_{n-1}}\\guf{\\beta_{n-1}}_{[x_{n},{x}_{n-1}]}\\phi^{\\beta_{n-1}\n\\alpha_{n-1}}_{{\\beta}_{n-2}}\\cdots\n\\guf{\\beta_{1}}_{[x_{1},{x}_{n}]}b_{(i)}^{\\beta_1}\\phi^{\\beta_{1}\n\\alpha_{1}}_{{\\beta}_{n}} \\times\\\\\n&&\\times\\muf{{\\beta}_{n}}^{-1}S(a_{(i)}^{{\\beta}_{n}}))\ntr_{V_{{\\beta}_{n-1}}}(S^2(a_{(j)}^{{\\beta}_{n-1}})\\psi_{\\beta_n\n\\alpha_n}^{{\\beta}_{n-1}}\nb_{(j)}^{\\beta_n} \\guf{{\\beta}_{n}}_{[{x'}_n,{x'}_1]} \\psi_{{\\beta}_{1}\n\\alpha_{1}}^{{\\beta'}_{n}}\\cdots \\psi_{\\beta_{n-1}\n\\alpha_{n-1}}^{{\\beta}_{n-2}}\\guf{\\beta_{n-1}}_{[{x'}_{n-1},{x'}_{n}]})=\\\\\n&&=\\sum_{\\beta_1,m_1,\\cdots,\\beta_n,m_n} \\Intri{\\beta_n\nm_n}{\\alpha_{n}}{x_{n}}{\\beta_1\nm_1}{\\alpha_1}{x_1} \\Outtri{{\\beta}_n m_n}{{\\alpha}_{n}}{{x}_{n}}{{\\beta}_1\nm_1}{{\\alpha}_1}{{x'}_1}\\\\ \\end{eqnarray*} This concludes the computation of ${\\cal\nA}^{elem}_{overcross}$,\n${\\cal A}^{elem}_{undercross}$, ${\\cal A}^{elem}_{creation}$, ${\\cal\nA}^{elem}_{annihil}$, and\n${\\cal A}^{elem}_{free}.$\\\\ The computations of ${\\cal A}^{elem}_{(n,m)(n+m)\ntri.}$ and ${\\cal\nA}^{elem}_{(n,m)(n+m) tri.}$ need one more step. It uses naturally the\nexpression of ${\\cal\nA}^{elem}_{free}$ as a basic object. Indeed we compute the element associated\nto the trinion by\ngluing one more plaquette to the cylinder with $n$ strands as it is shown in\nthe following figure.\n\n\n\\par\n\\centerline{\\psfig{figure=triangtrin.ps}}\n\\par\n\n\\medskip\nthe computation is realized by the usual techniques:\\\\\n\\begin{eqnarray*}\n&&{\\cal A}_{(n,m)(n+m) tri.}=\\int dh(U_{[x_1,x_{m+n}]})dh(U_{[x_{m+1},x_{m}]})\n\\delta_{[x_1,x_m,x_{m+1},x_{m+n}]}\\times\\\\\n&&\\times\\sum_{\\beta_1,\\cdots,\\beta_{m+n}}\n\\Intri{\\beta_{m+n}}{\\alpha_{n+m}}{x_{n+m}}{\\beta_1}{\\alpha_1}{x_1}\\Outtri{{\\beta}_{m+n}}{{\\alpha}_{n+m}}{{x'}_{n+m}}{{\\beta}_1}{{\\alpha}_1}{{x'}_1}=\\\\\n&&\\int dh(U_{[x_1,x_{m+n}]})dh(U_{[x_{m+1},x_{m}]})\n\\sum_{{\\beta'}_{m+n}}[d_{{\\beta'}_{m+n}}]v_{{\\beta'}_{m+n}}^{-1}\ntr_{V_{{\\beta'}_{m+n}}}(\\muf{{\\beta'}_{m+n}}\\guf{{\\beta'}_{m+n}}_{[x_{m+n},x_1]}\n\\guf{{\\beta'}_{m+n}}_{[x_{1},x_m]}\\times\\\\\n&&\\times\\guf{{\\beta'}_{m+n}}_{[x_{m},x_{m+1}]}\n\\guf{{\\beta'}_{m+n}}_{[x_{m+1},x_{m+n}]})\\;\n\\sum_{\\beta_{m+n},\\cdots,{\\beta}_{1}}\ntr_{V_{\\beta_{n+m}}}(\\muf{\\beta_{n+m}}\\Rff{\\beta_{m+n}}{\\alpha_1}\n\\guf{\\beta_{n+m}}_{[x_1,x_{n+m}]}\\phi^{\\beta_{m+n}\\alpha_{m+n}}_{{\\beta}_{m+n-1}}\\cdots\\\\\n&&\\cdots\\guf{\\beta_{1}}_{[x_{2},{x}_{1}]}\\phi^{\\beta_{1}\n\\alpha_{1}}_{{\\beta}_{m+n}})\n\\;\\Outtri{{\\beta}_{m+n}}{{\\alpha}_{n+m}}{{x'}_{n+m}}{{\\beta}_1}{{\\alpha}_1}{{x'}_1}=\\\\\n&&=\\int dh(U_{[x_1,x_{m+n}]})dh(U_{[x_{m+1},x_{m}]})\n\\sum_{{\\beta'}_{m+n},\\beta_{m+n},\\cdots,{\\beta}_{1}}\n[d_{{\\beta'}_{m+n}}]v_{{\\beta'}_{m+n}}^{-1}\\times\\\\\n&&\\times tr_{V_{{\\beta'}_{m+n}}\\otimes V_{\\beta_{n+m}}}\n(\\muf{{\\beta'}_{m+n}}\\muf{\\beta_{n+m}}\\Rff{\\beta_{n+m}}{\\alpha_1}\nS^{-1}(b^{\\beta_{m+n}}_{(j)})\\muf{\\beta_{n+m}}^{-1}\n(\\guf{{\\beta'}_{m+n}}_{[x_{m+n},x_1]})\\muf{\\beta_{n+m}}\\guf{\\beta_{n+m}}_{[x_1,x_{n+m}]})\\times\\\\\n&&\\times a_{(j)}^{{\\beta'}_{n+m}}a_{(i)}^{\\beta_{n+m}}\n\\guf{{\\beta'}_{m+n}}_{[x_{1},x_m]}\n\\phi^{\\beta_{m+n}\\alpha_{m+1}}_{{\\beta}_{m+n-1}}a_{(l)}^{{\\beta}_{n+m-1}}\\cdots\n\\phi_{\\beta_{m}}^{\\beta_{m+1}\\alpha_{m+1}}\nS^{-1}(b^{\\beta_{m}}_{(k)})\\muf{\\beta_{m+1}}^{-1}\\times\\\\\n&&\\times(\\guf{{\\beta'}_{m+n}}_{[x_{m},x_{m+1}]}\\muf{\\beta_{m+1}}\\guf{{\\beta}_{m+1}}_{[x_{m+1},x_{m}]})\na_{(k)}^{{\\beta'}_{n+m}}\\guf{{\\beta'}_{m+n}}_{[x_{m+1},x_{m+n}]}b_{(i)}^{{\\beta'}_{m+n}}\nS(b^{{\\beta'}_{m+n}}_{(l)})\n\\phi_{\\beta_{m-1}}^{\\beta_m \\alpha_m}\\cdots\\\\\n&&\\cdots\\guf{\\beta_{1}}_{[x_{2},{x}_{1}]}\\phi^{\\beta_{1}\n\\alpha_{1}}_{{\\beta}_{m+n}})\n\\Outtri{{\\beta}_{m+n}}{{\\alpha}_{n+m}}{{x'}_{n+m}}{{\\beta}_1}{{\\alpha}_1}{{x'}_1}=\\\\\n&&=\\sum_{{\\beta'}_{m+n},\\beta_{m+n},\\cdots,{\\beta}_{1}}v_{\\beta_m}\n[d_{{\\beta}_{m+n}}]^{-1} \\delta_{\\beta_{m+n},\\beta_m,{\\beta'}_{m+n}}\\times\\\\\n&&\\times tr_{V_{{\\beta}_{m}}}\n(\\muf{{\\beta}_{m}}\\muf{\\beta_{n+m}}\\Rff{\\beta_{m}}{\\alpha_1}\n\\guf{{\\beta}_{m}}_{[x_{1},x_m]}\n\\phi^{\\beta_{m}\\alpha_{m}}_{{\\beta}_{m-1}}\\cdots\\guf{{\\beta}_{1}}_{[x_{2},x_1]}\n\\phi_{\\beta_{m}}^{\\beta_{1}\\alpha_{1}})\\times\\\\\n&&\\times tr_{V_{{\\beta}_{m+n}}}(\\muf{\\beta_{m+n}}a_{(i)}^{{\\beta}_{n+m}}\n\\phi_{\\beta_{m+n-1}}^{\\beta_{m+n} \\alpha_{m+n}}a_{(l)}^{{\\beta}_{n+m-1}}\\cdots\n\\phi^{\\beta_{m+1}\n\\alpha_{m+1}}_{{\\beta}_{m+n}}\\guf{{\\beta}_{m+n}}_{[x_{m+1},x_{m+n}]}\nS(b^{{\\beta}_{m+n}}_{(l)})b_{(i)}^{{\\beta}_{m+n}})\\times\\\\\n&&\\times\\Outtri{{\\beta}_{m+n}}{{\\alpha}_{n+m}}{{x'}_{n+m}}{{\\beta}_1}{{\\alpha}_1}{{x'}_1}=\\\\\n&&= \\sum_{\\beta_1,\\cdots, \\beta_{n+m}} [d_{\\beta_{n+m}}]^{-1}\n\\Intri{{\\beta'}_n}{{\\alpha'}_n}{{x'}_n}{{\\beta'}_1}{{\\alpha'}_1}{{x'}_1}\\Intri{{\\beta''}_n}{{\\alpha''}_n}{{x''}_n}{{\\beta''}_1}{{\\alpha''}_1}{{x''}_1}\n\\times\\\\ &&\\times\n\\Outtri{{\\beta}_n}{{\\alpha}_n}{{x}_n}{{\\beta}_1}{{\\alpha}_1}{{x}_1}\n\\delta_{{\\beta'}_n, {\\beta''}_m, {\\beta}_{n+m}, \\beta_m}\n\\prod_{k=1}^{n}\\delta_{\\alpha_{m+k}, {\\alpha'}_{k}} \\prod_{k=1}^{m}\n\\delta_{\\alpha_{k}, {\\alpha''}_{k}}\n\\prod_{k=1}^{n-1}\\delta_{\\alpha_{m+k}, {\\alpha'}_{k}} \\prod_{k=1}^{m-1}\n\\delta_{\\alpha_{k}, {\\alpha''}_{k}}\\nonumber\\\\\n&&=\\sum_{\\beta_1,\\cdots, \\beta_{n+m}} [d_{\\beta_{n+m}}]^{-1}\n\\Intri{{\\beta}_n}{{\\alpha}_n}{{x}_n}{{\\beta}_1}{{\\alpha}_1}{{x}_1}\n\\Outtri{{\\beta'}_n}{{\\alpha'}_n}{{x'}_n}{{\\beta'}_1}{{\\alpha'}_1}{{x'}_1}\\times\\\\\n&&\\times\n\\Outtri{{\\beta''}_n}{{\\alpha''}_n}{{x''}_n}{{\\beta''}_1}{{\\alpha''}_1}{{x''}_1}\n\\delta_{{\\beta'}_n, {\\beta''}_m, {\\beta}_{n+m}, \\beta_m}\n\\prod_{k=1}^{n}\\delta_{\\alpha_{m+k}, {\\alpha'}_{k}} \\prod_{k=1}^{m}\n\\delta_{\\alpha_{k}, {\\alpha''}_{k}}\n\\prod_{k=1}^{n-1}\\delta_{\\alpha_{m+k}, {\\alpha'}_{k}}\n\\prod_{k=1}^{m-1}\\delta_{\\alpha_{k}, {\\alpha''}_{k}}\\nonumber\\\\\n\\end{eqnarray*}\nThis ends the proof of the Theorem.\\\\\n\n$\\Box$\n\nThese results can be easily generalized to the case where $q$ is a root of\nunity.\n It suffices to realize the following replacements:\\\\\nThe expression for the In state becomes\n\\begin{eqnarray*}\n&&\\Instate{\\beta_n}{\\alpha_n}{x_n}{\\beta_{n-1}}{\\alpha_{n-1}}{x_{n-1}}{\\beta_1}{\\alpha_1}{x_1}=\\\\\n&&=v_{\\beta_n}tr_{V_{\\beta_n}}(S(A)\\muf{\\beta_n}\n{\\buildrel {\\alpha_1} \\over {\\theta^{(1)}_l}}\n{\\buildrel {\\beta_n} \\over {\\theta^{(2)}_l}}\n\\Rff{\\beta_n}{\\alpha_1}\n{\\buildrel {\\beta_n} \\over {\\theta^{-1 (1)}_k}}\n{\\buildrel {\\alpha_1} \\over {\\theta^{-1 (2)}_k}}\n\\guf{\\beta_n}_{[x_1,x_{n}]}\nS({\\buildrel {\\beta_n} \\over {\\theta^{(1)}_m}}){\\buildrel {\\beta_n} \\over {A}}\n{\\buildrel {\\beta_n} \\over {\\theta^{(2)}_m}}\\otimes {\\buildrel {\\alpha_n} \\over\n{\\theta^{(3)}_m}}\n\\phi^{\\beta_n \\alpha_{n}}_{\\beta_{n-1}} \\cdots\\\\ &&\\cdots\n\\phi^{\\beta_2 \\alpha_2}_{\\beta_1}\\guf{\\beta_1}_{[x_2,x_{1}]} S({\\buildrel\n{\\beta_1} \\over\n{\\theta^{(1)}_i}}) S({\\buildrel {\\beta_1} \\over {\\theta^{(1)}_j}}){\\buildrel\n{\\beta_1} \\over\n{A}}{\\buildrel {\\beta_1} \\over {\\theta^{(2)}_j}}\\otimes {\\buildrel {\\alpha_1}\n\\over\n{\\theta^{(3)}_j}} \\phi^{\\beta_1 \\alpha_1}_{\\beta_n} {\\buildrel {\\beta_n} \\over\n{\\theta^{(2)}_i}}\n{\\buildrel {\\beta_n} \\over {B}}\nS({\\buildrel {\\beta_n} \\over {\\theta^{(3)}_i}})\nS({\\buildrel {\\beta_n} \\over {\\theta^{-1 (3)}_k}})\nS({\\buildrel {\\beta_n} \\over {\\theta^{(3)}_l}}))\n\\end{eqnarray*}\nand the analog formula for the Out state.\nThe $6-j$ associated to the crossing is changed to obtain an intertwiner.\\\\\nThe other objects remain unchanged and the summations are restricted to\nphysical representations\nonly.\\\\\n\n\\medskip\n\nWe have the following corollary:\n\\begin{proposition}\nIf $L$ is a link without boundaries in $D\\times[0,1]$ we have for any value of\n$q$:\n\\begin{equation}\n\\frac{_{q-YM(S^2)}}{<1>_{q-YM(S^2)}}= RT_{{\\cal U}_q({\\cal G})}(L)\n\\end{equation}\nwhere $RT$ is the Reshetikhin-Turaev's quantum invariant of coloured links.\nThis result has already been shown in \\cite{BR2}.\nand generally if $\\Sigma$ is a closed surface, the invariant associated to\n$\\Sigma+L$ is simply\na generalization to the case of a surface of the Reshetikhin-Kirillov invariant\nin the shadow world \\cite{KR}.\nThis theorem can be considered as a proof of the equivalence of Invariants\narising from Chern-Simons theory and Reshetikhin-Turaev quantum invariants.\n\\end{proposition}\n\n\\bigskip\n\n\n\\begin{definition}\nWe can also generalize our invariant to admit other objects called \"coupons\"\ndefined to be respectively represented on the following figure:\n\n\\par\n\\centerline{\\psfig{figure=coupons.ps}}\n\\par\n\nand which expressions are given by:\n\\begin{eqnarray*}\n{\\cal\nA}_{(n+k)(n)coupon}=\\sum_{}\\Intri{\\beta_{n+k}}{\\alpha_{n+k}}{x_{n+k}}{\\beta_1}{\\alpha_1}{x_1}\\times\nShadow(coupon)\\times\n\\Outtri{{\\beta'}_{n}}{{\\alpha'}_{n}}{{x'}_{n}}{\\beta_1}{\\alpha_1}{x_1}\\\\\n{\\cal\nA}_{(n)(n+k)coupon}=\\sum_{}\\Intri{\\beta_{n}}{\\alpha_{n}}{x_{n}}{\\beta_1}{\\alpha_1}{x_1}\\times\nShadow(coupon)\\times\n\\Outtri{{\\beta'}_{n+k}}{{\\alpha'}_{n+k}}{{x'}_{n+k}}{\\beta_1}{\\alpha_1}{x_1}\n\\end{eqnarray*}\nwith $Shadow(coupon)$ being defined as usual by the following rule:\n\\par\n\\centerline{\\psfig{figure=coupons2.ps}}\n\\par\n\\end{definition}\n\n\n\\bigskip\n\n\\section{A new description of invariants of three manifolds}\n\nIn this chapter $q$ will be a root of unity.\n\n\\subsection{Heegaard splitting and surgery of 3-manifolds}\n In the following ${\\cal M}$ is a compact orientable 3-manifold given\nby a simplicial complex $K$.Let us recall standard definitions that can be\nfound in \\cite{Si}.\n\\begin{definition}\nA {\\bf canonical region} ${\\cal R}$ of ${\\cal M}$ is a region within which\nthere are p non intersecting 2-cells $(E_i)_{i=1 \\cdots p}$(the {\\bf canonical\ncells}) with boundaries $e_i$ (the {\\bf canonical curves}) on the boundary\n${\\cal L}$ of ${\\cal R}$ such that we obtain a 3-cell by cutting ${\\cal R}$ at\neach $(E_i).$\n A surface ${\\cal L}$ is said to be a {\\bf canonical surface} of a 3-manifold\n${\\cal M}$ if it satisfies these conditions:\n\\begin{itemize}\n\\item ${\\cal L}$ is a subcomplex of ${\\cal M}$ and is a compact, connected\n2-dimensional manifold\n\\item ${\\cal M}= {\\cal R}_1 + {\\cal L} + {\\cal R}_2$ with ${\\cal R}_1,{\\cal\nR}_2$ canonical regions and\n${\\cal L}=\\partial {\\cal R}_1 = \\partial {\\cal R}_2$\n\\end{itemize}\nsuch a decomposition is called {\\bf canonical decomposition}.\nIt is important to recall that, if $g$ is the genus of ${\\cal L}$, in this\ncase, ${\\cal R}_1$ and ${\\cal R}_2$ are homeomorphic to a genus $g$ handlebody.\n\\end{definition}\n\n{\\bf Remark: }\nIt is easy to give, for each ${\\cal M}$, at least one canonical\ndecomposition.\nTo this aim, let us consider $\\{A_0^i\\}$, $\\{A_1^j\\}$, $\\{A_2^k\\}$, $\\{A_3^l\\}$\nthe sets of $0-,1-,2-,3-$cells of $K$.\nLet $\\{B_1^j\\}$, $\\{B_2^k\\}$, $\\{B_3^l\\}$ be respectively the middle of\n$\\{A_1^j\\}$, $\\{A_2^k\\}$, $\\{A_3^l\\}$. The complex $K'$ obtained by adding\nthe $B_{1}s$, the $B_{2}s$ and the $B_{3}s$ to the vertices of $K$ is called\nthe {\\bf first derived complex of K}, its 3-simplexes are of the form\n$(A_{0}B_{1}B_{2}B_{3})$.\nThe {\\bf second derived complex of K}, denoted $K\"$ is the complex complex\ngenerated from $K'$ by adding the vertices $C_{1}s,C_{2}s,C_{3}s$ middle of\nthe $1-,2-,3-$simplexes of $K'$.\nLet us denote by ${\\cal R}_1$ the set of all $3-$simplexes of $K''$ of the type\n$(A_{0}C_{1}C_{2}C_{3})$ or $(B_{1}C_{1}C_{2}C_{3})$, by ${\\cal R}_2$ the set\nof\nall $3-$simplexes of $K\"$ of the type $(B_{2}C_{1}C_{2}C_{3})$ or\n$(B_{3}C_{1}C_{2}C_{3})$ and by ${\\cal L}$ the common frontier of ${\\cal R}_1$\nand ${\\cal R}_2$.\nIf we call $G$ (resp.$G^{\\star}$) the linear graph\ngenerated by the 1-simplexes of $K$ (resp. its dual) we can see that ${\\cal\nR}_1$\nand ${\\cal R}_2$ are respectively $K''-$neighbourhood of $G$ and $G^{\\star}$.\nThen we have that ${\\cal M}= {\\cal R}_1 + {\\cal L} + {\\cal R}_2$ is a canonical\ndecomposition, it is\ncalled the {\\bf canonical decomposition derived from the triangulation}.\nIt is easy to check that for each canonical decomposition, there exits a\ntriangulation of ${\\cal M}$ such that the decomposition is in fact the\ncanonical decomposition derived from the triangulation.\n\\begin{definition} {\\bf Heegaard Splitting}\\\\\nA Heegaard splitting of a 3-manifold ${\\cal M}$ is a set $(g,f)$ where $g$\nis a non negative integer and $f$ is a diffeomorphism of a genus $g$ surface\n${\\cal L}_g$ such that ${\\cal M}$ is the manifold obtained by gluing two copies\nof the handlebody ${\\cal T}_g$ (the interior of ${\\cal L}_g$) along their\nboundaries after\nhaving acted on one of them by f:\n$${\\cal M} = {\\cal T}_g \\#_f {\\cal T}_g$$\nA Heegaard diagram is a set $({\\cal L},(e_i)_{i=1\\cdots g},(f_j)_{j=1 \\cdots\ng})$ where ${\\cal L}$ is a compact connected 2-dimensional manifold of genus\n$g$ and $(e_i)_{i=1\\cdots g}$ (resp.$(f_j)_{j=1 \\cdots g})$) are canonical\ncurves of ${\\cal R}_1$, the region interior to ${\\cal L}$ (resp. canonical\ncurves of ${\\cal R}_2$ the\nexterior of ${\\cal L}$).This data is sufficient to reconstruct an element $f$\nof $Diff({\\cal L})$ such that $f(e_j)=f_j.$\nTwo Heegaard diagrams are said to be equivalent if they describe homeomorphic\n3-manifolds.\nLet $({f'}_i)_{i=1...g}$ be $g$ other canonical curves in ${\\cal L}$ such that\n$({\\cal L},(e_i)_{i=1\\cdots g},({f'}_j)_{j=1 \\cdots g})$ is a Heegaard diagram\nof the sphere $S^3$ then $({\\cal L},(e_i)_{i=1\\cdots g},(f_j)_{j=1 \\cdots\ng},({f'}_j)_{j=1 \\cdots g})$ is said to be an augmented Heegaard diagram.\n\\end{definition}\nWe must recall that any element of the moduli space of a surface can be written\nas the composition of Dehn twists. A Dehn twist can be described by the\nfollowing replacement of a regular neighbourhood of the corresponding curve:\n\n\\par\n\\centerline{\\psfig{figure=dehn.ps}}\n\\par\n\n\nThere is an important theorem due to Singer \\cite{Si} describing the\nrelation between equivalent Heegaard diagrams.\n\n\\begin{definition}{\\bf Singer's elementary moves}\nLet us describe a set of elementary moves on the Heegaard diagrams:\n\\begin{itemize}\n{\\bf type 0: trivial moves}\n\\item replace a curve by another curve isotopic to it, or to its inverse, or\nreembedded the canonical surface ${\\cal L}$ in a different way in $S^3$.\\\\\n\n{\\bf type 1: solid handlebodies diffeomorphisms}\n\\item replace one canonical curve of the set $(e_i)_{i=1\\cdots g}$ (resp.\n$(f_j)_{j=1\\cdots g}$ ) by the composition\nof this curve with another one in this set.\n\\item making a Dehn twist along one of the $e_{i}s$.\\\\\n\n{\\bf type 2: $g \\rightarrow g+1$ moves}\n\\item add a handle to ${\\cal L}$ and define $e_{p+1}$ (resp. $f_{p+1}$) to be\nthe $a-$cycle (resp. the $b-$cycle) of this handle, or erase a handle with\ncycles for which $e_{p+1}$ is the $a-$cycle (resp. $f_{p+1}$ is the $b-$cycle).\n\\end{itemize}\n\\end{definition}\n\nThen we have the following classification theorem \\cite{Si}:\n\n\\begin{proposition}[ Singer's Theorem]\nIf the diagrams $D$ and $D'$, related by a finite number of Singer's moves,\ngive rise to the manifolds ${\\cal M}$ and ${\\cal M}'$ then ${\\cal M}$ and\n${\\cal M}'$ are homeomorphic.\nConversely, if $D$ and $D'$ are any two Heegaard diagrams whatsoever arising\nfrom a manifold ${\\cal M}$ then $D$ and $D'$ are related by a finite number of\nSinger's moves.\n\\end{proposition}\nA more generally used description of three manifolds is \"the surgery\npresentation\". Let us recall some facts about this description \\cite{Li}.\n\\begin{definition}[Surgery presentation of 3-manifolds]\nLet $(R,r)=\\cup_{i=1}^{n}(R_i,r_i)$ be a framed link in the oriented sphere\n$S^3.$ We can define a manifold ${\\cal M}$\nby \"surgery\" from $(R,r)$ using the following procedure:\\\\\nremove from $S^3$ pairwise disjoint tubular neighbourhoods $V_i$ of the curves\n$R_i$ and resew them identifying a meridian $z_i$ in $\\partial V_i$ with a\ncurve $y_i \\in \\partial(S^3\\setminus V_i^{int})$ which links $R_i$ exactly\n$r_i$ times.\\\\\nMoreover, every 3-manifold ${\\cal M}$ can be obtained from a certain framed\nlink by this procedure \\cite{Li}.\n\\end{definition}\nIt is relatively easy to relate the Heegaard and Surgery points of view\n\\cite{NL}. Let us consider a Heegaard diagram based on a gluing diffeomorphism\n$f$ described in terms of Dehn twists of the surface. We first remark that\nsplitting $S^3$ along ${\\cal L}$ then doing a Dehn twist along a certain curve\nand resewing the handlebody is equivalent to do a surgery along the ribbon\nglued on the surface along this curve as it can be seen on the figure:\n\n\\par\n\\centerline{\\psfig{figure=annulus.ps}}\n\\par\n\nLet $(R_i)_{i=1..n}$ be a set of ribbons trivially embedded on the surface\n${\\cal L}$, $f_i$ the corresponding Dehn twists. We want also define the framed\nlink $L$ defined to be the set of ribbons $(R_i \\times \\epsilon_i)_{i=1 \\cdots\nn}$ for $0 \\le \\epsilon_1 \\le \\cdots \\le \\epsilon_n \\le 1.$\nWe consider a partition of $S^3$ in three pieces: ${\\cal L}\\times [0,1]$, the\nhandlebody ${\\cal H}_g$ interior to ${\\cal L}\\times \\{0\\}$ and the handlebody\n${\\cal H}^{'}_g$ exterior to ${\\cal L}\\times \\{1\\}.$ Let us consider the\nmanifold ${\\cal M}(f_1,f_2,\\cdots,f_n)$ obtained by gluing the manifolds ${\\cal\nH}_g$, ${\\cal H}^{'}_g$, $({\\cal L}\\times [\\epsilon_{i-1},\\epsilon_i])_{i=1\n\\cdots n}$ with the gluing diffeomorphisms $id, f_1, \\cdots, f_n.$ Obviously\nthe manifold ${\\cal M}(f_1,f_2,\\cdots,f_n)$ is the manifold defined by the\nHeegaard data $({\\cal L}, f_n \\circ f_{n-1} \\circ \\cdots \\circ f_1)$, but it is\nalso obvious that this manifold is that defined by the surgery data $R.$ We\nwill say that these surgery and Heegaard presentation of the same manifold are\n\"related\" description of ${\\cal M}.$\n\n\\subsection{Invariants associated to Heegaard diagrams and Lattice q-gauge\ntheory}\n\nOur principal aim in this section is to prove the following theorem:\n\n\\begin{proposition}[Invariants of three manifolds and Heegaard diagrams]\nLet $({\\cal T}_g, (x_j)_{j=1,...,g}, (y_j)_{j=1,...,g}, (z_j)_{j=1,...,g})$ be\nan augmented Heegaard diagram associated to a manifold ${\\cal M}$ then the\nexpectation value :\n\\begin{equation}\n{\\cal J}_{{\\cal M}}=\\frac{<\n\\prod_{i=1}^{g}\\delta_{y_i}\\prod_{i=1}^{g}\\delta_{x_i}>_{q-YM({\\cal T}_g)}}{<\n\\prod_{i=1}^{g}\\delta_{z_i}\\prod_{i=1}^{g}\\delta_{x_i}>_{q-YM({\\cal T}_g)}}\n\\end{equation}\n is an invariant of the manifold ${\\cal M}.$ Moreover this value is equal to\nthe Reshetikhin-Turaev invariant associated to the manifold ${\\cal M}.$\n\\end{proposition}\nThe normalization by the expectation value associated to the sphere is chosen\nto obtain an $3-$manifold invariant equal to $1$ for the sphere.\n{\\bf Remark 1:}\nThe latter definition of the correlation function is in fact very natural from\nthe general construction of q-gauge theory. Indeed putting a delta function\nassociated to a plaquette $P$ corresponds to imposing that any ribbon, i.e.\nholonomy defined in terms of the gauge fields algebra, can be displaced\nthrough $P$ without torsion and without changing the expectation value. So,\nadding to the projector associated to the surface some\ndelta functions corresponding to the $x_i$s, i.e. canonical curves of the\ninterior handlebody, and, at a future time, the delta functions of the $y_i$s,\ni.e. canonical curves of the exterior handlebody, allows us to displace any\ncurve through a handle of any of the two Heegaard components.This is exactly\nwhat we want to do in the framework of Chern-Simons theory.\\\\\n{\\bf Remark 2:}\nUsing the properties\n\\begin{eqnarray}\n&&(\\frac{\\delta_{C}}{\\sum_{\\alpha}[d_{\\alpha}]^2})^2=(\\frac{\\delta_{C}}{\\sum_{\\alpha}[d_{\\alpha}]^2})\\\\\n&&(\\frac{\\delta_{C_1}}{\\sum_{\\alpha}[d_{\\alpha}]^2})(\\frac{\\delta_{C_2}}{\\sum_{\\alpha}[d_{\\alpha}]^2})=(\\frac{\\delta_{C_1}}{\\sum_{\\alpha}[d_{\\alpha}]^2}) (\\frac{\\delta_{C_1 \\# C_2}}{\\sum_{\\alpha}[d_{\\alpha}]^2})\\nonumber\n\\end{eqnarray}\nwe can replace easily the correlation function by one where we put all\nLickorish generators rather than the canonical curves only. This fact will be\nuseful in the next section.\\\\\n\n\n\n\nWe are going to proove the last theorem through two lemmas describing some\nproperties of this invariant.\n\n\\begin{lemma}\nThe expectation value\n$<\\prod_{i=1}^{g}\\delta_{y_i}\\prod_{i=1}^{g}\\delta_{x_i}>_{q-YM({\\cal T}_g)}$\nassociated to a Heegaard diagram $({\\cal T}_g, (x_j)_{j=1,...,g},\n(y_j)_{j=1,...,g})$ of a manifold ${\\cal M}$ is invariant under any Singer's\nmove applied to the diagram.\n\\end{lemma}\n{\\sl Proof:}\\,\\\\\n{\\bf Trivial moves:}\\\\\nit is a fact already established that the expectation value is invariant under\nany isotopic deformation of any curve in the surface, simply because of the\nflatness condition.\\\\\nWe have the formula $\\delta_C=\\delta_{C^{-1}}$, which is exactly the second\ntrivial move.\\\\\n{\\bf Handlebodies Diffeomorphisms:}\\\\\nThe flatness condition implies trivially the following property for any curves\n$C_1,C_2$:\n\\begin{equation}\n\\delta_{C_1}\\delta_{C_2}=\\delta_{C_1}\\delta_{C_1 \\# C_2}\n\\end{equation}\nmoreover the $\\delta_{e_i}$s (resp.$\\delta_{f_i}$s ) commute one with the\nothers. We then obtain the invariance under the first type 1 move.\\\\\nIt is easy to see, with the expression of the block element ${\\cal A}_{free}$,\nthat we can do the following replacement along any curve $x_i$.:\n\\par\n\\centerline{\\psfig{figure=dehn2.ps}}\n\\par\nwhich implies the invariance under the second type 1 move.\\\\\n{\\bf $g \\rightarrow g+1$ moves:}\\\\\nLet us consider a Heegaard diagram with one handle with its $a-$ and\n$b-$cycles. We cut the surface along a certain $2-$cell to obtain a torus with\na puncture on which are drawn the two cycles as in the following figure. We\nchoose the minimal fat graph describing this object to describe the partial\nintegration of the expectation value over the edges of the latter object.\n\\par\n\\centerline{\\psfig{figure=sing2.ps}}\n\\par\nwe obtain easily:\n\\begin{eqnarray*}\n&&\\int dh(U_{[y,t]})dh(U_{[x,z]}) \\delta_{[y,t,x,z,t,y,z,x]}\\delta_{[x,y,z,x]}\n\\delta_{[x,y,t,x]} =\\\\\n&&= \\int dh(U_{[y,t]})dh(U_{[x,z]}) \\delta_{[y,t,x,z,t]}\\delta_{[x,y,z,x]}\n\\delta_{[x,y,t,x]}\\\\\n&&= \\int dh(U_{[y,t]})dh(U_{[x,z]}) \\delta_{[t,x,z]}\\delta_{[x,y,z,x]}\n\\delta_{[x,y,t,x]}\\\\\n&&= \\int dh(U_{[y,t]})dh(U_{[x,z]}) \\delta_{[t,x,y,z]}\\delta_{[x,y,z,x]}\n\\delta_{[x,y,t,x]}\\\\\n&&=\\delta_{[t,x,y,z]}\\\\\n\\end{eqnarray*}\nthe last line is easily obtained by using the property that the integration\njust \"pick\" the zero component associated to a link.\nThe latter result establishes the invariance under the type 2 Singer move.\nThis ends the proof of the lemma and shows that\nthe expectation value is an invariant of the manifold ${\\cal M}.$\n$\\Box$\n\n\\begin{lemma}\nFor any augmented Heegaard diagram $({\\cal L},(x_i)_{i=1 \\cdots g}, (y_i)_{i=1\n\\cdots g}, (z_i)_{i=1 \\cdots g})$ describing a manifold ${\\cal M}$ there exists\na framed link $L$ which is a surgery data describing the same manifold ${\\cal\nM}$ and verifying:\n\\begin{equation}\n\\frac{<\\prod_{i=1}^{g}\\delta_{y_i}\\prod_{i=1}^{g}\\delta_{x_i}>_{q-YM({\\cal\nA})}} {<\\prod_{i=1}^{g}\\delta_{z_i}\\prod_{i=1}^{g}\\delta_{x_i}>_{q-YM({\\cal\nA})} }=\nRT({\\cal M})\n\\end{equation}\nwhere RT is the Reshetikhin Turaev invariant of the manifold computed from $L.$\n\\end{lemma}\n\n{\\sl Proof:}\\,\nThe trick already used in the proof of the invariance under the second Singer\nmove can be used also here. We first use a natural property of delta functions\nthat can be described by the following figure :\n\\par\n\\centerline{\\psfig{figure=flatness.ps}}\n\\par\n\\begin{equation}\n\\delta_{{C}_2}\\delta_{{C}_1}=\\delta_{{C'}_2}\\delta_{{C}_1}\n\\end{equation}\nto transform the correlation function in a new one\\\\\n $<(\\sum_{\\alpha_1,\\cdots, \\alpha_g}(\\prod_{i}\n[d_{\\alpha_i}])W((R_i,\\alpha_i)_{i=1\\cdots\ng})\\prod_{i=1}^{g}\\delta_{x_i}>_{q-YM({\\cal L})}$\n where the $R_i$s are ribbons glued on the surface with the same framings and\nknotted in $S^3$ in the same way as the $y_i$s but with a support now included\nin the area described in the following figure:\n\\par\n\\centerline{\\psfig{figure=zone.ps}}\n\\par\nUsing now the usual flatness property (\\ref{flatness}) we can deform again the\nlatter knot to put its crossings in the \"discs\", the rest of the knot being\ncomposed of parallel strands along handles zones.\\\\\n Then we are able to do the same calculus as in\nthe verification of the invariance under type 2 Singer's move. The integration\n\"picks\" again the zero component on each segment of the skeleton. We then\nobtain the equality:\n\\begin{equation}\n<\\prod_i \\delta_{y_i} \\prod_i \\delta_{x_i}>_{q-YM({\\cal\nA})}=\\sum_{\\alpha_1,\\cdots, \\alpha_g}(\\prod_{i} [d_{\\alpha_i}])(\\prod_{j}\nI_{disc_j})\n\\end{equation}\nwith $I_{disc_j}$ being the invariant associated, by our construction, to the\nknot contained in the j-th disc, placed on the sphere $S^2$ and with four\ncoupons picking the zero component on the boundary of the disc.\\\\\nIt is then easy to see, using the equivalence already established in section\n(3) between our invariant on the sphere and the Reshetikhin invariant of link ,\nthat the quantity $I_{disc_j}$ is exactly the Reshetikhin invariant associated\nto this framed link with coupons. \\\\\nNow the proof can be achieved by establishing, using the Reshetikhin-Turaev\nframework, that the latter data is a surgery data of the manifold ${\\cal M}.$\n Let us first recall that, using the \"related\" surgery and Heegaard\ndescriptions, we can replace the set of curves $y_i$ describing the manifold\n${\\cal M}$ by a link composed of the curves $z_i$ associated to the Heegaard\ndescription of $S^3$ placed at a time $t$ and the curves $R_i \\times {t_i}$ (\nwith $t_i \\le t$ ) associated to the composition of Dehn twists describing the\nHeegaard gluing diffeomorphism encoded in the $y_i$s. If we compute, with the\nnotations of Reshetikhin and Turaev in \\cite{RT2}, the invariant associated to\nthe framed link described before, with an insertion of two \"coupons\" for each\nhandle picking the zero component, we obtain easily that this invariant is\nequal to the invariant associated to the link $L=\\cup_i R_i \\times {t_i}$ only.\nThis property uses trivially the fact that :\n\\begin{eqnarray*}\n\\sum_{\\alpha,{\\alpha'}_1,{\\alpha'}_n}tr_{V_{\\alpha}}(\\buildrel\\alpha \\over \\mu\\phi^{\\alpha_n\n\\alpha_{n-1}}_{{\\alpha'}_{n-1}} \\cdots \\phi^{\\alpha_2\n\\alpha_1}_{{\\alpha'}_1}\\phi^{{\\alpha'}_1 \\alpha}_{0}\n\\psi_{{\\alpha'}_1 \\alpha}^{0}\\psi_{\\alpha_2\n\\alpha_1}^{{\\alpha'}_1}\\cdots\\psi_{\\alpha_n \\alpha_{n-1}}^{{\\alpha'}_{n-1}})=\nid_{V_{\\alpha_1}} \\otimes \\cdots \\otimes id_{V_{\\alpha_n}}\n\\end{eqnarray*}\nWe then obtain:\n\\begin{eqnarray*}\n\\frac{<\\prod_{i=1}^{g}\\delta_{y_i}\\prod_{i=1}^{g}\\delta_{x_i}>_{q-YM({\\cal\nL})}}{<\\prod_{i=1}^{g}\\delta_{z_i}\\prod_{i=1}^{g}\\delta_{x_i}>_{q-YM({\\cal\nL})}}=\\sum_{(\\alpha)}(\\prod_{i}[d_{\\alpha_i}]) RT((R_i,\\alpha_i)_{i=1 \\cdots\nn})\n\\end{eqnarray*}\nUsing now the celebrated result of \\cite{RT2} this object is a non trivial\ninvariant of the 3-manifold ${\\cal M},$ it is the Reshetikhin-Turaev's\ninvariant of three manifolds. \\\\\n$\\Box$\n\n\n\\subsection{Chern-Simons theory on a lattice and Three dimensional Lattice\nq-gauge theory }\nWe will define here a three dimensional gauge theory which extends in some\nsense\nthe previous construction on a surface. The definition of this theory is based\non a choice of a simplicial presentation of the manifold which exhibits\nnaturally a\ncanonical decomposition of the manifold.\nLet us consider a 3-manifold ${\\cal M}$ given by a complex $K$. We impose here\nthat all vertices of $K$ are tetravalent. We will denote $K^{\\star}$ the dual\ncomplex of $K$.\nWe will denote as before $A_0^i,A_1^j,A_2^k,A_3^l$ the $0-,1-,2-,3-$simplexes\nof $K$\n, $A_0^{\\star i},A_1^{\\star j},A_2^{\\star k},A_3^{\\star l}$ the\n$0-,1-,2-,3-$simplexes\nof $K^{\\star}$ and $B_1^j$ (resp.$B_1^{\\star j}$) the middle of the $A_1^j$\n(resp. $A_1^{\\star j}$).\n\n\n\\begin{definition}[canonical thickening of a graph]\nLet us define another tetravalent complex $ K^{\\#}$ build up from the previous\none as follows:\nA couple of $B_1^j$ and $B_1^{\\star j}$ are said\nto be a couple of neighbours if $B_1^j$ is the middle of an edge of a certain\n$A_2^k$ and $B_1^{\\star j}$ is in the middle of this $A_2^k.$\nWe denote by $J$ the set of $1-$simplexes defined by the set of couples of\nneighbour\npoints. We now define the $0-simplexes$ of $K^{\\#}$ to be the middles of the\nelements\nof $J$. The $1-simplexes$ of $K^{\\#}$ are then given by the set of couples\nof $0-simplexes$ corresponding to elements of $J$ having one vertex in common,\nif\nthis vertex is a $B_1^j$ (resp. a $B_1^{\\star j}$) then this $1-$simplex is\nsaid \"of type $K$\"\n(resp. \"of type $K^{\\star}$\").\nNow the $2-$simplexes are defined to be of three types: one $2-$simplex is\nassociated to each closed curve formed by type $K$ $1-$simplexes only, one to\neach closed\ncurve formed by type $K^{\\star}$ $1-$simplexes only, and one to each closed\ncurve formed\nalternatively by type $K$ and type $K^{\\star}$ $1-$simplexes.We will refer us\nto \"the $e_{i}$s\"\n, \"the $f_{j}$s\", and \"the $P$s\" to denote respectively these three types of\n$2-$simplexes.\nFinally the $3-$simplexes are defined in an obvious way by considering each\nconnected region\naround the vertices of $K$ and $K^{\\star}$.\n\\end{definition}\nWe will denote by $K^{\\#}_0,K^{\\#}_1,K^{\\#}_2,K^{\\#}_3$ the sets of\n$0-,1-,2-,3-$simplexes respectively. A piece of this new complex is shown in\nthe following figure:\n\n\\par\n\\centerline{\\psfig{figure=thick.ps}}\n\\par\n\n\n\nThe graph $K^{\\#}$ build up from any triangulation $K$ describing a manifold\n${\\cal M}$ owns the\nfollowing properties:\n\n\\begin{lemma}\nIf we denote by ${\\cal R}_1$ (resp. ${\\cal R}_2$) the region defined by the set\nof $3-$cells associated to vertices of $K$\n(resp.$K^{\\star}$) and by ${\\cal L}$ the surface defined by the set of Ps.The\ndecomposition:\n${\\cal M}={\\cal R}_1 + {\\cal L} + {\\cal R}_2$ is a canonical decomposition and\n$K^{\\#}$ is homeomorphic to $K$.\nThe set formed by the elements of $K^{\\#}_0$,the elements of $ K^{\\#}_1$ and\nall $P$s forms the complex L\nassociated to the triangulation of the canonical surface {\\cal L}(for this\nreason these sets of $0-,1-$ and\n$2-$simplexes will be also denoted respectively by $L_0,L_1,L_2$)\n\nThe set of canonical $2-$cells of ${\\cal R}_1$ (resp ${\\cal R}_2$) is a subset\nof the $e_i$s (resp.the $f_j$s).\n\\end{lemma}\n\n{\\sl Proof:}\\,\nThis decomposition is equivalent to the Heegaard decomposition\n\"derived\" from the complex $K.$\n$\\Box$\n\n\\begin{definition}[3-dimensional lattice q-gauge theory]\nAs a consequence of the property that $K^{\\#}_0=L_0$ and\n$K^{\\#}_1=L_1$, we can define as before the\nexchange algebra associated to the elements of $K^{\\#}_1$\nby imposing the coaction of the gauge symmetry algebra at each element of\n$K^{\\#}_0$ and by choosing\na cilium order on the surface.We can define as before\nthe Wilson loops attached to each closed path formed by elements of $K^{\\#}_1$\n(i.e. drawn on L)\nand delta functions associated to each $2-$cell.\nIn fact we define the Yang-Mills weight associated to a 2-cell $P$ of area\n$A_P$ to be:\n\\begin{equation}\n\\delta^{\\beta}_P=\\sum_{\\alpha \\in Phys(A)}[d_{\\alpha}] e^{- \\frac{A_P\nC_{\\alpha}}{2 \\beta}}\\buildrel\\alpha \\over W_{P}\n\\end{equation}\nwhere $C_{\\alpha}$ is the quadratic casimir of the representation $\\alpha$ and\n$\\beta$ is a coupling constant of the Yang-Mills theory.\nWe define the expectation value associated to any element ${\\cal A}$ of\n$\\Lambda^{inv}$ in the 3 dimensional q-Yang Mills theory to be:\n\\begin{equation}\n<{\\cal A}>_{{\\cal M}}:= \\int \\prod_{l\\in K^{\\#}_1} dh(U_l) (\\prod_{j}\n\\delta^{\\beta}_{f_j})(\\prod_{P\\in L_2}\\delta^{\\beta}_P)\\;\\; {\\cal A}\\;\\; (\n\\prod_{i} \\delta^{\\beta}_{e_i})\n\\end{equation}\nin the limit $q \\rightarrow 1$ this theory becomes the well known Yang-Mills\ntheory on a lattice associated to a manifold ${\\cal M}.$\n\\end{definition}\n\n\\begin{proposition}\nLet $L$ be a link drawn on the 1-skeleton $K^{\\#}_1$ of ${\\cal M}$. Using again\nthe properties of the complex $K$, $L$ is in fact drawn on the canonical\nsurface\nand we can define $W_L$ in the framework defined in this article. The\ncorrelation function associated to $L$ in the limit $\\beta \\rightarrow 0$ is\nthen\n\\begin{equation}\n{lim}_{\\beta \\rightarrow \\infty}\\frac{_{{\\cal M}}}{<1>_{{\\cal M}}} =\n\\frac{RT({\\cal M}, L)}{RT({\\cal M})}\n\\end{equation}\nthis formula can be considered as a description of Reshetikhin-Turaev\ninvariants, i.e. of Chern-Simons invariants in term of a well defined lattice\ngauge theory and a definition of the Witten's path integral formulas.\n\\end{proposition}\n\n{\\sl Proof:}\\,\\\\\nThe expectation value is simply the same as that introduced in the last\nsubsection but with a very special Heegaard decomposition where the gluing\ndiffeomorphism is simply the identity. \\\\\n$\\Box$\n\n\\medskip\n\n{\\bf Acknowledgements:} It is a pleasure to thank my friend P.Roche for his\nconstant support. I want also aknowledge\nilluminating discussions with N.Reshetikhin and C.Mercat.\n\n\\bibliographystyle{unsrt}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} {"text":"\\section{Introduction}\n\\label{intro}\n\nGamma-ray bursts (GRBs) are intense flashes of gamma radiation which originate from distant galaxies and typically last from a fraction of a second to several minutes \\cite{vedrenne2009}.\nThousands of GRBs have been detected to date by high-energy astronomy missions such as the Compton Gamma Ray Observatory\\cite{goldstein2013}, High Energy Transient Explorer \\cite{hete}, Neil Gehrels Swift Telescope \\cite{Gehrels_2004}, Fermi Space Telescope \\cite{Meeg2009ApJ...702..791M,Atwood_2009} and INTEGRAL \\cite{Wink2003A&A...411L...1W}.\nThe distribution of GRB durations demonstrates a bi-modality, with two classes of GRBs associated with different generation mechanisms \\cite{kouveliotou1993}.\nLong GRBs (\\textgreater2\\,s) are produced in the core-collapse of massive stars \\cite{macfayden1998} while short GRBs (\\textless2\\,s) are associated with mergers of compact binary systems \\cite{1984SvAL...10..177B}.\n\nThe detection of the short gamma-ray burst GRB 170817A~\\cite{goldstein2017} in coincidence with the gravitational wave (GW) signal GW170817 from a binary neutron star inspiral~\\cite{abbott2017a} marked the beginning of a new era of multi-messenger astronomy and experimentally confirmed that at least some of the progenitor systems for short GRBs are binary neutron star mergers.\nThe joint localisation of the GRB and GW event led to multi-wavelength follow-up observations of the GRB afterglow, providing information on the orientation of the binary system, and the detection of the associated kilonova in ultraviolet, optical and infrared band~\\cite{followup_Abbott_2017}. \nTo date, 67 GW candidates have been recorded by the LIGO~\\cite{2015CQGra..32g4001L} and Virgo~\\cite{2015CQGra..32b4001A} interferometers, including several neutron star--neutron star and neutron star--black hole merger candidates~\\cite{2020arXiv201014527A,gws_nature}.\nHowever, no electromagnetic counterparts have been detected for any of these events except the aforementioned GW170817.\n\nThe GW170817\/GRB 170817A discovery and its follow-up campaign emphasised the importance of simultaneous gamma-ray and GW observations.\nThis demand will be further increased by future major upgrades of the GW observatories \\cite{2018LRR....21....3A} which will improve their sensitivity and the detection rate of GW events.\nThe detection of $10_{-10}^{+52}$ binary neutron star mergers, of $1^{+91}_{-1}$ neutron star--black hole mergers and $79^{+89}_{-44}$ binary black hole mergers in one calendar year is predicted~\\cite{gw_prospects_2020} for the next operating run (O4) of LIGO~\\cite{2015CQGra..32g4001L}, Virgo~\\cite{2015CQGra..32b4001A} and KAGRA~\\cite{2019NatAs...3...35K}, which is planned to commence in July 2022.\n\nHowever, a major challenge to the detection of electromagnetic counterparts is the potential lack of future gamma-ray missions.\nMany of the current missions, including the Neil Gehrels Swift Telescope, Fermi Space Telescope and INTEGRAL, are all approaching or have exceeded their nominal mission lifetimes. Two major GRB-related missions are in a study phase, THESEUS~\\cite{amati2018} and AMEGO~\\cite{mcenery2019}.\n\nCurrent and planned near-future large scale missions, including SVOM \\cite{svom_2015arXiv151203323C}, a Chinese-French mission due for launch in 2022, will not provide the full-sky coverage required for efficient detection of electromagnetic counterparts to GW events \\cite{perkins2017}.\nThis potential gap has led to a search for alternative solutions, such as a fleet of small satellites with gamma-ray detecting technology.\nA Chinese mission, GECAM \\cite{zhang2019}, utilises two GRB detecting small satellites to provide full-sky coverage.\nEach satellite uses a hemispherical array of LaBr$_3$ and SiPM instruments, achieving GRB localisation capability of a few degrees.\nThe pair launched in December 2020 and detected its first GRB in January 2021 \\cite{2021GCN.29331....1A}.\nOther agencies are in the process of developing CubeSats with GRB detection and localisation capabilities e.g. BurstCube \\cite{perkins2017}, MoonBEAM \\cite{moon_hui_briggs_2018}, HERMES \\cite{fiore2020,evangelista2020,sanna2020,campana2020}, CAMELOT \\cite{werner2018} and GRID \\cite{wen2019}.\nGRBAlpha \\cite{pal2020} is an in-orbit demonstration for CAMELOT which has recently been launched \\cite{grbalphalaunch}.\nTo detect gamma-rays, all of the above mentioned CubeSat missions use inorganic scintillator crystals (CsI:Tl, NaI:Tl, GAGG:Ce) read out by silicon photomultipliers (or silicon drift detectors in the case of HERMES).\nThe miniature size and low weight of these photosensors allow the CubeSats to carry relatively large detectors with an effective area on the order of 100\\,cm$^2$, comparable to the effective area of the Fermi GBM detectors \\cite{Meeg2009ApJ...702..791M}.\nCubeSats are relatively low cost and have short launch timescales \\cite{OnthevergeofanastronomyCubeSatrevolution} making them ideal candidates to bridge potential gaps in coverage by the large gamma-ray missions.\nIn addition, a fleet of GRB detecting CubeSats could independently provide detection and localisation of GRBs during future gravitational wave instrumentation operating runs \\cite{fuschino2019}. \n\nCubeSats are a class of nano-satellite (usually defined as having a mass of less than 10\\,kg) which approximately conform to the Cal Poly CubeSat specification \\cite{cubesat_spec_rev13}.\nThey may be a number of different, approximately cuboid, sizes which are all built from predefined configurations of multiples of the 1U CubeSat unit.\nThe base CubeSat unit (the 1U CubeSat) has at its core a 10\\,cm $\\times$ 10\\,cm $\\times$ 10\\,cm cubic volume.\nThe Cal Poly specification allocates a mass of 1.33\\,kg per 1U volume for most CubeSat sizes, though the new 6U specification allows for a total mass of 12\\,kg. \n\nThe Educational Irish Research Satellite 1 (EIRSAT-1) is a 2U CubeSat which will be Ireland's first satellite~\\cite{ssea18_eirsat}.\nEIRSAT-1 was proposed in response to an ESA announcement of opportunity as part of their educational Fly Your Satellite! (FYS!) programme and was accepted into that programme in 2017.\nThe FYS! programme supports university student teams to build, launch, and operate their own CubeSat and has launched more than 11 CubeSats since 2008.\nIt is an initiative of the ESA Academy which provides educational opportunities for university students \\cite{ssea18_eirsat,Doyle2020,Dunwoody2020,Walsh2020,ssea18_adm,ssea18_wbc}.\nThe main goal of the proposal was to fly and space-qualify a novel gamma-ray detector that was already under development~\\cite{ulyanov2016,ulyanov2017}.\nThis detector had a technologically mature design using a cerium bromide (CeBr$_3$) scintillator and silicon photomultipliers (SiPMs), and had demonstrated compatibility with a CubeSat form factor.\nThe detector was capable of measuring gamma-rays in 30\\,keV--10\\,MeV energy range which would make it suitable for GRB detection.\nHowever, the detector would require a compact low-power readout system, such as the SIPHRA ASIC, to fit in a CubeSat and therefore the EIRSAT-1 Gamma-ray Module (GMOD) was envisaged as a demonstrator to combine all the necessary components in a CubeSat payload. \nSuch a payload is seen as a significant advance over legacy instrumentation which typically relied on classical photomultiplier tube scintillator detection and discrete control and readout electronics.\nA prototype configuration of the payload was evaluated on a balloon flight \\cite{murphy2021} and used to test the radiation harness of the SiPMs \\cite{ulyanov2020}.\n\nThis paper gives an overview of the GMOD instrument to be flown on board EIRSAT-1 and presents its expected in-orbit performance obtained from Monte Carlo simulations.\nThe detailed detector design and experimental characterisation of the module are described in a separate paper~\\cite{gmod2}.\nThe final detector design has been environmentally qualified \\cite{Mangan2021}.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{EIRSAT-1}\n\\label{sec:eirsat}\n\nEIRSAT-1 is a 2U CubeSat measuring 22.7\\,cm\\hskip0.16667em\\relax$\\times$\\,10\\,cm\\hskip0.16667em\\relax$\\times$\\,10\\,cm, incorporating GMOD as its primary payload along with two other novel payloads --- EMOD (discussed below) and Wave-Based Control \\cite{ssea18_wbc,thompson2016}, a novel control algorithm which will be used in an experimental attitude control capacity.\nTo ensure mission success, design decisions were made to prioritise mission reliability and lifetime over a complex mission profile.\nAs part of the FYS! programme, EIRSAT-1 is planned to be deployed from the International Space Station (ISS), resulting in an expected mission lifetime of approximately one year (9--18\\,months depending on Solar activity).\n\n\n\\begin{figure}\n\\includegraphics[width=\\textwidth]{figures\/exploded.jpg}\n\\caption{Exploded isometric view of the EIRSAT-1 spacecraft. GMOD, comprising a motherboard PCB and a large cuboid detector assembly, is the top subsystem in the main stack shown at the centre of the drawing. The EMOD experiment can be seen above the stack.}\n\\label{fig:eirsat-1exploded}\n\\end{figure}\n\nThe overall spacecraft design is presented in Figure~\\ref{fig:eirsat-1exploded} and is based around the requirements of accommodating the three payloads on a CubeSat platform, launched into an ISS-like orbit at 400\\,km altitude.\nThe spacecraft is built around a core stack of electronic subsystems manufactured by \\r{A}AC Clyde Space.\nDetails of the particular subsystems in use can be found in \\cite{ssea18_eirsat}.\nThis subsystem stack is supported by a 2U \\r{A}AC Clyde Space structure which has been heavily modified to suit the needs of the EIRSAT-1 mission, particularly the +Z end-cap which has been entirely replaced by a custom structural element to accommodate the EMOD payload.\n\nThe structure is surrounded by body-mounted solar arrays on four sides.\nDue to the anticipated launch of the spacecraft from the ISS, achieving the maximum possible mission lifetime requires minimising drag, meaning that the use of deployable solar arrays is not possible and power generation is therefore limited.\nThe limited power budget precludes the use of reaction wheels, limiting EIRSAT-1 to magnetic attitude actuators, which in turn requires that the communication system work with the spacecraft in any orientation.\nThe communication is therefore based on a VHF\/UHF system utilising omni-directional deployable antennas \\cite{ssea18_adm} which can be seen at the -Z end of the spacecraft illustrated `below' the stack in Figure~\\ref{fig:eirsat-1exploded}.\n\nThe GMOD detector sits at the `top' of the main stack.\nIt is surrounded on four sides by the body-mounted solar arrays.\nBetween GMOD and the solar arrays are 1\\,mm thick aluminium structural shear panels though these are skeletonised and are primarily a clear aperture having little effect on GMOD.\n\nThe Thermal Coupon Assembly (TCA) of the EMOD payload is located in the spacecraft `above' GMOD.\nThis payload is an experiment to perform measurements of two thermal management coatings known as SolarBlack \\cite{doherty2016} and SolarWhite \\cite{doherty2016a} which were developed in support of ESA's Solar Orbiter mission.\nThe TCA contains samples of the coatings which are measured using resistance temperature detectors and which must be placed on the exterior of the spacecraft where they will be exposed to solar radiation and must be thermally isolated from the spacecraft.\nTo ensure good isolation, the samples are mounted on a 1\\,mm thick titanium baseplate via PEEK columns with a multi-layer insulation blanket included between the samples and the baseplate.\nAt energies below approximately 50\\,keV, this construction is expected to provide more effective shielding than the body-mounted solar arrays leading to larger effective area for GMOD in the X and Y directions than in the Z direction.\n\n\n\n\\section{GMOD --- The Gamma-ray Module}\n\\label{sec:gmod}\n\nGMOD is a scintillator-based gamma-ray detector which utilises a cerium bromide (CeBr$_3$) crystal scintillator, ON Semiconductor (formerly SensL) J-series silicon photomultipliers and the SIPHRA application specific integrated circuit (ASIC).\nThe instrument consists of a motherboard and a detector assembly which hosts the scintillator, SiPMs, and SIPHRA in a light-tight enclosure.\nA cutaway view of GMOD illustrating its design can be seen in Figure~\\ref{fig:gmodchopped} and a detailed description of the design and assembly can be found in \\cite{gmod2}.\n\n\\begin{figure}\n\\includegraphics[width=0.8\\textwidth]{figures\/GMODAssemblyChoppedAnnotated.png}\n\\caption{Cut-away view of the GMOD payload. The design comprises a detector assembly mounted on a motherboard PCB. The detector assembly includes a CeBr$_3$ scintillator, SiPM array, and the SIPHRA ASIC within a light-tight enclosure. Red surfaces indicate a component which has been cut to reveal internal features.}\n\\label{fig:gmodchopped}\n\\end{figure}\n\nThe scintillator is a 25\\,mm\\hskip0.16667em\\relax$\\times$\\,25\\,mm\\hskip0.16667em\\relax$\\times$\\,40\\,mm CeBr$_3$ scintillator produced by Scionix.\nAs CeBr$_3$ is hygroscopic, the scintillator is supplied in an aluminium hermetically sealed unit with a quartz window.\nThe scintillation light from the crystal is measured by a custom 4$\\times$4 array of 6\\,mm J-series SiPMs \\cite{jseries} which are readout and digitised by the SIPHRA ASIC \\cite{ide3380} from Integrated Detector Electronics AS (IDEAS).\nThe SiPM array is assembled on one side of a dedicated `SiPM Array PCB'.\nThe reverse side of this PCB features supporting passive electronics for the SiPMs, the interface connectors for the board, and a PT100 temperature sensor which allows the array temperature to be monitored.\nAn `Interface PCB' sits between the SiPM array and SIPHRA and provides connectors for a harnessed connection to the motherboard.\nPTFE spacers support the detector components within the aluminium light-tight enclosure, forming the 75\\,mm\\hskip0.16667em\\relax$\\times$\\,51\\,mm\\hskip0.16667em\\relax$\\times$\\,42\\,mm detector assembly which is bolted to the motherboard.\nThese spacers additionally provide good thermal isolation between the enclosure and the internal components, stabilising the scintillator and SiPM array temperature \\cite{Mangan2021}.\n\nThe motherboard includes the functionality necessary to interface the detector assembly to the spacecraft, including control and readout of the ASIC and regulation of the various voltages required by the detector including the SiPM bias voltage.\nThe SiPM bias PSU is adjustable between $-25$\\,V and $-28.3$\\,V allowing the SiPM over-voltage, and therefore the SiPM gain, to the adjusted in flight.\nThis voltage can also be automatically varied in response to the measured array temperature to maintain a constant gain, accounting for temperature-dependent variations of the SiPM breakdown voltage \\cite{jseries}.\nThe motherboard has been designed to be compatible with the standard CubeSat PC-104 form factor and the whole GMOD assembly is placed at the top of the main spacecraft stack of subsystems.\nThe GMOD detector assembly and motherboard can be seen in the stack assembly in Figure~\\ref{fig:eirsat-1exploded}.\n\nSiPMs are known to receive damage from proton radiation leading to an increase in operating current and detector noise. These effects were evaluated using a prototype of the GMOD detector irradiated with a proton beam~\\cite{ulyanov2020}. After one year of operation in the ISS-like orbit, the detector is still expected to detect gamma rays with energy above 30\\,keV, which exceeds the mission requirement of 50\\,keV. The total current of 16 SiPMs may increase to 500--1200S\\,$\\mu$A depending on temperature, which is easily handled by the power supply and SIPHRA readout. The SiPM radiation damage is not expected to be a problem for the GMOD detector as it uses a bright scintillator with a relatively small SiPM array and the SiPM exposure to protons in the ISS orbit will be relatively low (1\\,MeV neutron equivalent fluence of $4.3\\times 10^8$\\,cm$^{-2}$ over the mission lifetime of one year). With a higher energy threshold of 50\\,keV, the detector can achieve a one year operation in higher orbits.\n\n\n\n\n\\section{Simulations}\n\\label{ch:GMODsimulations}\n\nTo understand the performance of GMOD as a detector for gamma-ray bursts, its response was simulated using the Medium Energy Gamma-ray Astronomy Library (MEGAlib) \\cite{zoglauer2006}. MEGAlib is a collection of software tools designed to simulate the performance of various gamma-ray instruments using the Geant4 simulation toolkit~\\cite{AGOSTINELLI2003250} and to perform event reconstruction and analysis for Compton and pair production telescopes. This section describes the EIRSAT-1 mass model and the GRB and background models used in the MEGAlib simulations. The analysis of simulated data is discussed in Section~\\ref{sec:simanalysis}.\n\n\n\n\\subsection{Mass Model}\n\\label{sec:massmodel}\n\nThe full EIRSAT-1 spacecraft has been included in the mass model as it is important to characterise the capabilities of the GMOD detector in the environment in which it will operate.\nThe mass model is derived from a very high-fidelity 3D CAD model of EIRSAT-1.\nAn exploded render of this model is shown in Figure~\\ref{fig:eirsat-1exploded}.\nThe high-fidelity model is made up of contributions from various sources.\nThe models of the commercial off-the-shelf components were provided by \\r{A}AC Clyde Space.\nModels of the custom structural components were created in Autodesk AutoCAD.\nThe models of all GMOD mechanical components, including the scintillator crystal and hermetic enclosure (see Section~\\ref{sec:gmod}) were also modelled in AutoCAD.\nThese models were also the basis for the manufacturing drawings that were sent to the workshop for fabrication.\nTo generate 3D models of EIRSAT-1's custom circuit boards, an EIRSAT-1 electronic components library was created in Autodesk EAGLE.\nAll components beyond extremely standard ones such as resistors and capacitors were managed using this library, primarily to ensure that all component footprints were correct based on manufacturer's drawings, but it also allowed 3D models of components to be added.\nWith all circuit components managed in EAGLE, it was possible to export high-fidelity 3D representations of the PCBs.\nThe 3D models of the Antenna Deployment Module and EMOD coupon assembly were provided by the respective sub-teams.\n\nDue to the text-based mass model format required by MEGAlib, it is not possible to import 3D models from standard CAD formats and instead the volumes need to be defined manually.\nFurthermore, only a limited number of volumes types are available in MEGAlib, including cuboid, sphere and cylinder (both of which may be solid or hollow and may be segmented), cone (which may also be hollow and be truncated), and various types of trapezoids.\n\nThe high-fidelity model of EIRSAT-1 was therefore simplified to the point that it contained only solids which could be represented using these supported MEGAlib volumes.\nWhile this has the effect of removing finer details from the model, the overall mass distribution is not significantly changed and provides equivalent effective shielding. \nThis simplification process was performed manually using AutoCAD, as it was significantly easier to place the volumes in a 3D environment and ensure that they correctly represented the EIRSAT-1 geometry using the AutoCAD interface than using the MEGAlib text-based volume description format.\nFor EIRSAT-1, the cuboid and trapezoid proved to be the most useful of the available volumes with many of the complex components of EIRSAT-1 being made of a combination of cuboids and trapezoids.\nAn example of this is given in Figure~\\ref{fig:simplifiedcap}, which compares the $+$Z end-cap of the spacecraft before and after the simplification process and shows how the simplified version is constructed from multiple cuboids and trapezoids.\nThe full spacecraft simplified model is shown in Figure~\\ref{fig:simplifiedsat}.\n\n\n\\begin{figure}\n\\begin{center}\n \\includegraphics[width=0.8\\textwidth]{figures\/simplifiedcap.pdf}\n \\caption[Simplified +Z End-cap for MEGAlib]{A comparison of the high-fidelity (left) and simplified for MEGAlib (right) models of the $+Z$ end-cap. The top row shows an isometric view of the top of the end-cap while the bottom row shows an isometric view of the underside of the end-cap.}\n \\label{fig:simplifiedcap}\n\\end{center}\n\\end{figure}\n\n\n\\begin{figure}\n\\begin{center}\n \\includegraphics[width=0.8\\textwidth]{figures\/simplifiedsat.pdf}\n \\caption[Simplified EIRSAT-1 for MEGAlib]{Partially exploded isometric view of the simplified AutoCAD 3D model of EIRSAT-1. The model has been designed such that all of the solids which comprise it can be represented as MEGAlib volumes.}\n \\label{fig:simplifiedsat}\n\\end{center}\n\\end{figure}\n\n\nFollowing the simplification process, the spacecraft geometry was manually transcribed into the MEGAlib format by defining the position, orientation, size and material of each volume as well as relationships between volumes and mother volumes.\nRenderings of the final mass model are shown in Figure~\\ref{fig:massmodel}.\nThe mass model can be downloaded from \\cite{eirsatmegalib}.\n\n\n\\begin{figure}\n\\centering\n \\includegraphics[width=0.4\\textwidth]{figures\/MEGAlib1.png}\n \\hspace{1cm}\n \\includegraphics[width=0.4\\textwidth]{figures\/MEGAlib2.png}\n \\par\\bigskip\\bigskip\n \\includegraphics[width=0.4\\textwidth]{figures\/MEGAlib3.png}\n \\hspace{1cm}\n \\includegraphics[width=0.4\\textwidth]{figures\/MEGAlib4.png}\n \\caption[EIRSAT-1 MEGAlib Mass Model]{Various views of the EIRSAT-1 MEGAlib mass model showing the internal subsystems and constituent components of the GMOD detector assembly. \\emph{Top-left:} Outside view of the full mass model. \\emph{Top-right:} +X and $-$Y solar arrays and shear panels removed, showing internal components. \\emph{Bottom-left:} Diagonal vertical slice through the spacecraft. \\emph{Bottom-right:} Horizontal slice through the centre of the GMOD detector assembly.}\n \\label{fig:massmodel}\n\\end{figure}\n\n\nA GMOD MEGAlib detector was defined as the `Scintillator' type with a sensitive volume of 25\\,mm\\,$\\times$\\,25\\,mm\\,$\\times$\\,40\\,mm which represented the CeBr$_3$ crystal.\nThe energy resolution of the detector was defined in the MEGAlib geometry file as Gaussian with the sigma values shown in Table~\\ref{table:megalibresolution}.\nThese values were based on spectral measurements of gamma-rays from $^{137}$Cs and $^{22}$Na sources using a development model of GMOD and are in good agreement with later measurements performed with the GMOD engineering qualification model \\cite{gmod2}. The trigger threshold was set to 30\\,keV.\n\n\n\\begin{table}[h]\n\\caption{GMOD energy resolution in the MEGAlib model}\n\\label{table:megalibresolution}\n\n\\begin{tabular}{l l}\n\\hline\\noalign{\\smallskip}\nEnergy (keV) & 1$\\sigma$ (keV)\\\\\n\\noalign{\\smallskip}\\hline\\noalign{\\smallskip}\n20 & 8.5 \\\\\n100 & 9.4 \\\\\n350 & 12.1 \\\\\n511 & 13.6 \\\\\n662 & 14.9 \\\\\n1000 & 17.6 \\\\\n5000 & 41.5 \\\\\n10000 & 67.0 \\\\\n\\noalign{\\smallskip}\\hline\n\\end{tabular}\n\\end{table}\n\n\n\n\\subsection{GRB Source Model}\n\\label{sec:grbmodel}\n\nFor the vast majority of GRBs, the spectral distribution of gamma-rays can be described by a smoothly broken power-law known as the Band function~\\cite{band1993}:\n\\begin{equation}\n \\label{eq:band}\n \\frac{\\mathrm{d}N}{\\mathrm{d}E} =\n \\begin{cases}\n A E^\\alpha \\exp\\left(-\\frac{(2+\\alpha)E}{E_\\mathrm{peak}}\\right) & \\text{if } E < \\frac{\\alpha-\\beta}{2+\\alpha}E_\\mathrm{peak} \\\\\n A \\left(\\frac{\\alpha-\\beta}{2+\\alpha}E_\\mathrm{peak}\\right)^{\\alpha-\\beta} \\exp(\\beta-\\alpha) E^\\beta & \\text{if } E \\geq \\frac{\\alpha-\\beta}{2+\\alpha}E_\\mathrm{peak},\n \\end{cases}\n\\end{equation}\nwhere $A$ is the amplitude (normalisation factor of the spectrum), $\\alpha$ is the low energy index, $\\beta$ is the high energy index and $E_\\mathrm{peak}$ is the peak energy of the power density spectrum $E^2\\mathrm{d}N\/\\mathrm{d}E$.\n\nIn this study, a Band function with parameter values $\\alpha = -1.1$, $\\beta = -2.3$ and $E_\\mathrm{peak} = 300$\\,keV was used to represent the spectrum of an average GRB, based on the BATSE spectral analysis of bright bursts~\\cite{kaneko2006}.\nThe GRB spectrum was simulated within an energy range of 20\\,keV--300\\,MeV, as GMOD will only detect photons with measured energies above 20\\,keV. \n\nTo simulate the detector response to GRBs from all possible directions, the simulation was performed using an isotropic distribution of source gamma-rays over the full sky (using a source type `FarFieldAreaSource' in MEGAlib).\nDirectional point-like sources were then created in the analysis stage by filtering the simulation outputs and restricting the directions of the source gamma rays to small areas of the sky.\n\n\\subsection{Background Model}\n\\label{sec:bgmodel}\n\nOperation of gamma-ray detectors in low Earth orbits is affected by many sources of background radiation, including cosmic gamma-rays, cosmic-ray particles and secondary particles generated in Earth's atmosphere. A recent summary of various background components and models can be found in \\cite{cumani2019}. According to simulations~\\cite{galgoczi2021}, the count rate of a scintillator detector in low Earth orbit outside the Van Allen radiation belts is dominated by the extragalactic gamma-ray background and atmospheric (albedo) gamma-ray emission. This is consistent with a background model for Fermi GBM based on real data~\\cite{biltzinger2020}. The relative importance of cosmic and albedo gamma-rays depends on the photon energy band, with the cosmic gamma-ray background playing a major role at low energies and albedo photons dominating the gamma-ray spectrum above about 200\\,keV, depending on the orbit altitude and inclination.\n\nOnly the two main background sources, that is cosmic and albedo gamma-rays, were considered in this work. The cosmic gamma-ray background was simulated using the model described in \\cite{turler2010}, with the photon flux calculated as\n\\begin{equation}\n \\label{eq:cgbmodel}\n F = \\frac{0.109}{(E\/28\\,\\mathrm{keV})^{1.40}+(E\/28\\,\\mathrm{keV})^{2.88}}\\ \\mathrm{ph}\\, \\mathrm{cm}^{-2} \\mathrm{s}^{-1} \\mathrm{sr}^{-1} \\mathrm{keV}^{-1}\n\\end{equation}\n\nAlbedo gamma-rays are generated through interactions of cosmic rays with the Earth atmosphere. Due to the effects of the geomagnetic field and solar activity on cosmic rays, the albedo flux seen by a satellite in a low Earth orbit can vary by a factor of 4--5 depending on the geomagnetic latitude and solar activity~\\cite{imhof1976,sazonov2007}. For this study we used the models given in \\cite{sazonov2007} for $E<1$\\,MeV and \\cite{mizuno2004} for $E\\geq1$\\,MeV, with the photon flux in units of ph\\,cm$^{-2}$\\,s$^{-1}$\\,sr$^{-1}$\\,keV$^{-1}$ given by\n\\begin{equation}\n \\label{eq:albedomodel}\n F =\n \\begin{cases}\n \\frac{C}{(E\/44\\,\\mathrm{keV})^{-5}+(E\/44\\,\\mathrm{keV})^{1.4}}\\ & \\text{if } E < 1\\,\\text{MeV} \\\\\n 1.01\\times 10^{-4} (\\frac{E}{1000\\,\\mathrm{keV}})^{-1.34} & \\text{if } 1\\,\\text{MeV} \\leq E < 20\\,\\text{MeV} \\\\\n 7.29\\times 10^{-4} (\\frac{E}{1000\\,\\mathrm{keV}})^{-2.0} & \\text{if } E \\geq 20\\,\\text{MeV},\n \\end{cases}\n\\end{equation}\nwhere the constant $C$ was set to 0.0080 to avoid discontinuity at 1\\,MeV.\nEquation~\\ref{eq:albedomodel} gives the albedo photon flux for a geomagnetic latitude of about 40--50\\textdegree{} and minimum solar activity which represent a relatively conservative (large background) scenario. In this study, the flux given by Equation~\\ref{eq:albedomodel} was additionally multiplied by a factor of 1.5, corresponding to the most conservative case of albedo in the polar regions. \n\nThe angular distribution of albedo photons depends on energy, exhibiting central brightening for low photon energies $E<1$\\,MeV and limb brightening for $E\\geq1$\\,MeV~\\cite{sazonov2007,mizuno2004}. These effects are not very important for an instrument with a wide field of view and were ignored in this study: the albedo emission as seen by the satellite was assumed to be uniformly distributed over the Earth disk.\n\nThe background spectra were simulated within an energy range of 20\\,keV--100\\,MeV.\nBoth background sources were defined in the MEGAlib simulations as isotropic emission over the full sphere.\nIn the analysis stage, the photon directions were constrained to the sky region occupied by Earth for albedo gamma-rays and to the unocculted sky for cosmic gamma rays.\nThis approach helps to calculate the background count rates for different satellite orientations without rerunning the full Monte Carlo simulations for each particular case.\nIt does, however, require that the simulated flux is appropriately scaled accounting for the difference between the solid angle of the directional source and the full sphere of the isotropic simulated flux.\n\n\n\n\\section{Simulation Analysis}\n\\label{sec:simanalysis}\n\nEach of the three sources (GRB, cosmic background, Earth albedo background) were separately simulated against the EIRSAT-1 geometry using MEGAlib. In addition, an isotropic photon distribution with a flat spectrum having constant intensity between 20\\,keV and 3\\,MeV was simulated to calculate the GMOD effective area in different energy bands and ensure sufficient photons at higher energies. \n\nFor each simulated photon that has some energy deposited in the sensitive detector volume (scintillator), MEGAlib records a number of parameters including the event ID, the position, direction and energy of the initial photon, the actual energy that was deposited in the scintillator and the energy that was ``measured'' by the detector taking its energy resolution into account. Two custom file parsers were written to import the useful parameters from MEGAlib's \\texttt{.sim} and \\texttt{.tra} output files into a Python Pandas dataframe. The data were then analysed in Python to produce effective area plots, estimate the background count rate and the number of photons detected per unit GRB flux assuming the typical spectrum, and finally to calculate the GRB detection rate based on the distribution of GRBs found in the BATSE 4B catalog~\\cite{paciesas1999}.\n\n\n\n\n\\subsection{Effective Area}\n\\label{sec:energyeffectivearea}\n\n\\begin{figure}\n\\includegraphics[width=0.7\\textwidth]{figures\/effectiveareaenergygraph_P1.pdf}\n\\caption{The simulated effective area of GMOD in EIRSAT-1, averaged over all directions, as a function of energy. The effective areas were calculated in 10\\,keV energy bins. The blue line represents the photo peak efficiency while the grey line includes photons which were detected at a lower energy.}\n\\label{fig:effectiveareaenergygraph}\n\\end{figure}\n\nThe effective area of GMOD, averaged over all directions, as a function of energy is shown in Figure~\\ref{fig:effectiveareaenergygraph}. The effective area in each energy bin was simulated using a flat photon spectrum.\nThe blue line gives the photo peak efficiency, while the grey line includes photons which were detected at a lower energy than their initial energy due to scattering or escape effects.\nThe grey line is a good representation of the effective area for burst detection purposes as the spectral range used for GRB triggering tends to be quite wide and therefore photons recorded at lower than their true energy will still contribute to the count rate in the detector.\n\nThe total effective area peaks at approximately 120\\,keV reaching a value of 10 cm$^2$. It decreases at lower energies due to the absorption of gamma rays by the spacecraft structure and the aluminium housing of the detector. \nFigure~\\ref{fig:effectiveareaenergygraph} indicates that the instrument is most sensitive in the tens of keV up to a few MeV energy range.\n\n\n\\begin{figure}\n\\includegraphics[width=0.9\\textwidth]{figures\/effectivearea3050_P1.pdf}\n\\begin{center}\n(a) 30--50\\,keV.\n\\end{center}\n\n\\includegraphics[width=0.9\\textwidth]{figures\/effectivearea50100_P1.pdf}\n\\begin{center}\n(b) 50--100\\,keV.\n\\end{center}\n\n\\includegraphics[width=0.9\\textwidth]{figures\/effectivearea100300_P1.pdf}\n\n\\begin{center}\n(c) 100--300\\,keV.\n\\end{center}\n\n\\caption[GMOD Effective Area vs Direction in Various Energy Bands]{The simulated effective area of GMOD in EIRSAT-1 as a function of direction in various energy bands. }\n\\label{fig:effectiveareaenergybands}\n\\end{figure}\n\n\\begin{figure}\n\\includegraphics[width=0.9\\textwidth]{figures\/effectivearea300500_P1.pdf}\n\\begin{center}\n(a) 300--500\\,keV. \n\\end{center}\n\n\\includegraphics[width=0.9\\textwidth]{figures\/effectivearea5001000_P1.pdf}\n\\begin{center}\n(b) 500\\,keV--1\\,MeV.\n\\end{center}\n\n\\includegraphics[width=0.9\\textwidth]{figures\/effectivearea10003000_P1.pdf}\n\\begin{center}\n(c) 1--3\\,MeV.\n\\end{center}\n\n\\caption{The simulated effective area of GMOD in EIRSAT-1 as a function of direction in various energy bands (cont).}\n\\label{fig:effectiveareaenergybands2}\n\\end{figure}\n\nThe effective area as a function of direction is shown in Figures~\\ref{fig:effectiveareaenergybands} and~\\ref{fig:effectiveareaenergybands2} for six energy bands. \nThe entire field of view is divided into bins (pixels) by azimuthal angle $\\phi$ and polar angle $\\theta$ and the effective area is calculated for each bin. A 5\\textdegree{} bin size is used in both directions. \nThis binning suffers from poor simulation statistics at the poles caused by the fact that the solid angles represented by each pixel becomes very small in these regions.\nAn adaptive rebinning technique was therefore used to increase the size of pixels at the poles such that no pixel had a solid angle more than 1.5 times smaller that those at the equator.\nThe effect of this rebinning can be seen in the plots below a polar angle of 40\\textdegree{} where pixels get progressively wider as they approach 0\\textdegree{} and pixels above 140\\textdegree{} get progressively wider as they approach 180\\textdegree{}. For each pixel, the effective area is the average value calculated using a flat photon spectrum in the given energy band.\n\n\nFigure~\\ref{fig:effectiveareaenergybands} reveals many details about the effect of the structure of the GMOD detector and the EIRSAT-1 spacecraft.\nThe two most striking features are the high effective area region at $(\\theta=90\\degree{}$, $\\phi=-90\\degree{})$, the $-$Y direction (the spacecraft coordinate reference frame is indicated in Figures~\\ref{fig:eirsat-1exploded} \\& \\ref{fig:gmodchopped}), and the very poor effective area at $\\theta=180\\degree{}$, the $-$Z direction.\nThe $-$Z direction is blocked by the spacecraft bus and so poor performance in this direction was expected.\n\nThe high effective area region in the $-$Y direction corresponds to one of the long faces of the CeBr$_3$ crystal and this direction in particular is where the crystal is closest to edge of the spacecraft.\nThere is a corresponding smaller region of high effective area in the opposite $+$Y direction.\nThis region has a cut off at a smaller polar angle than the $-$Y due the effect of the PCB stack attenuating photons.\nThis imbalance between $\\pm$Y would likely be resolved if GMOD were to be placed centrally within the EIRSAT-1 structure.\nThe geometry explaining this effect can be most clearly seen in the bottom-right part of Figure~\\ref{fig:massmodel} which shows a horizontal cross-section through the centre of the crystal.\n\nThe $+$Z direction which is shown at $\\theta=0\\degree{}$ retains reasonably high effective area in the 100--300\\,keV band as it also corresponds to one of the long faces of the CeBr$_3$ crystal.\nAttenuation through the EMOD thermal coupon assembly (TCA) is however higher than through the solar arrays found in the $\\pm$Y directions. This effect is particularly strong in the 30-50\\,keV and 50-100\\,keV bands and less obvious in the higher energy bands.\nA line of lower effective area can also be seen snaking around the $+$Z face of the spacecraft between the adjacent faces at polar angles between 30\\textdegree{} and 60\\textdegree{}.\nThis corresponds to the bulk of aluminium and titanium around the perimeter of the TCA which can be seen in Figures~\\ref{fig:eirsat-1exploded} and~\\ref{fig:simplifiedcap}.\n\nThe $+$X direction at $(\\theta=90\\degree{}$, $\\phi=0\\degree{})$ shows a relatively average effective area.\nThis direction corresponds to one of the smaller, square sides of the crystal and in particular this side is also the one which is coupled to the SiPM array.\nThis direction therefore exhibits attenuation from both the SiPM array PCB and the ASIC PCB.\nThere is a matching region in the opposite $-$X direction at $(\\theta=90\\degree{}$, $\\phi=\\pm180\\degree{})$ direction, also corresponding to a square side of the crystal.\nIn this case however, there are no SiPMs or PCBs and therefore the effective area remains higher, though not as high as those directions with a long side of the crystal.\n\nFigure~\\ref{fig:effectiveareaenergybands2} shows that the effect of the spacecraft structure is smaller at higher energies, as expected.\nIn the higher energy bands, the photons are energetic enough to penetrate the spacecraft structure from almost any angle equally, though the effective area is reduced as the scintillator absorption is also lower.\n\n\n\n\\subsection{GRB Effective Area}\n\\label{sec:grbeffectivearea}\nSimilar to BATSE and GBM, GMOD will detect GRBs by registering a statistically significant increase in the detector count rate. \nFor the purposes of burst detection in this study, the GMOD count rate is integrated over an energy range of 50--300\\,keV. This is the range of maximum effective area for GMOD (Figure~\\ref{fig:effectiveareaenergygraph}) comprising the majority of detected photons from a typical GRB. It was also the energy range of the nominal BATSE on-board burst trigger used during the first several years of BATSE operation~\\cite{paciesas1999}. The GRB peak fluxes in the BATSE catalog are given in this range. Calculation of the GMOD count rate for these GRBs is less sensitive to their spectra if performed in the identical energy band, 50--300\\,keV. \n\nThe GRB effective area, $A_{\\textrm{GRB}}$, is defined here as the total number of photons detected with a reconstructed energy in the 50--300\\,keV energy band, $N_{\\textrm{det, band}}$, divided by the total number of photons simulated per cm$^2$ with an initial energy in that same energy band, $H_{\\textrm{band}}$. \n\\begin{equation}\n A_{\\textrm{GRB}} = \\frac{N_{\\textrm{det, band}}}{H_{\\textrm{band}}}\n\\label{eq:effareasimple}\n\\end{equation}\nIn other words, this quantity represents the detector count rate in the 50--300\\,keV energy band per unit GRB flux in the 50--300\\,keV band. Unlike the effective area described in Section~\\ref{sec:energyeffectivearea} (which was simply averaged over photons simulated in the given energy bands), this quantity includes contributions to the 50--300\\,keV count rate from the entire GRB spectrum. \n\nThe GRB effective area of GMOD was calculated for the typical GRB spectrum described in Section~\\ref{sec:grbmodel}. The simulated fluence in the 50--300\\,keV band can be calculated from the total simulated fluence $H_{\\textrm{tot}}$ as \n\\begin{equation}\n H_{\\textrm{band}} = R_{\\textrm{band}} H_{\\textrm{tot}},\n\\label{eq:simfluence}\n\\end{equation}\nwhere $R_{\\textrm{band}}$=0.493 is the ratio of photons in the simulated source spectrum that have an energy in the 50--300\\,keV band. \n\nTherefore the effective area can be rewritten in terms of known parameters from the simulation as\n\\begin{equation}\n A_{\\textrm{GRB}} = \\frac{ N_{\\textrm{det, band}} }{ R_{\\textrm{band}} H_{\\textrm{tot}} }\n\\label{eq:effareaknown}\n\\end{equation}\n\nThe GRB effective area of GMOD on EIRSAT-1 as a function of azimuth and polar angle is shown in Figure~\\ref{fig:effectivearea}.\nAs expected, the plot displays similar structural effects as were seen in Figure~\\ref{fig:effectiveareaenergybands} for the effective area in the 50--100\\,keV and 100--300\\,keV energy bands. The average value of the GRB effective area calculated over all directions is 8.75\\,cm$^2$. \n\n\n\\begin{figure}\n\\includegraphics[width=0.9\\textwidth]{figures\/effectivearea_P1.pdf}\n\\caption[GMOD Effective Area vs Direction]{The GRB effective area of GMOD in EIRSAT-1 as a function of direction. The effective area is calculated in the 50--300\\,keV range for a typical GRB spectrum ($\\alpha=-1.1$, $\\beta=-2.3$, $E_\\mathrm{peak}=300$\\,keV).}\n\\label{fig:effectivearea}\n\\end{figure}\n\n\n\\subsection{Effects of Spacecraft Spin and Earth Occultation}\n\\label{sec:spin}\n\nAs the Earth occults a large part of the sky (approximately 33\\% at 400\\,km altitude), the average effective area of GMOD over the unnocculted sky depends on satellite orientation with respect to the planet. The orientation also affects the detector count rate from cosmic and albedo gamma rays. \n\nDue to the constraints of the magnetorquer based attitude control system on EIRSAT-1, the spacecraft will be spin stabilised at a few to tens of revolutions per minute. The effective area and background, averaged over the rotation period, depend only on the angle between the +Z spin-axis of the spacecraft and Zenith. \n\nThe effect of the spin of EIRSAT-1 can be accounted for by aligning the azimuth axis of the spherical coordinates used in the simulation with the +Z spin-axis, averaging over all azimuthal angles and applying a weight based on the fraction of time that any given polar angle relative to the spacecraft is oriented toward or away from Earth.\nIn the cases of Earth-occulted sources, such as GRBs or cosmic background, the weight will be the fraction of the revolution that a given polar angle relative to the spacecraft spends oriented away from the Earth, while for albedo background it will be the fraction spent pointed towards the Earth.\n\nIn order to calculate these weights, the Earth is modelled as a disc of a given angular size.\nThe extents of this disc form a circle projected onto a sphere in the spherical coordinate reference frame of the spacecraft.\nThe centre of the disc is the spacecraft-Earth vector and its radius, being the shortest distance between the centre and the edge of the disc, is therefore a great circle on the sphere.\n\n\\begin{figure}\n\\begin{center}\n \\includegraphics[width=0.5\\textwidth]{figures\/spin_paper.pdf}\n \\caption{Geometry used to calculate the fraction of time any direction from the spacecraft is oriented towards the Earth. The blue vector ($\\theta_E$, $\\phi_E$) points towards Earth, with the blue circle giving the maximum extents of the Earth and its angular radius in red. For a vector ($\\theta_E$, $\\phi_E$) such as that shown in black, the solid black line inside the blue circle represents the fraction of the full revolution spent pointed towards Earth. Note: for clarity, the Earth disc is shown as a smaller angular size than reality.}\n \\label{fig:spin}\n\\end{center}\n\\end{figure}\n\nThis geometry is shown in Figure~\\ref{fig:spin}.\nThe blue vector is the spacecraft-Earth vector ($\\theta_E$, $\\phi_E$), with the blue circle indicating the extents of the apparent disc of the Earth and the red arc indicating the great circle distance radius of the Earth.\nThe black vector is an orientation in the spacecraft frame from which a photon may originate ($\\theta$, $\\phi$), with the thicker dashed black circle indicating the arc which it sweeps out across the sky.\nThe portion of the revolution that that vector spends oriented towards the Earth is the solid black arc inside the blue circle.\nThis angle swept out in this portion is twice the difference between the azimuthal angles, $\\Delta\\phi$.\nThe weight, $W$, corresponding to the fraction of the revolution spent pointed towards the Earth is therefore\n\\begin{equation}\n W = \\frac{2\\left(\\phi_E - \\phi\\right)}{2\\pi} = \\frac{\\Delta\\phi}{\\pi}\n\\label{eq:weight}\n\\end{equation}\n\nThe great circle distance formula gives the shortest distance between two points on a sphere.\nFor a unit sphere, this distance is equivalent to the angle subtended on the sphere between two points, or in this problem the angular radius of the Earth as seen from the spacecraft.\n\nThe great circle distance, $\\psi$, between two points ($\\theta_E,\\phi_E$) and ($\\theta,\\phi$) in spherical coordinates is given by:\n\\begin{equation}\n \\psi = \\cos^{-1} \\left( \\cos\\theta_E \\cos\\theta + \\sin\\theta_E \\sin\\theta\\cos\\left( \\phi_E - \\phi \\right) \\right)\n\\label{eq:greatcircle}\n\\end{equation}\n\nEquation~\\ref{eq:greatcircle} is thus rearranged to give:\n\\begin{equation}\n \\Delta\\phi = \\cos^{-1} \\left( \\frac{\\cos\\psi - \\cos\\theta_E \\cos\\theta}{\\sin\\theta_E \\sin\\theta} \\right),\n\\label{eq:phifrac}\n\\end{equation}\nwhich is only valid when $|\\theta - \\theta_E| \\leq \\psi\/2$, representing parts of the spacecraft which see the Earth for only part of the revolution.\nWhere the Earth is seen for the full revolution, the bracketed term in Equation~\\ref{eq:phifrac} evaluates as $<-1$ and where the Earth is not seen at all, the bracketed term evaluates as $>1$, both of which are undefined for real values for the inverse cosine function.\nClipping the range of the bracketed term to between -1 and 1 gives the appropriate $\\Delta\\phi$ for the fully Earth or fully space orientations.\nFinally, dividing $\\Delta\\phi$ by $\\pi$ as per Equation~\\ref{eq:weight} gives $W$, the fraction of a revolution that any part of the spacecraft, with a polar angle of $\\theta$ in the spacecraft's spherical coordinate reference frame, spends pointed towards the Earth.\n\n\n\n\\subsection{GRB Effective Area as a Function of Spacecraft Attitude}\n\\label{sec:attitudeeffectivearea}\n\nUtilising the technique discussed in Section~\\ref{sec:spin}, it is possible to simulate the effect of Earth occultation on the average effective area of GMOD.\nIn this analysis the Earth is placed at a range of polar angles in the spacecraft's reference frame, $180\\degree{} \\geq \\theta_E \\geq 0\\degree{}$, representing pointing angles 0\\degree{}--180\\degree{} between the $+$Z axis and Zenith.\nFor an altitude of approximately 400\\,km above the Earth, the Earth appears as a disc which subtends an angular radius of 70\\textdegree{} and occults 33\\% of the sky.\n\nThe GRB effective area is calculated over all directions as before but rather than simply counting the detected photons, each photon carries a weight and it is the weights of detected photons that are summed.\nThe photon weights are the fraction of a revolution for which the direction in the spacecraft's rotating frame from which they originated, was pointed towards space. The total simulated fluence $H_{\\textrm{tot}}$ in Equation~\\ref{eq:effareaknown} is replaced by $0.67H_{\\textrm{tot}}$ representing the total fluence of unocculted photons.\n\nThe resulting average effective area of the instrument as a function of spacecraft attitude is shown in Figure~\\ref{fig:effectiveareaattitude}. As expected, the average effective area is maximised when the least sensitive ($-$Z) side of the detector is facing the Earth. However, this effect is relatively small. \n\n\\begin{figure}\n\\includegraphics[width=0.8\\textwidth]{figures\/effectiveattitude_P1.pdf}\n\\caption[GMOD Effective Area vs Spacecraft Attitude]{The GRB effective area of GMOD in EIRSAT-1, averaged over the entire unocculted sky, as a function of spacecraft attitude. The effective area is calculated in the 50--300\\,keV range for a typical GRB spectrum.}\n\\label{fig:effectiveareaattitude}\n\\end{figure}\n\n\n\\subsection{Background Rate}\n\\label{sec:bgrate}\nThe GMOD background count rate was simulated using the cosmic gamma-ray background and Earth albedo sources as described in Section~\\ref{sec:bgmodel}.\nThe background rate averaged over the spacecraft rotation period was calculated for spacecraft pointing angles 0\\textdegree{}--180\\textdegree{} between the $+$Z axis and Zenith using the technique described in Section~\\ref{sec:spin}.\nThe detected cosmic background photons were weighted by the fraction of a spacecraft revolution spent pointed towards space while the Earth albedo photons were weighted by the fraction spent pointed towards Earth.\nThe resulting count rates are shown in Figure~\\ref{fig:attitudenoise}.\n\n\n\\begin{figure}\n \\includegraphics[width=0.8\\textwidth]{figures\/noiseattitude_P1.pdf}\n \\caption[GMOD Background Count Rates vs Spacecraft Attitude]{The simulated background count rates in GMOD in the 50--300\\,keV range with contributions from cosmic gamma-ray and Earth albedo backgrounds, as a function of spacecraft attitude.}\n \\label{fig:attitudenoise}\n\\end{figure}\n\n\n\n\\subsection{GRB Detection}\n\\label{sec:grbdetection}\n\nGMOD's ability to detect GRBs was analysed by calculating the detection significance which would be expected for GRBs with the same fluxes as those found in the BATSE 4B catalog~\\cite{paciesas1999}. \n\nThe number of photons which would be detected by GMOD from a given GRB depends on the GRB position in the spacecraft coordinate frame. To simplify the calculations, \nthe number of detected counts was determined using the average GRB effective area of the detector as a function of spacecraft attitude as described in Section~\\ref{sec:attitudeeffectivearea}.\nFor each GRB in the BATSE 4B catalog, the flux for the 64\\,ms, 256\\,ms, and 1024\\,ms trigger bins was multiplied by the effective area and then scaled by the bin duration to get the number of counts detected from the GRB.\nThe GMOD background count rates were also scaled by the bin duration.\nThe detection significance was then calculated as the GRB counts divided by the square root of the background counts.\nThe detection significance, $\\sigma$ is \n\\begin{equation}\n \\sigma = \\frac{ A_{\\textrm{GRB}} F t_\\textrm{bin} }{ \\sqrt{B t_\\textrm{bin} } },\n\\label{eq:significance}\n\\end{equation}\nwhere $A_{\\textrm{GRB}}$ is the GMOD effective area for GRBs, $F$ is the flux reported by BATSE, $t_\\textrm{bin}$ is the duration of the trigger bin, and $B$ is the background count rate for GMOD. Using the effective area simulated with the average GRB spectrum, this approach ignores variations in GRB spectral properties.\nThe significance in each of the three trigger bins as well as the best calculated significance for each GRB was recorded.\n\nThe BATSE 4B catalog covers a duration of 1960 days or 5.3 years.\nThe average exposure factor for the 4B catalog, accounting for Earth occultation and instrument down time due to telemetry gaps or South Atlantic Anomaly (SAA) passage for example, is 0.483.\nTherefore if BATSE was able to view the entire sky at once without the effects of Earth occultation and was operational continuously, it would have detected the 1637 GRBs found in the 4B catalog in 2.56 years.\n\nThe detection significances are binned using a cumulative histogram with each bin containing the number of GRBs with a significance greater than the lower bin edge value.\nThe values in each bin are then divided by the duration of the catalog, divided by the average BATSE exposure factor and multiplied by the GMOD exposure factor of 0.496 to give the GMOD GRB detection rate as a function of significance.\nThis GMOD exposure factor accounts for Earth occultation of $\\sim$33\\% and the effects of instrument\/trigger down-time which will average $\\sim$10\\% due to transits of the SAA and $\\sim$16\\% due to transits of the outer Van Allen belt at high latitudes.\n\n\nThe GMOD GRB detection rates for the nominal mission lifetime of one year are shown in Figure~\\ref{fig:grbdetection}.\nAt a detection significance of 10$\\sigma$ GMOD is expected to detect between 11 and 14 GRBs per year, depending of spacecraft attitude.\nMany more GRBs will be detected at lower significance.\nAt 5$\\sigma$, the detection rate would be between 28 and 32 GRBs per year.\n\n\nThe detection rates for the shorter trigger bins are noticeably lower, with the 1024\\,ms bin very closely matching the detection rate when the best response from all bins is chosen.\nThis is a particularly interesting result as it indicates that in the event that the on-board processing resources are too constrained to support multiple bins for triggering, the longer 1024\\,ms bin is capable of providing a trigger for most GRBs that would be detected using the shorter bins. It should be noted, however, that triggering with the 1024\\,ms bin would be inefficient for very short bursts. \n\n\\begin{figure}\n\\includegraphics[width=0.8\\textwidth]{figures\/detection_saa_P1.pdf}\n\\caption[GMOD GRB Detection Rate]{The GRB detection rate for GMOD as a function of detection significance for the one-year nominal mission lifetime. The detection rates based on three different trigger bin durations is shown as well as when the best response across the three bins is chosen. The solid lines indicate the detection rate if the spacecraft is always zenith pointing while the dashed lines indicate a pointing angle of 148\\textdegree{} away from zenith which is where GMOD's response is most impacted by Earth occultation.}\n\\label{fig:grbdetection}\n\\end{figure}\n\n\n\n\\section{Summary}\n\nGMOD is a small gamma-ray detector designed for a 2U CubeSat. The main purpose of the instrument is to qualify the new detector technology using a CeBr$_3$ scintillator, silicon photomultipliers and the SIPHRA readout ASIC for space applications and to validate its capability to detect GRBs in low Earth orbit.\n\nThe sky-average effective area of GMOD, reaching a peak value of 10\\,cm$^2$ at 120\\,keV, is an order of magnitude lower than that of the GBM NaI detectors or detectors in recently proposed larger GRB-detecting CubeSat missions such as HERMES, CAMELOT, MoonBeam, GRID and BurstCube. The shape of the scintillator in GMOD results in omni-directional sensitivity which allows for a nearly all-sky field of view but also limits the GRB sensitivity in any given direction. Thin scintillators used in other missions provide better sensitivity for directions close to the detector normal. The limited field of view of such detectors can be compensated by using several detectors oriented in different directions on board a single spacecraft and\/or using a network of satellites viewing different parts of the sky.\n\nDespite the relatively small effective area, GMOD is expected to detect between 11 and 14 GRBs, at a significance greater than 10$\\sigma$ (and up to 32 at 5$\\sigma$), during a nominal one-year mission.\nIt will be able to record the light curves of the brightest bursts and measure their spectra in an energy range from tens of keV up to about 1\\,MeV. This will be an important step in technology qualification which will prepare the ground for design of larger detectors and future instruments.\n\n\n\\begin{acknowledgements}\nThe EIRSAT-1 project is carried out with the support of ESA's Education Office under the Fly Your Satellite!\\,2 programme. This study was supported by The European Space Agency's Science Programme under contract 4000104771\/11\/NL\/CBi. JM, AU, DM, and SMB acknowledge support from Science Foundation Ireland (SFI) under grant number 17\/CDA\/4723. LH acknowledges support from SFI under grant 19\/FFP\/6777 and the EU AHEAD2020 project (grant agreement 871158). DM, RD, MD and JT acknowledge support from the Irish Research Council (IRC) under grants GOIPG\/2014\/453, GOIPG\/2019\/2033, GOIPG\/2018\/2564 and GOIPG\/2014\/685. We acknowledge all students who have contributed to EIRSAT-1.\n\\end{acknowledgements}\n\n\\bibliographystyle{spphys} \n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} {"text":"\\section{Introduction}\n\nHit phenomena in entertainments are very popular dynamical phenomena in economics. In contrast to the usual economic activities, there are no balances of demand and supply in hit phenomena. Thus, for analysis of hit phenomena, we consider it as time-dependent non-equilibrium phenomena. Moreover, especially for movie hits, the marketing share is not important. The significant factors for movie hit are the movie itself is attractive or not. Therefore, we can consider the hit for each movie individually. \n\n In this study, we consider the hit phenomena of movies with both experimental and theoretical ways. For experimental viewpoint, we observe daily revenue data and daily blog-writing data for many movies. We also obtain the advertisement cost for the movies. For theoretical viewpoint, we present here a mathematical model for hit phenomena as non-equilibrium, nonlinear and dynamical phenomena. Comparing the simulated revenue with the observed revenue for the movies checks the model. The model presented in this paper is based on the previous works.\\cite{ishii2005},\\cite{ishii2007}\n \n\\section{Observation of hit evidence}\n\n We observe three data of recent 25 popular movies for daily revenue, daily advertisement costs and the entry number of blog writing in Japanese market. The revenue data is obtained from Nikkan Kogyo Tsushinsya in Japan. The advertisement cost data are obtained from Dentsu inc. The blog writing data is collected using the site \"Kizasi\". \n \n First, as shown in fig.1, we found that the revenues decrease almost exponentially. This evidence is very natural, because the number of audience decrease monotonically due to the effect that person who watched the movie does not watch the same movie again. In fig.1, only the data E shows the sudden decrease. It happens due to the sudden accident of the actress of the movie. From the data, we found that the decay factors for most of all movies in fig.1 seems to be similar. The decay factor of the exponential decay is nearly 0.06 per day. It agrees well with the empirical rule of the Japanese movie market that the number of audience decrease roughly 6 percents.\n \n\\begin{figure}[htbp]\n\\begin{center}\n\\includegraphics[width=12cm,bb=0 0 667 471]{fig1.jpg}\n\\caption{Revenues of the movies are shown as functions of date. Revenue data is the value in Japanese market. The movie titles are as follows. A:Death Note, B:Death Note the last name, C: zoku Always, D: Close zero, E: Closed Note, F:Biohazard 3, G:Bushi no Ichibun, H:Ratatouille, I:fra-girl, J:Dainihonjin, K:Maiko haaaan!!, L:Koizora, M:HERO.}\n\\label{fig:fig1}\n\\end{center}\n\\end{figure} \n\n \n This exponential decrease can be explained easily by using a simple mathematical model. First, we define that the number of potential audience to be $N_0$ and the number of integrated audience at the time t to be $N(t)$. Thus, the number of people who have interest on the movie is $N_0-N(t)$. When we assume that the probability to watch the movie per one day is $a$, we obtain the equation to describe the number of audience to be \n\\begin{equation} \n\\label{eq:eq1}\n\\frac{dN(t)}{dt} = a(N_0 - N(t))\n\\end{equation}\nThe solution of the equation is\n\\begin{equation}\nN(t) = N_0 - Ce^{-at}\n\\end{equation}\nThus, when the initial condition is $N(t)=0$ at $t=0$, we obtain\n\\begin{equation}\nN(t) = N_0(1-e^{-at})\n\\end{equation}\nIt is clear that the result can be explain roughly the exponential decay of the audience shown in fig.1.\n\nNext, we compare the daily change of the revenue data with the number of the entry of the blog writing. The typical results are shown in figs.2-5. The data shows the exponential decay. The quasi-periodic enhancement corresponds to the weekend effect. The revenue of each movies increases at Saturday and Sunday. The weak peaks in the middle of the week correspond to the ladies day discount system at Wednesday that is very popular in Japan. From these figures, we found that the daily change of the revenue and the blog writing is very similar. This feature is found for all 25 movies. It means that people write their opinion on their blog with a certain probability. Figs2-3 shows us that the probability is almost constant.\n\nAccording to the similarity, we propose here that the number of blog-writing entry can be use as {\\it quasi-revenue} for each movie. This quasi-revenue is very useful, because the quasi-revenue can be defined even before the release day. Thus, we can catch the increasing purchase intention signal using the quasi-revenue.\n\n\\begin{figure}[htbp]\n\\begin{center}\n\\includegraphics[width=12cm,bb=0 0 300 240]{davinchi_best2br.jpg}\n\\caption{The daily data of revenue and the blog-writing entry number for the Da Vinci Code in Japanese market. The blog data are normalized to adjust the peak of the revenue at the release day.}\n\\label{fig:fig2}\n\\end{center}\n\\end{figure} \n\n\\begin{figure}[htbp]\n\\begin{center}\n\\includegraphics[width=12cm,bb=0 0 834 667]{transformer31br.jpg}\n\\caption{The daily data of revenue and the blog-writing entry number for Transformer in Japanese market. The blog data are normalized to adjust the peak of the revenue at the release day.}\n\\label{fig:fig5}\n\\end{center}\n\\end{figure} \n\n\n\\section{Mathematical Model}\n\n\\subsection{Purchase intention for individual person}\n\nBased on the observation of the movie hit phenomena in Japanes market, we present a mathematical model to explain and predict hit phenomena. First, instead of the number of audience $N(t)$, we introduce here the integrated purchase intention of individual customer, $J_i(t)$ defined as follows,\n\n\\begin{equation} \n\\label{eq:eq3}\nN(t) = \\sum_i J_i(t)\n\\end{equation}\n\nhere the suffix j corresponds to individual person who has attention to the movie. Substitute (\\ref{eq:eq3}) into (\\ref{eq:eq1}), we obtain,\n\n\\begin{equation}\n\\frac{d}{dt}(\\sum_i J_i(t)) = a(N_0 - \\sum_i J_i(t)) = a\\sum_i (1-J_i(t))\n\\end{equation}\n\nThus, we obtain the equation for $J_i(t)$ as \n\\begin{equation}\n\\frac{dJ_i (t)}{dt} =a(1-J_i (t))\n\\end{equation}\n\nwhen $J_i (t)$ at$t=0$, we have the solution as\n\n\\begin{equation}\nJ_i(t) = 1-e^{-at}\n\\end{equation}\n\n\nThe daily purchase intention is defined from $J_i(t)$ as follows, \n\\begin{equation}\n\\frac{dJ_i(t)}{dt} = I_i(t)\n\\end{equation}\n\nThus, we can rewrite the equation (\\ref{eq:eq1}) to the equation for the purchase intention as the following way.\n\n\\begin{equation}\n\\frac{dI_i(t)}{dt} = -a I_i(t)\n\\end{equation}\n\nThis equation is the base of our mathematical model presented in the next subsection. The number of integrated audience can be calculated using the purchase intention as follows,\n\n\\begin{equation}\nN(t) = \\int_0^t \\sum_i I_i(\\tau) d\\tau\n\\end{equation}\n\nWe extend the purchase intention to the range before the movie release date, because the customer has each purchase intention even before the movie release because of the advertisement of the movie. The purchase intention of the individual person will increase rapidly toward the release date of the movie because of the concentrated advertising campaign. Since we pointed out in the previous section that the number of the entry of the blog-writing can be considered as the quasi-revenue, it is interesting to compare the quasi-revenue and the extended purchase intention before the release of the movie.\n\n Since the purchase intention of the individual customer increase due to both the advertisement and the communication with other persons, we construct a mathematical model for hit phenomena as the following equation. \n\n\\begin{equation}\n\\frac{dI_i(t)}{dt} = -aI_i(t) + advertisement + communication\n\\end{equation}\n\n\n\\subsection{advertisement}\n\nAdvertisement is the very important factor to increase the purchase intention of the customer in the market. Usually, the advertisement campaign is done at TV, newspaper and other medias. We consider the advertisement effect as an external force to the purchase intention as follows,\n\n\\begin{eqnarray}\n\\label{eq:eq12}\n\\frac{dI_i(t)}{dt} = -aI_i(t) + A(t) + \\sum_j D_{ij} I_j(t) \\nonumber \\\\\n+ \\sum_j \\sum_k P_{ijk} I_j(t) I_k(t)\n\\end{eqnarray}\n\nwhere $D_{ij}$ is the factor for the direct communication and $P_{ijk}$ is the factor for the indirect communication. Because of the term of the indirect communication, this equation is a nonlinear equation.\n\n\n\\subsection{Mean field approximation}\n\nTo solve the equation (\\ref{eq:eq12}), we introduce here the mean field approximation for simplicity. Namely, we assume that the every person moves equally so that we can introduce the averaged value of the individual purchase intention. \n\n\\begin{equation}\nI = \\frac{1}{N_p} \\sum_j I_j(t)\n\\end{equation}\n\nwhere introducing the number of potential audience $N_p$. We obtain the direct communication term from the person who do not watch the movie as follows,\n\n\\begin{eqnarray}\n\\sum_j D_{ij} I_j(t) = N_p \\frac{1}{N_p} \\sum_j D_{ij} I_j(t) \\nonumber \\\\\n\\Rightarrow \\frac{N_p -N(t)}{N_p} (N_p-N(t))D^{nn}I\n\\end{eqnarray}\n\nwhere $D^{nn}$ is the factor of the direct communication between the persons who do not watch the movie at the time $t$. \n\nSimilarly, we obtain the indirect communication term due to the communication between the person who do not watch the movie at the time $t$, \n\n\\begin{eqnarray}\n\\sum_j \\sum_k P_{ijk} I_j(t) I_k(t) = N_p \\frac{1}{N_p} \\sum_j N_p \\frac{1}{N_p} \\sum_k \\nonumber \\\\\n\\Rightarrow (\\frac{N_p-N(t)}{N_p})^3 N_p^2 P^{nn} I^2 = \\frac{(N_p-N(t))^3}{N_p} P^{nn} I^2\n\\end{eqnarray}\n\nwhere $P^{nn}$ is the factor of the indirect communication between the persons who do not watch the movie at the time $t$.\n\n For the direct communication between the watched person and the unwatched person can be written as follows,\n \n\\begin{equation}\n\\sum_j D_{ij} = N_p \\frac{1}{N_p} \\sum_j D_{ij} I_j(t) \\Rightarrow \\frac{N(t)}{N_p} (N_p-N(t)) D^{ny} I\n\\end{equation} \n\nwhere $D^{ny}$ is the factor of the direct communication between the watched person and the unwatched person. For the indirect communication, we obtain more two terms corresponding to the indirect communication due to the communication between the watched persons and that between the watched person and the unwatched person as follows,\n\n\\begin{equation}\n\\frac{(N(t))^2 (N_p-N(t))}{N_p} P^{yy} I^2 + \\frac{N(t) (N_p-N(t))^2}{N_p} P^{ny} I^2\n\\end{equation}\n\nwhere $P^{yy}$ is the factor of the indirect communication between the watched persons and $P^{ny}$ is the factor of the indirect communication due to the communication between the watched person and the unwatched person at the time $t$. \n\n Finally, we obtain the equation of the mathematical model for hit phenomena within the mean field approximation as follows,\n\n\\begin{eqnarray}\n\\label{eq:eq22}\n\\frac{I(t)}{dt} &=& -aI(t) + A(t) + \\frac{N_p -N(t)}{N_p} (N_p-N(t))D^{nn}I \\nonumber \\\\\n&& + \\frac{N(t)}{N_p} (N_p-N(t)) D^{ny} I \\nonumber \\\\ \n&& + \\frac{(N_p-N(t))^3}{N_p} P^{nn} I^2 \\nonumber \\\\\n&& + \\frac{(N(t))^2 (N_p-N(t))}{N_p} P^{yy} I^2 \\nonumber \\\\\n&&+ \\frac{N(t) (N_p-N(t))^2}{N_p} P^{ny} I^2\n\\end{eqnarray}\n\nwhere\n\n\\begin{equation}\n\\label{eq:eq23}\nN(t) = N_p \\int_0^t I(\\tau) d\\tau\n\\end{equation}\n\nThus, the equation (\\ref{eq:eq22}) with (\\ref{eq:eq23}) is the nonlinear integrodifferential equation. However, since the handling data is daily, the time difference is one day, we can solve the equation numerically as a difference equation.\n\n\n\\section{Calculated Results}\n\n Using the equation (\\ref{eq:eq22}) and (\\ref{eq:eq23}), we calculate the purchase intention for several movies where the advertisement cost presented from Dentsu is inputted into $A(t)$ with the unit of 1000 yen. The calculated results are shown in figs.6-8. The results are compared with the counts of the blog-writing entry as the quasi-revenue. We found that the agreement of the calculation with the quasi-revenue (blog) is very well. For the Da Vinci Code and the Pirates of Carribean at World's End, we have used almost same parameters except the initial value of the purchase intension. \n \n\\begin{figure}[htbp]\n\\begin{center}\n\\includegraphics[width=12cm,bb=0 0 1257 1015]{davinci12.jpg}\n\\caption{Daily data of the calculated purchase intention (green) and the observed number of blog-writing entry(red) for the Da Vinci Code in Japanese market. .}\n\\label{fig:fig6}\n\\end{center}\n\\end{figure} \n \n\\begin{figure}[htbp]\n\\begin{center}\n\\includegraphics[width=12cm,bb=0 0 1250 1001]{pirates02.jpg}\n\\caption{Daily data of the calculated purchase intention (green) and the observed number of blog-writing entry(red) for Pirates of Carribean At World's End in Japanese market. .}\n\\label{fig:fig7}\n\\end{center}\n\\end{figure} \n\n \n Interesting results are shown in figs.6 and 7. In these figures, we present the calculation for \"Always\" and \"Zoku Always\". The two movies are the part 1 and the part 2 of the same story. Though Always is the very successful movie, the information for the movie before the opening was almost nothing. Thus, the factors of the direct and indirect communication have very large values after the opening day. For the part 2, the Zoku Always, the factors are large values even before the opening. \n \n Finally, we show the components in the simulation of the Da Vince Code. The components shown in figure 8 correspond to the terms of eq(\\ref{eq:eq22}). the results shows us that the indirect communication terms are effective for the case of the Da Vinci Code.\n \n \n\\begin{figure}[htbp]\n\\begin{center}\n\\includegraphics[width=12cm,bb=0 0 1250 1001]{always05.jpg}\n\\caption{Daily data of the calculated purchase intention (green) and the observed number of blog-writing entry(red) for \"Always\" in Japanese market.}\n\\label{fig:fig9}\n\\end{center}\n\\end{figure} \n\n\\begin{figure}[htbp]\n\\begin{center}\n\\includegraphics[width=12cm,bb=0 0 1250 1001]{zoku_always03.jpg}\n\\caption{Daily data of the calculated purchase intention (green) and the observed number of blog-writing entry(red) for \"Zoku Always\" in Japanese market.}\n\\label{fig:fig10}\n\\end{center}\n\\end{figure} \n\n\\begin{figure}[htbp]\n\\begin{center}\n\\includegraphics[width=12cm,bb=0 0 642 482]{always_components.jpg}\n\\caption{The decoupled components of the calculation of the daily purchase intention and the daily revenue data are shown for the Da Vinci Code in Japanese market. adv, d1, d2, in1, in2 and in3 correpond to the terms of eq(\\ref{eq:eq22})}\n\\label{fig:fig11}\n\\end{center}\n\\end{figure} \n\n\n\\section{Disucssion}\n\n With the agreement in figs4 and 5, we found that our model can reproduce the purchase intention very well. If we can assume that the attractions of the two movies are same, this success of the reproduction of the purchase intention means that our model can predict the revenue of the movies as a function of the advertisement cost. \n \n The validity of the mean field approximation we introduced can be confirmed in fig.9 where we compare the present calculation for Always with the calculation of the mathematical model for hit phenomena using the scale free network model as the network of the direct communication instead of the mean field approximation. The detail of model using the scale free network will be published elsewhere\\cite{Matsuda}. The indirect communication is not included in the calculation of the scale free network. The result shows us that the mean field approximation is farly good approximation, though the tail of the counts of blog entry can have the information of the human communication in the society.\n \n\\begin{figure}[htbp]\n\\begin{center}\n\\includegraphics[width=12cm,bb=0 0 978 811]{scalefree.jpg}\n\\caption{The comparison of the calculation of the mathematical model for hit phenomena using the mean field approximation and the scale free network are shown for the Da Vinci Code in Japanese market. }\n\\label{fig:fig12}\n\\end{center}\n\\end{figure} \n\n Therefore, we can conclude that the mathematical model presented in this paper can simulate hit phenomena at least for movie entertainment. It is very important meanings in the marketing science, because it is possible to predict whether movie will hit or not. However, we should mind that the case of the movie entertainment is very simple case as hit phenomena, because there ara no competition with other movies, in principle. No one who cannot watch the Da Vinci Code because of occupied seats tries to watch Harry Potter. Each movie can be considered almost individually. In order to predict hit of other products like car, computer, foods or fashions, we should take into account the competition with other rival products in the same market. \n \n Our mathematical model of hit phenomena include the indirect communication term. As we shown in ref.\\cite{ishii2007}, if we neglect the indirect communication term in our model, we obtain easily the Bass model which has been known as the model of diffusion of informations by word of mouth\\cite{Bass1, Bass2, Bass3}. Since the effect of the indirect communication cannot be neglect in the adjustment of the simulation with the real blog data, we found that the Bass model cannot reproduce the big hit phenomena because of the lack of the nonlinear indirect communication term.\n\n\\clearpage \n\n\n\\section{Conclusion}\n\n We found the counts of blog-writing entry is very similar to the revenue of corresponding movie. The counts of blog-writing entry can be used as quasi-revenue. The mathematical model for hit phenomena is presented including the advertisement cost and the communication effect. In the communication effect, we include both the direct communication and the indirect communication. The results calculated with the model can predict the revenue of corresponding movie very well. The conclusion presented in this paper is very useful also in marketing science.\n \n \n \n \n\n\\section*{Acknowledgements}\nThe advertisement cost data is presented by Dentsu Inc. The revenue data is presented from the Nikkan Kogyo Tsushinsya. The research is partially supported by the Hit Content Laboratory Inc.\n\n\n\n\n\n\n\n\\section{Introduction}\n\nThis is the author's guide to \\revtex~4, the preferred submission\nformat for all APS journals. This guide is intended to be a concise\nintroduction to \\revtex~4. The documentation has been separated out\ninto smaller units to make it easier to locate essential\ninformation.\n\nThe following documentation is also part of the APS \\revtex~4\ndistribution. Updated versions of these will be maintained at\nthe \\revtex~4 homepage located at \\url{http:\/\/publish.aps.org\/revtex4\/}.\n\\begin{itemize}\n\\item \\textit{APS Compuscript Guide for \\revtex~4}\n\\item \\textit{\\revtex~4 Command and Options Summary}\n\\item \\textit{\\revtex~4 Bib\\TeX\\ Guide}\n\\item \\textit{Differences between \\revtex~4 and \\revtex~3}\n\\end{itemize}\nThis guide assumes a working \\revtex~4\ninstallation. Please see the installation guide included with the\ndistribution.\n\nThe \\revtex\\ system for \\LaTeX\\ began its development in 1986 and has\ngone through three major revisions since then. All versions prior to\n\\revtex~4 were based on \\LaTeX2.09 and, until now, \\revtex\\ did not\nkeep pace with the advances of the \\LaTeX\\ community and thus became\ninconvenient to work with. \\revtex~4 is designed to remedy this by\nincorporating the following design goals:\n\n\\begin{itemize}\n\\item\nMake \\revtex\\ fully compatible with \\LaTeXe; it is now a \\LaTeXe\\\ndocument class, similar in function to the standard\n\\classname{article} class.\n\n\\item\nRely on standard \\LaTeXe\\ packages for common tasks, e.g,\n\\classname{graphicx},\n\\classname{color}, and\n\\classname{hyperref}.\n\n\\item\nAdd or improve macros to support translation to tagged formats such as\nXML and SGML. This added markup will be key to enhancing the\npeer-review process and lowering production costs.\n\n\\item\nProvide a closer approximation to the typesetting style used in\n\\emph{Physical Review}.\n\n\\item\nIncorporate new features, such as hypertext, to make \\revtex\\ a\nconvenient and desirable e-print format.\n\n\\item\nRelax the restrictions in \\revtex\\ that had only been necessary for\ntypesetting journal camera-ready copy.\n\\end{itemize}\n\nTo meet these goals, \\revtex~4 is a complete rewrite with an emphasis\non maintainability so that it will be easier to provide enhancements.\n\nThe \\revtex~4 distribution includes both a template\n(\\file{template.aps}) and a sample document (\\file{apssamp.tex}).\nThe template is a good starting point for a manuscript. In the\nfollowing sections are instructions that should be sufficient for\ncreating a paper using \\revtex~4.\n\n\\subsection{Submitting to APS Journals}\n\nAuthors using \\revtex~4 to prepare a manuscript for submission to\n\\textit{Physical Review} or \\textit{Reviews of Modern Physics} \nmust also read the companion document \\textit{APS Compuscript Guide\nfor \\revtex~4}\ndistributed with \\revtex\\ and follow the guidelines detailed there.\n\nFurther information about the compuscript program of the American\nPhysical Society may be found at \\url{http:\/\/publish.aps.org\/ESUB\/}.\n\n\\subsection{Contact Information}\\label{sec:resources}%\nAny bugs, problems, or inconsistencies should reported to\n\\revtex\\ support at \\verb+revtex@aps.org+.\nReports should include information on the error and a \\textit{small}\nsample document that manifests the problem if possible (please don't\nsend large files!).\n\n\\section{Some \\LaTeXe\\ Basics}\nA primary design goal of \\revtex~4 was to make it as compatible with\nstandard \\LaTeXe\\ as possible so that authors may take advantage of all\nthat \\LaTeXe\\ offers. In keeping with this goal, much of the special\nformatting that was built in to earlier versions of \\revtex\\ is now\naccomplished through standard \\LaTeXe\\ macros or packages. The books\nin the bibliography provide extensive coverage of all topics\npertaining to preparing documents under \\LaTeXe. They are highly recommended.\n\nTo accomplish its goals, \\revtex~4 must sometimes patch the underlying\n\\LaTeX\\ kernel. This means that \\revtex~4 requires a fairly recent version of\n\\LaTeXe. Versions prior to 1996\/12\/01 may not work\ncorrectly. \\revtex~4 will be maintained to be compatible with future\nversions of \\LaTeXe.\n\n\\subsection{Useful \\LaTeXe\\ Markup}\n\\LaTeXe\\ markup is the preferred way to accomplish many basic tasks.\n\n\\subsubsection{Fonts}\n\nBecause \\revtex~4 is based upon \\LaTeXe, it inherits all of the\nmacros used for controlling fonts. Of particular importance are the\n\\LaTeXe\\ macros \\cmd{\\textit}, \\cmd{\\textbf}, \\cmd{\\texttt} for changing to\nan italic, bold, or typewriter font respectively. One should always\nuse these macros rather than the lower-level \\TeX\\ macros \\cmd{\\it},\n\\cmd{\\bf}, and \\cmd{\\tt}. The \\LaTeXe\\ macros offer\nimprovements such as better italic correction and scaling in super-\nand subscripts for example. Table~\\ref{tab:fonts}\nsummarizes the font selection commands in \\LaTeXe.\n\n\\begin{table}\n\\caption{\\label{tab:fonts}\\LaTeXe\\ font commands}\n\\begin{ruledtabular}\n\\begin{tabular}{ll}\n\\multicolumn{2}{c}{\\textbf{Text Fonts}}\\\\\n\\textbf{Font command} & \\textbf{Explanation} \\\\\n\\cmd\\textit\\marg{text} & Italics\\\\\n\\cmd\\textbf\\marg{text} & Boldface\\\\\n\\cmd\\texttt\\marg{text} & Typewriter\\\\\n\\cmd\\textrm\\marg{text} & Roman\\\\\n\\cmd\\textsl\\marg{text} & Slanted\\\\\n\\cmd\\textsf\\marg{text} & Sans Serif\\\\\n\\cmd\\textsc\\marg{text} & Small Caps\\\\\n\\cmd\\textmd\\marg{text} & Medium Series\\\\\n\\cmd\\textnormal\\marg{text} & Normal Series\\\\\n\\cmd\\textup\\marg{text} & Upright Series\\\\\n &\\\\\n\\multicolumn{2}{c}{\\textbf{Math Fonts}}\\\\\n\\cmd\\mathit\\marg{text} & Math Italics\\\\\n\\cmd\\mathbf\\marg{text} & Math Boldface\\\\\n\\cmd\\mathtt\\marg{text} & Math Typewriter\\\\\n\\cmd\\mathsf\\marg{text} & Math Sans Serif\\\\\n\\cmd\\mathcal\\marg{text} & Calligraphic\\\\\n\\cmd\\mathnormal\\marg{text} & Math Normal\\\\\n\\cmd\\bm\\marg{text}& Bold math for Greek letters\\\\\n & and other symbols\\\\\n\\cmd\\mathfrak\\marg{text}\\footnotemark[1] & Fraktur\\\\\n\\cmd\\mathbb\\marg{text}\\footnotemark[1] & Blackboard Bold\\\\\n\\end{tabular}\n\\end{ruledtabular}\n\\footnotetext[1]{Requires \\classname{amsfonts} or \\classname{amssymb} class option}\n\\end{table}\n\n\\subsubsection{User-defined macros}\n\\LaTeXe\\ provides several macros that enable users to easily create new\nmacros for use in their manuscripts:\n\\begin{itemize}\n\\footnotesize\n\\item \\cmd\\newcommand\\marg{\\\\command}\\oarg{narg}\\oarg{opt}\\marg{def} \n\\item \\cmd\\newcommand\\verb+*+\\marg{\\\\command}\\oarg{narg}\\oarg{opt}\\marg{def}\n\\item \\cmd\\renewcommand\\marg{\\\\command}\\oarg{narg}\\oarg{opt}\\marg{def}\n\\item \\cmd\\renewcommand\\verb+*+\\marg{\\\\command}\\oarg{narg}\\oarg{opt}\\marg{def}\n\\item \\cmd\\providecommand\\marg{\\\\command}\\oarg{narg}\\oarg{opt}\\marg{def}\n\\item \\cmd\\providecommand\\verb+*+\\marg{\\\\command}\\oarg{narg}\\oarg{opt}\\marg{def}\n\\end{itemize}\nHere \\meta{\\\\command} is the name of the macro being defined,\n\\meta{narg} is the number of arguments the macro takes,\n\\meta{opt} are optional default values for the arguments, and\n\\meta{def} is the actually macro definiton. \\cmd\\newcommand\\ creates a\nnew macro, \\cmd\\renewcommand\\ redefines a previously defined macro,\nand \\cmd\\providecommand\\ will define a macro only if it hasn't\nbeen defined previously. The *-ed versions are an optimization that\nindicates that the macro arguments will always be ``short'' arguments. This is\nalmost always the case, so the *-ed versions should be used whenver\npossible.\n\nThe use of these macros is preferred over using plain \\TeX's low-level\nmacros such as\n\\cmd\\def{},\\cmd\\edef{}, and \\cmd\\gdef{}. APS authors must follow the\n\\textit{APS Compuscript Guide for \\revtex~4} when defining macros.\n\n\\subsubsection{Symbols}\n\n\\LaTeXe\\ has added some convenient commands for some special symbols\nand effects. These are summarized in Table~\\ref{tab:special}. See\n\\cite{Guide} for details.\n\n\\begin{table}\n\\caption{\\label{tab:special}\\LaTeXe\\ commands for special symbols and effects}\n\\begin{ruledtabular}\n\\begin{tabular}{lc}\nCommand & Symbol\/Effect\\\\\n\\cmd\\textemdash & \\textemdash\\\\\n\\cmd\\textendash & \\textendash\\\\\n\\cmd\\textexclamdown & \\textexclamdown\\\\\n\\cmd\\textquestiondown & \\textquestiondown\\\\\n\\cmd\\textquotedblleft & \\textquotedblleft\\\\\n\\cmd\\textquotedblright & \\textquotedblright\\\\\n\\cmd\\textquoteleft & \\textquoteleft\\\\\n\\cmd\\textquoteright & \\textquoteright\\\\\n\\cmd\\textbullet & \\textbullet\\\\\n\\cmd\\textperiodcentered & \\textperiodcentered\\\\\n\\cmd\\textvisiblespace & \\textvisiblespace\\\\\n\\cmd\\textcompworkmark & Break a ligature\\\\\n\\cmd\\textcircled\\marg{char} & Circle a character\\\\\n\\end{tabular}\n\\end{ruledtabular}\n\\end{table}\n\n\\LaTeXe\\ also removed some symbols that were previously automatically\navailable in \\LaTeX 2.09. These symbols are now contained in a\nseparate package \\classname{latexsym}. To use these symbols, include\nthe package using:\n\\begin{verbatim}\n\\usepackage{latexsym}\n\\end{verbatim}\n\n\\subsection{Using \\LaTeXe\\ packages with \\revtex}\\label{sec:usepackage}%\n\nMany \\LaTeXe\\ packages are available, for instance, on CTAN at\n\\url{ftp:\/\/ctan.tug.org\/tex-archive\/macros\/latex\/required\/}\nand at\n\\url{ftp:\/\/ctan.tug.org\/tex-archive\/macros\/latex\/contrib\/}\nor may be available on other distribution media, such as the \\TeX\\\nLive CD-ROM \\url{http:\/\/www.tug.org\/texlive\/}. Some of these packages\nare automatically loaded by \\revtex~4 when certain class options are\ninvoked and are, thus, ``required''. They will either be distributed\nwith \\revtex\\ or are already included with a standard \\LaTeXe\\\ndistribution.\n\nRequired packages are automatically loaded by \\revtex\\ on an as-needed\nbasis. Other packages should be loaded using the\n\\cmd\\usepackage\\ command. To load the\n\\classname{hyperref} package, the document preamble might look like:\n\\begin{verbatim}\n\\documentclass{revtex}\n\\usepackage{hyperref}\n\\end{verbatim}\n\nSome common (and very useful) \\LaTeXe\\ packages are \\textit{a priori}\nimportant enough that \\revtex~4 has been designed to be specifically\ncompatible with them. \nA bug stemming from the use of one of these packages in\nconjunction with any of the APS journals may be reported by contacting\n\\revtex\\ support.\n\\begin{description}\n\\item[\\textbf{AMS packages}] \\revtex~4 is compatible with and depends\n upon the AMS packages\n\\classname{amsfonts},\n\\classname{amssymb}, and\n\\classname{amsmath}. In fact, \\revtex~4 requires use of these packages\nto accomplish some common tasks. See Section~\\ref{sec:math} for more.\n\\revtex~4 requires version 2.0 or higher of the AMS-\\LaTeX\\ package.\n\n\\item[\\textbf{array and dcolumn}]\nThe \\classname{array} and \\classname{dcolumn} packages are part of\n\\LaTeX's required suite of packages. \\classname{dcolumn} is required\nto align table columns on decimal points (and it in turn depends upon\nthe \\classname{array} package).\n\n\\item[\\textbf{longtable}]\n\\file{longtable.sty} may be used for large tables that will span more than one\npage. \\revtex~4 dynamically applies patches to longtable.sty so that\nit will work in two-column mode.\n\n\\item[\\textbf{hyperref}] \\file{hyperref.sty} is a package by Sebastian Rahtz that is\nused for putting hypertext links into \\LaTeXe\\ documents.\n\\revtex~4 has hooks to allow e-mail addresses and URL's to become\nhyperlinks if \\classname{hyperref} is loaded.\n\\end{description}\n\nOther packages will conflict with \\revtex~4 and should be\navoided. Usually such a conflict arises because the package adds\nenhancements that \\revtex~4 already includes. Here are some common\npackages that clash with \\revtex~4:\n\\begin{description}\n\\item[\\textbf{multicol}] \\file{multicol.sty} is a package by Frank Mittelbach\nthat adds support for multiple columns. In fact, early versions of\n\\revtex~4 used \\file{multicol.sty} for precisely this. However, to\nimprove the handling of floats, \\revtex~4 now has its own macros for\ntwo-column layout. Thus, it is not necessary to use \\file{multicol.sty}.\n\n\\item[\\textbf{cite}] Donald Arseneau's \\file{cite.sty} is often used to provide\nsupport for sorting a \\cmd\\cite\\ command's arguments into numerical\norder and to collapse consecutive runs of reference numbers. \\revtex~4\nhas this functionality built-in already via the \\classname{natbib} package.\n\n\\item[\\textbf{endfloat}] The same functionality can be accomplished\nusing the \\classoption{endfloats} class option.\n\n\\item[\\textbf{float}] \\revtex~4 already contains a lot of this\nfunctionality.\n\\end{description}\n\n\\section{The Document Preamble}\n\nThe preamble of a \\LaTeX\\ document is the set of commands that precede\nthe \\envb{document} line. It contains a\n\\cmd\\documentclass\\ line to load the \\revtex~4 class (\\textit{i.~e.},\nall of the \\revtex~4 macro definitions), \\cmd\\usepackage\\ macros to\nload other macro packages, and other macro definitions.\n\n\\subsection{The \\emph{documentclass} line}\nThe basic formatting of the manuscript is controlled by setting\n\\emph{class options} using\n\\cmd\\documentclass\\oarg{options}\\aarg{\\classname{revtex4}}.\nThe macro \\cmd\\documentclass\\ \nreplaces the \\cmd\\documentstyle\\ macro of \\LaTeX2.09. The optional\narguments that appear in the square brackets control the layout of the\ndocument. At this point, one only needs to choose a journal style\n(\\classoption{pra}, \\classoption{prb},\n\\classoption{prc}, \\classoption{prd},\n\\classoption{pre}, \\classoption{prl}, \\classoption{prstab},\nand \\classoption{rmp}) and either \\classoption{preprint} or\n\\classoption{twocolumn}. Usually, one would want to use\n\\classoption{preprint} for draft papers. \\classoption{twocolumn} gives\nthe \\emph{Physical Review} look and feel. Paper size options are also\navailable as well. In particular, \\classoption{a4paper} is available\nas well as the rest of the standard \\LaTeX\\ paper sizes. A\nfull list of class options is given in the \\textit{\\revtex~4 Command\nand Options Summary}.\n\n\\subsection{Loading other packages}\nOther packages may be loaded into a \\revtex~4 document by using the\nstandard \\LaTeXe\\ \\cmd\\usepackage\\ command. For instance, to load\nthe \\classoption{graphics} package, one would use\n\\verb+\\usepackage{graphics}+.\n\n\\section{The Front Matter}\\label{sec:front}\n\nAfter choosing the basic look and feel of the document by selecting\nthe appropriate class options and loading in whatever other macros are\nneeded, one is ready to move on to creating a new manuscript. After\nthe preamble, be sure to put in a \\envb{document} line (and put\nin an \\enve{document} as well). This section describes the macros\n\\revtex~4 provides for formatting the front matter of the\narticle. The behavior and usage of these macros can be quite\ndifferent from those provided in either \\revtex~3 or \\LaTeXe. See the\nincluded document \\textit{Differences between \\revtex~4 and \\revtex~3} for an\noverview of these differences.\n\n\\subsection{Setting the title}\n\nThe title of the manuscript is simply specified by using the\n\\cmd\\title\\aarg{title} macro. A \\verb+\\\\+ may be used to put a line\nbreak in a long title.\n\n\\subsection{Specifying a date}%\n\nThe \\cmd\\date\\marg{date} command outputs the date on the\nmanuscript. Using \\cmd\\today\\ will cause \\LaTeX{} to insert the\ncurrent date whenever the file is run:\n\\begin{verbatim}\n\\date{\\today}\n\\end{verbatim}\n\n\\subsection{Specifying authors and affiliations}\n\nThe macros for specifying authors and their affiliations have\nchanged significantly for \\revtex~4. They have been improved to save\nlabor for authors and in production. Authors and affiliations are\narranged into groupings called, appropriately enough, \\emph{author\ngroups}. Each author group is a set of authors who share the same set\nof affiliations. Author names are specified with the \\cmd\\author\\\nmacro while affiliations (or addresses) are specified with the\n\\cmd\\affiliation\\ macro. Author groups are specified by sequences of\n\\cmd\\author\\ macros followed by \\cmd\\affiliation\\ macros. An\n\\cmd\\affiliation\\ macro applies to all previously specified\n\\cmd\\author\\ macros which don't already have an affiliation supplied.\n\nFor example, if Bugs Bunny and Roger Rabbit are both at Looney Tune\nStudios, while Mickey Mouse is at Disney World, the markup would be:\n\\begin{verbatim}\n\\author{Bugs Bunny}\n\\author{Roger Rabbit}\n\\affiliation{Looney Tune Studios}\n\\author{Mickey Mouse}\n\\affiliation{Disney World}\n\\end{verbatim}\nThe default is to display this as \n\\begin{center}\nBugs Bunny and Roger Rabbit\\\\\n\\emph{Looney Tune Studios}\\\\\nMickey Mouse\\\\\n\\emph{Disney World}\\\\\n\\end{center}\nThis layout style for displaying authors and their affiliations is\nchosen by selecting the class option\n\\classoption{groupedaddress}. This option is the default for all APS\njournal styles, so it does not need to be specified explicitly.\nThe other major way of displaying this\ninformation is to use superscripts on the authors and\naffiliations. This can be accomplished by selecting the class option\n\\classoption{superscriptaddress}. To achieve the display\n\\begin{center}\nBugs Bunny,$^{1}$ Roger Rabbit,$^{1,2}$ and Mickey Mouse$^{2}$\\\\\n\\emph{$^{1}$Looney Tune Studios}\\\\\n\\emph{$^{2}$Disney World}\\\\\n\\end{center}\none would use the markup\n\\begin{verbatim}\n\\author{Bugs Bunny}\n\\affiliation{Looney Tune Studios}\n\\author{Roger Rabbit}\n\\affiliation{Looney Tune Studios}\n\\affiliation{Disney World}\n\\author{Mickey Mouse}\n\\affiliation{Disney World}\n\\end{verbatim}\n\nNote that \\revtex~4 takes care of any commas and \\emph{and}'s that join\nthe author names together and font selection, as well as any\nsuperscript numbering. Only the author names and affiliations should\nbe given within their respective macros.\n\nThere is a third class option, \\classoption{unsortedaddress}, for\ncontrolling author\/affiliation display. The default\n\\classoption{groupedaddress} will actually sort authors into the\napproriate author groups if one chooses to specify an affiliation for\neach author. The markup:\n\\begin{verbatim}\n\\author{Bugs Bunny}\n\\affiliation{Looney Tune Studios}\n\\author{Mickey Mouse}\n\\affiliation{Disney World}\n\\author{Roger Rabbit}\n\\affiliation{Looney Tune Studios}\n\\end{verbatim}\nwill result in the same display as for the first case given\nabove even though Roger Rabbit is specified after Mickey Mouse. To\navoid Roger Rabbit being moved into the same author group as Bugs\nBunny, use the\n\\classoption{unsortedaddress} option instead. In general, it is safest\nto list authors in the order they should appear and specify\naffiliations for multiple authors rather than one at a time. This will\nafford the most independence for choosing the display option. Finally,\nit should be mentioned that the affiliations for the\n\\classoption{superscriptaddress} are presented and numbered \nin the order that they are encountered. These means that the order\nwill usually follow the order of the authors. An alternative ordering\ncan be forced by including a list of \\cmd\\affiliation\\ commands before\nthe first \\cmd{\\author} in the desired order. Then use the exact same\ntext for each affilation when specifying them for each author.\n\nIf an author doesn't have an affiliation, the \\cmd\\noaffiliation\\\nmacro may be used in the place of an \\cmd\\affiliation\\ macro.\n\n\n\\subsubsection{Collaborations}\n\nA collaboration name can be specified with the \\cmd\\collaboration\\\nmacro. This is very similar to the \\cmd\\author\\ macro, but it can only\nbe used with the class option \\classoption{superscriptaddress}. The\n\\cmd\\collaboration\\ macro should appear at the end of the list of\nauthors. The collaboration name will be appear centered in parentheses\nbetween the list of authors and the list of\naffiliations. Because collaborations\ndon't normally have affiliations, one needs to follow the\n\\cmd\\collaboration\\ with \\cmd\\noaffiliation.\n\n\\subsubsection{Footnotes for authors, collaborations, affiliations or title}\\label{sec:footau}\n\nOften one wants to specify additional information associated with an\nauthor, collaboration, or affiliation such an e-mail address, an\nalternate affiliation, or some other anicillary information. \n\\revtex~4 introduces several new macros just for this purpose. They\nare:\n\\begin{itemize}\n\\item\\cmd\\email\\oarg{optional text}\\aarg{e-mail address}\n\\item\\cmd\\homepage\\oarg{optional text}\\aarg{URL}\n\\item\\cmd\\altaffiliation\\oarg{optional text}\\aarg{affiliation}\n\\item\\cmd\\thanks\\aarg{miscellaneous text}\n\\end{itemize}\nIn the first three, the \\emph{optional text} will be prepended before the\nactual information specified in the required argument. \\cmd\\email\\ and\n\\cmd\\homepage\\ each have a default text for their optional arguments\n(`Electronic address:' and `URL:' respectively). The \\cmd\\thanks\\\nmacro should only be used if one of the other three do not apply. Any\nauthor name can have multiple occurences of these four macros. Note\nthat unlike the\n\\cmd\\affiliation\\ macro, these macros only apply to the \\cmd\\author\\\nthat directly precedes it. Any \\cmd\\affiliation\\ \\emph{must} follow\nthe other author-specific macros. A typical usage might be as follows:\n\\begin{verbatim}\n\\author{Bugs Bunny}\n\\email[E-mail me at: ]{bugs@looney.com}\n\\homepage[Visit: ]{http:\/\/looney.com\/}\n\\altaffiliation[Permanent address: ]\n {Warner Brothers}\n\\affiliation{Looney Tunes}\n\\end{verbatim}\nThis would result in the footnote ``E-mail me at: \\texttt{bugs@looney.com},\nVisit: \\texttt{http:\/\/looney.com\/}, Permanent address: Warner\nBrothers'' being attached to Bugs Bunny. Note that:\n\\begin{itemize}\n\\item Only an e-mail address, URL, or affiliation should go in the\nrequired argument in the curly braces.\n\\item The font is automatically taken care of.\n\\item An explicit space is needed at the end of the optional text if one is\ndesired in the output.\n\\item Use the optional arguments to provide customized\ntext only if there is a good reason to.\n\\end{itemize}\n\nThe \\cmd\\collaboration\\ , \\cmd\\affiliation\\ , or even \\cmd\\title\\ can\nalso have footnotes attached via these commands. If any ancillary data\n(\\cmd\\thanks, \\cmd\\email, \\cmd\\homepage, or\n\\cmd\\altaffiliation) are given in the wrong context (e.g., before any\n\\cmd\\title, \\cmd\\author, \\cmd\\collaboration, or \\cmd\\affiliation\\\ncommand has been given), then a warning is given in the \\TeX\\ log, and\nthe command is ignored.\n\nDuplicate sets of ancillary data are merged, giving rise to a single\nshared footnote. However, this only applies if the ancillary data are\nidentical: even the order of the commands specifying the data must be\nidentical. Thus, for example, two authors can share a single footnote\nindicating a group e-mail address.\n\nDuplicate \\cmd\\affiliation\\ commands may be given in the course of the\nfront matter, without the danger of producing extraneous affiliations\non the title page. However, ancillary data should be specified for\nonly the first instance of any particular institution's\n\\cmd\\affiliation\\ command; a later instance with different ancillary\ndata will result in a warning in the \\TeX\\ log.\n\nIt is preferable to arrange authors into\nsets. Within each set all the authors share the same group of\naffiliations. For each author, give the \\cmd\\author\\ (and appropriate\nancillary data), then follow this author group with the needed group\nof \\cmd\\affiliation\\ commands.\n\nIf affiliations have been listed before the first\n\\cmd\\author\\ macro to ensure a particular ordering, be sure\nthat any later \\cmd\\affiliation\\ command for the given institution is\nan exact copy of the first, and also ensure that no ancillary data is\ngiven in these later instances.\n\n\nEach APS journal has a default behavior for the placement of these\nancillary information footnotes. The \\classoption{prb} option puts all\nsuch footnotes at the start of the bibliography while the other\njournal styles display them on the first page. One can override a\njournal style's default behavior by specifying explicitly the class\noption\n\\classoption{bibnotes} (puts the footnotes at the start of the\nbibliography) or \\classoption{nobibnotes} (puts them on the first page).\n\n\\subsubsection{Specifying first names and surnames}\n\nMany APS authors have names in which either the surname appears first\nor in which the surname is made up of more than one name. To ensure\nthat such names are accurately captured for indexing and other\npurposes, the \\cmd\\surname\\ macro should be used to indicate which portion\nof a name is the surname. Similarly, there is a \\cmd\\firstname\\ macro\nas well, although usage of \\cmd\\surname\\ should be sufficient. If an\nauthor's surname is a single name and written last, it is not\nnecessary to use these macros. These macros do nothing but indicate\nhow a name should be indexed. Here are some examples;\n\\begin{verbatim}\n\\author{Andrew \\surname{Lloyd Weber}}\n\\author{\\surname{Mao} Tse-Tung}\n\\end{verbatim}\n\n\\subsection{The abstract}\nAn abstract for a paper is specified by using the \\env{abstract}\nenvironment:\n\\begin{verbatim}\n\\begin{abstract}\nText of abstract\n\\end{abstract}\n\\end{verbatim}\nNote that in \\revtex~4 the abstract must be specified before the\n\\cmd\\maketitle\\ command and there is no need to embed it in an explicit\nminipage environment.\n\n\\subsection{PACS codes}\nAPS authors are asked to supply suggested PACS codes with their\nsubmissions. The \\cmd\\pacs\\ macro is provided as a way to do this:\n\\begin{verbatim}\n\\pacs{23.23.+x, 56.65.Dy}\n\\end{verbatim}\nThe actual display of the PACS numbers below the abstract is\ncontrolled by two class options: \\classoption{showpacs} and\n\\classoption{noshowpacs}. In particular, this is now independent of\nthe \\classoption{preprint} option. \\classoption{showpacs} must be\nexplicitly included in the class options to display the PACS codes.\n\n\\subsection{Keywords}\nA \\cmd\\keywords\\ macro may also be used to indicate keywords for the\narticle. \n\\begin{verbatim}\n\\keywords{nuclear form; yrast level}\n\\end{verbatim}\nThis will be displayed below the abstract and PACS (if supplied). Like\nPACS codes, the actual display of the the keywords is controlled by\ntwo classoptions: \\classoption{showkeys} and\n\\classoption{noshowkeys}. An explicit \\classoption{showkeys} must be\nincluded in the \\cmd\\documentclass\\ line to display the keywords.\n\n\\subsection{Institutional report numbers}\nInstitutional report numbers can be specified using the \\cmd\\preprint\\\nmacro. These will be displayed in the upper lefthand corner of the\nfirst page. Multiple \\cmd\\preprint\\ macros maybe supplied (space is\nlimited though, so only three or less may actually fit). \n\n\\subsection{maketitle}\nAfter specifying the title, authors, affiliations, abstract, PACS\ncodes, and report numbers, the final step for formatting the front\nmatter of the manuscript is to execute the \\cmd\\maketitle\\ macro by\nsimply including it:\n\\begin{verbatim}\n\\maketitle\n\\end{verbatim}\nThe \\cmd\\maketitle\\ macro must follow all of the macros listed\nabove. The macro will format the front matter in accordance with the various\nclass options that were specified in the\n\\cmd\\documentclass\\ line (either implicitly through defaults or\nexplicitly).\n\n\\section{The body of the paper}\n\nFor typesetting the body of a paper, \\revtex~4 relies heavily on\nstandard \\LaTeXe\\ and other packages (particulary those that are part\nof AMS-\\LaTeX). Users unfamiliar with these packages should read the\nfollowing sections carefully. \n\n\\subsection{Section headings}\n\nSection headings are input as in \\LaTeX.\nThe output is similar, with a few extra features.\n\nFour levels of headings are available in \\revtex{}:\n\\begin{quote}\n\\cmd\\section\\marg{title text}\\\\\n\\cmd\\subsection\\marg{title text}\\\\\n\\cmd\\subsubsection\\marg{title text}\\\\\n\\cmd\\paragraph\\marg{title text}\n\\end{quote}\n\nUse the starred form of the command to suppress the automatic numbering; e.g.,\n\\begin{verbatim}\n\\section*{Introduction}\n\\end{verbatim}\n\nTo label a section heading for cross referencing, best practice is to\nplace the \\cmd\\label\\marg{key} within the argument specifying the heading:\n\\begin{verbatim}\n\\section{\\label{sec:intro}Introduction}\n\\end{verbatim}\n\nIn the some journal substyles, such as those of the APS,\nall text in the \\cmd\\section\\ command is automatically set uppercase.\nIf a lowercase letter is needed, use \\cmd\\lowercase\\aarg{x}.\nFor example, to use ``He'' for helium in a \\cmd\\section\\marg{title text} command, type\n\\verb+H+\\cmd\\lowercase\\aarg{e} in \\marg{title text}.\n\nUse \\cmd\\protect\\verb+\\\\+ to force a line break in a section heading.\n(Fragile commands must be protected in section headings, captions, and\nfootnotes and \\verb+\\\\+ is a fragile command.)\n\n\\subsection{Paragraphs and General Text}\n\nParagraphs always end with a blank input line. Because \\TeX\\\nautomatically calculates linebreaks and word hyphenation in a\nparagraph, it is not necessary to force linebreaks or hyphenation. Of\ncourse, compound words should still be explicitly hyphenated, e.g.,\n``author-prepared copy.''\n\nUse directional quotes for quotation marks around quoted text\n(\\texttt{``xxx''}), not straight double quotes (\\texttt{\"xxx\"}).\nFor opening quotes, use one or two backquotes; for closing quotes,\nuse one or two forward quotes (apostrophes).\n\n\\subsection{One-column vs. two-column}\\label{sec:widetext}\n\nOne of the hallmarks of \\textit{Physical Review} is its two-column\nformatting and so one of the \\revtex~4 design goals is to make it easier to\nacheive the \\textit{Physical Review} look and feel. In particular, the\n\\classoption{twocolumn} option will take care of formatting the front matter\n(including the abstract) as a single column. \\revtex~4 has its own\nbuilt-in two-column formatting macros to provide well-balanced columns\nas well as reasonable control over the placement of floats in either\none- or two-column modes.\n\nOccasionally it is necessary to change the formatting from two-column to\none-column to better accomodate very long equations that are more\neasily read when typeset to the full width of the page. This is\naccomplished using the \\env{widetext} environment:\n\\begin{verbatim}\n\\begin{widetext}\nlong equation goes here\n\\end{widetext}\n\\end{verbatim}\nIn two-column mode, this will temporarily return to one-column mode,\nbalancing the text before the environment into two short columns, and\nreturning to two-column mode after the environment has\nfinished. \\revtex~4 will also add horizontal rules to guide the\nreader's eye through what may otherwise be a confusing break in the\nflow of text. The\n\\env{widetext} environment has no effect on the output under the \n\\classoption{preprint} class option because this already uses\none-column formatting.\n\nUse of the \\env{widetext} environment should be restricted to the bare\nminimum of text that needs to be typeset this way. However short pieces\nof paragraph text and\/or math between nearly contiguous wide equations\nshould be incorporated into the surrounding wide sections.\n\nLow-level control over the column grid can be accomplished with the\n\\cmd\\onecolumngrid\\ and \\cmd\\twocolumngrid\\ commands. Using these, one\ncan avoid the horizontal rules added by \\env{widetext}. These commands\nshould only be used if absolutely necessary. Wide figures and tables\nshould be accomodated using the proper \\verb+*+ environments.\n\n\\subsection{Cross-referencing}\\label{sec:xrefs}\n\n\\revtex{} inherits the \\LaTeXe\\ features for labeling and cross-referencing\nsection headings, equations, tables, and figures. This section\ncontains a simplified explanation of these cross-referencing features.\nThe proper usage in the context of section headings, equations,\ntables, and figures is discussed in the appropriate sections.\n\nCross-referencing depends upon the use of ``tags,'' which are defined by\nthe user. The \\cmd\\label\\marg{key} command is used to identify tags for\n\\revtex. Tags are strings of characters that serve to label section\nheadings, equations, tables, and figures that replace explicit,\nby-hand numbering.\n\nFiles that use cross-referencing (and almost all manuscripts do)\nneed to be processed through \\revtex\\ at least twice to\nensure that the tags have been properly linked to appropriate numbers.\nIf any tags are added in subsequent editing sessions, \n\\LaTeX{} will display a warning message in the log file that ends with\n\\texttt{... Rerun to get cross-references right}.\nRunning the file through \\revtex\\ again (possibly more than once) will\nresolve the cross-references. If the error message persists, check\nthe labels; the same \\marg{key} may have been used to label more than one\nobject.\n\nAnother \\LaTeX\\ warning is \\texttt{There were undefined references},\nwhich indicates the use of a key in a \\cmd\\ref\\ without ever\nusing it in a \\cmd\\label\\ statement.\n\n\\revtex{} performs autonumbering exactly as in standard \\LaTeX.\nWhen the file is processed for the first time,\n\\LaTeX\\ creates an auxiliary file (with the \\file{.aux} extension) that \nrecords the value of each \\meta{key}. Each subsequent run retrieves\nthe proper number from the auxiliary file and updates the auxiliary\nfile. At the end of each run, any change in the value of a \\meta{key}\nproduces a \\LaTeX\\ warning message.\n\nNote that with footnotes appearing in the bibliography, extra passes\nof \\LaTeX\\ may be needed to resolve all cross-references. For\ninstance, putting a \\cmd\\cite\\ inside a \\cmd\\footnote\\ will require at\nleast three passes.\n\nUsing the \\classname{hyperref} package to create hyperlinked PDF files\nwill cause reference ranges to be expanded to list every\nreference in the range. This behavior can be avoided by using the\n\\classname{hypernat} package available from \\url{www.ctan.org}.\n\n\\subsection{Acknowledgments}\nUse the \\env{acknowledgments} environment for an acknowledgments\nsection. Depending on the journal substyle, this element may be\nformatted as an unnumbered section title \\textit{Acknowledgments} or\nsimply as a paragraph. Please note the spelling of\n``acknowledgments''.\n\\begin{verbatim}\n\\begin{acknowlegments}\nThe authors would like to thank...\n\\end{acknowlegments}\n\\end{verbatim}\n\n\\subsection{Appendices}\nThe \\cmd\n\\section{Introduction}\nThis document gives a brief summary of how \\revtex~4 is different from\nwhat authors may already be familiar with. The two primary design\ngoals for \\revtex~4 are to 1) move to \\LaTeXe\\ and 2) improve the\nmarkup so that infomation can be more reliably extracted for the\neditorial and production processes. Both of these goals require that\nauthors comfortable with earlier versions of \\revtex\\ change their\nhabits. In addition, authors may already be familiar with the standard\n\\classname{article.cls} in \\LaTeXe. \\revtex~4 differs in some\nimportant ways from this class as well. For more complete\ndocumentation on \\revtex~4, see the main \\textit{\\revtex~4 Author's\nGuide}. The most important changes are in the markup of the front\nmatter (title, authors, affiliations, abstract, etc.). Please see\nSec.~\\ref{sec:front}.\n\n\\section{Version of \\LaTeX}\nThe most obvious difference between \\revtex~4 and \\revtex~3 is that\n\\revtex~4 works solely with \\LaTeXe; it is not useable as a \\LaTeX2.09 package.\nFurthermore, \\revtex~4 requires an up-to-date \\LaTeX\\ installation\n(1996\/06\/01 or later); its use under older versions is not supported.\n\n\\section{Class Options and Defaults}\nMany of the class options in \\revtex~3 have been retained in\n\\revtex~4. However, the default behavior for these options can be\ndifferent than in \\revtex~3. Currently, there is only one society\noption, \\classoption{aps}, and this is the default. Furthermore, the\nselection of a journal (such as \\classoption{prl}) will automatically\nset the society as well (this will be true even after other societies\nare added).\n\nIn \\revtex~3, it was necessary to invoke the \\classoption{floats}, but\nthis is the default for \\classoption{aps} journal in\n\\revtex~4. \\revtex~4 introduces two new class options,\n\\classoption{endfloats} and \\classoption{endfloats*} for moving floats\nto the end of the paper.\n\nThe preamble commands \\cmd{\\draft} and \\cmd{\\tighten} have been replaced\nwith new class options \\classoption{draft} and\n\\classoption{tightenlines}, respectively. The \\cmd{\\preprint} command\nis now used only for specifying institutional report numbers (typeset\nin the upper-righthand corner of the first page); it no longer\ninfluences whether PACS numbers are displayed below the abstract. PACS\ndisplay is controlled by the \\classoption{showpacs} and\n\\classoption{noshowpacs} (default) class options.\n\nPaper size options (\\classoption{letter}, \\classoption{a4paper}, etc.)\nwork in \\revtex~4. The text ``Typeset by \\revtex'' no longer appears\nby default - the option \\classoption{byrevtex} will place this text in\nthe lower-lefthand corner of the first page.\n\n\\section{One- and Two-column formatting}\n\n\\revtex~4 has excellent support for achieving the two-column\nformatting in the \\textit{Physical~Review} and \\textit{Reviews of\nModern Physics} styles. It will balance the columns\nautomatically. Whereas \\revtex~3 had the \\cmd{\\widetext} and\n\\cmd{\\narrowtext} commands for switching between one- and two-cloumn\nmodes, \\revtex~4 simply has a \\env{widetext} environment,\n\\envb{widetext} \\dots \\enve{widetext}. One-column formatting can be\nspecified by choosing either the \\classoption{onecolumn} or\n\\classoption{preprint} class option (the \\revtex~3 option\n\\classoption{manuscript} no longer exists). Two-column formatting is\nthe default for most journal styles, but can be specified with the\n\\classoption{twocolumn} option. Note that the spacing for\n\\classoption{preprint} is now set to 1.5, rather than full\ndouble-spacing. The \\classoption{tightenlines} option can be used to\nreduce this to single spacing.\n\n\n\\section{Front Matter Markup}\n\\label{sec:front}\n\n\\revtex~4 has substantially changed how the front matter for an article\nis marked up. These are the most significant differences between\n\\revtex~4 and other systems for typesetting manuscripts. It is\nessential that authors new to \\revtex~4 be familiar with these changes.\n\n\\subsection{Authors, Affiliations, and Author Notes}\n\\revtex~4 has substantially changed the markup of author names,\naffiliations, and author notes (footnotes giving additional\ninformation about the author such as a permanent address or an email\naddress).\n\\begin{itemize}\n\\item Each author name should appear separately in\nindividual \\cmd\\author\\ macros. \n\n\\item Email addresses should be marked up using the \\cmd\\email\\ macro.\n\n\\item Alternative affiliation information should be marked up using\nthe \\cmd\\altaffiliation\\ macro.\n\n\\item URLs for author home pages can be specified with a\n\\cmd\\homepage\\ macro.\n\n\\item The \\cmd\\thanks\\ macro should only be used if one of the above\ndon't apply.\n\n\\item \\cmd{\\email}, \\cmd{\\homepage}, \\cmd{\\altaffiliation}, and\n\\cmd{\\thanks} commands are grouped together under a single footnote for\neach author. These footnotes can either appear at the bottom of the\nfirst page of the article or as the first entries in the\nbibliography. The journal style controls this placement, but it may be\noverridden by using the class options \\classoption{bibnotes} and\n\\classoption{nobibnotes}. Note that these footnotes are treated\ndifferently than the other footnotes in the article.\n\n\\item The grouping of authors by affiliations is accomplished\nautomatically. Each affiliation should be in its own\n\\cmd{\\affiliation} command. Multiple \\cmd{\\affiliation},\n\\cmd{\\email}, \\cmd{\\homepage}, \\cmd{\\altaffiliation}, and \\cmd{\\thanks}\ncommands can be applied to each author. The macro \\cmd\\and\\ has been\neliminated.\n\n\\item \\cmd\\affiliation\\ commmands apply to all previous authors that\ndon't have an affiliation already declared. Furthermore, for any\nparticular author, the \\cmd\\affilation\\ must follow any \\cmd{\\email},\n\\cmd{\\homepage}, \\cmd{\\altaffiliation}, or \\cmd{\\thanks} commands for\nthat author.\n\n\\item Footnote-style associations of authors with affilitations should\nnot be done via explicit superscripts; rather, the class option\n\\classoption{superscriptaddress} should be used to accomplish this\nautomatically.\n\n\\item A collaboration for a group of authors can be given using the\n\\cmd\\collaboration\\ command.\n\n\\end{itemize}\n\nTable~\\ref{tab:front} summarizes some common pitfalls in moving from\n\\revtex~3 to \\revtex~4.\n\\begin{table*}\n\\begin{ruledtabular}\n\\begin{tabular}{lll}\n\\textbf{\\revtex~3 Markup} & \\textbf{\\revtex~4 Markup} & \\textbf{Explanation}\\\\\n& & \\\\\n\\verb+\\author{Author One and Author Two}+ & \\verb+\\author{Author One}+ & One name per\\\\\n& \\verb+\\author{Author Two}+ & \\verb+\\author+ \\\\\n& & \\\\\n\\verb+\\author{Author One$^{1}$}+ & \\verb+\\author{Author One}+& Use \\classoption{superscriptaddress}\\\\\n\\dots &\\dots & class option \\\\\n\\verb+\\address{$^{1}$APS}+ &\\verb+\\affiliation{APS}+ & \\\\\n& & \\\\\n\\verb+\\thanks{Permanent address...}+ & \\verb+\\altaffiliation{}+& Use most\nspecific macro \\\\\n\\verb+\\thanks{Electronic address: user@domain.edu}+ &\n\\verb+\\email{user@domain.edu}+& available\\\\\n\\verb+\\thanks{http:\/\/publish.aps.org\/}+ &\n\\verb+\\homepage{http:\/\/publish.aps.org\/}+& \\\\\n\\end{tabular}\n\\end{ruledtabular}\n\\caption{Common mistakes in marking up the front matter}\n\\label{tab:front}\n\\end{table*}\n\n\n\\subsection{Abstracts}\n\\revtex~4, like \\revtex~3, uses the \\env{abstract} environment\n\\envb{abstract} \\dots \\enve{abstract} for the abstract. The\n\\env{abstract} environment must appear before the \\cmd{\\maketitle}\ncommand in \\revtex~4. The abstract will be formatted\nappropriately for either one-column (preprint) or two-column\nformatting. In particular, in the two-column case, the abstract will\nautomatically be placed in a single column that spans the width of the\npage. It is unnecessary to use a \\cmd{\\minipage} or any other macro to\nachieve this result.\n\n\n\\section{Citations and References}\n\n\\revtex~4 uses the same \\cmd{\\cite},\\cmd{\\ref}, and \\cmd{\\bibitem}\ncommmands as standard \\LaTeX\\ and \\revtex~3. Citation handling is\nbased upon Patick Daly's \\classname{natbib} package. The\n\\env{references} environment is no longer used. Instead, use the\nstandard \\LaTeXe\\ environment \\env{thebibliography}.\n\nTwo new \\BibTeX\\ files have been included with \\revtex~4,\n\\file{apsrev.bst} and \\file{apsrmp.bst}. These will format references\nin the style of \\textit{Physical Review} and \\textit{Reviews of Modern\nPhysics} respectively. In addition, these \\BibTeX\\ styles\nautomatically apply a special macro \\cmd{\\bibinfo} to each element of the\nbibliography to make it easier to extract information for use in the\neditorial and production processes. Authors are strongly urged to use\n\\BibTeX\\ to manage their bibliographies so that the \\cmd{\\bibinfo}\ndirectives will be automatically included. Other bibliography styles\ncan be specified by using the \\cmd\\bibliographystyle\\ command, but\nunlike standard \\LaTeXe, you must give this command \\emph{before} the\n\\envb{document} statement.\n\nPlease note that the package \\classname{cite.sty} is not needed with\n\\revtex~4 and is incompatible.\n\n\\section{Footnotes and Tablenotes}\n\\label{sec:foot}\n\n\\revtex~4 uses the standard \\cmd{\\footnote} macro for\nfootnotes. Footnotes can either appear on the bottom of the page on\nwhich they occur or they can appear as entries at the end of the\nbibliography. As with author notes, the journal style option controls\nthe placement; however, this can be overridden with the class options\n\\classoption{footinbib} and \\classoption{nofootinbib}.\n\nWithin a table, the \\cmd{\\footnote} command behaves differently. Footnotes\nappear at the bottom of the table. \\cmd{\\footnotemark} and\n\\cmd{\\footnotetext} are also available within the table environment so\nthat multiple table entries can share the same footnote text. There\nis no longer a need to use a \\cmd{\\tablenote}, \\cmd{\\tablenotemark},\nand \\cmd{\\tablenotetext} macros.\n\n\\section{Section Commands}\n\nThe title in a \\cmd\\section\\marg{title} command will be automatically\nuppercased in \\revtex~4. To prevent a particular letter from being\nuppercased, enclose it in curly braces.\n\n\\section{Figures}\n\nFigures should be enclosed within either a \\env{figure} or \\env{figure*}\nenvironment (the latter will cause the figure to span the full width\nof the page in two-column mode). \\LaTeXe\\ has two convenient packages\nfor including the figure file itself: \\classname{graphics} and\n\\classname{graphicx}. These two packages both define a macro\n\\cmd{\\includegraphics} which calls in the figure. They differ in how\narguments for rotation, translation, and scaling are specified. The\npackage \\classname{epsfig} has been re-implemented to use these\n\\classname{graphicx} package. The package \\classname{epsfig} provides\nan interface similar to that under the \\revtex~3 \\classoption{epsf}\nclass option. Authors should use these standard\n\\LaTeXe\\ packages rather than some other alternative.\n\n\\section{Tables}\n\nShort tables should be enclosed within either a \\env{table} or \\env{table*}\nenvironmnent (the latter will cause the table to span the full width\nof the page in two-column mode). The heart of the table is the\n\\env{tabular} environment. This will behave for the most part as in\nstandard \\LaTeXe. Note that \\revtex~4 no longer automatically adds\ndouble (Scotch) rules around tables. Nor does the \\env{tabular}\nenvironment set various table parameters as before. Instead, a new\nenvironment \\env{ruledtabular} provides this functionality. This\nenvironment should surround the \\env{tabular} environment:\n\\begin{verbatim}\n\\begin{table}\n\\caption{...}\n\\label{tab:...}\n\\begin{ruledtabular}\n\\begin{tabular}\n...\n\\end{tabular}\n\\end{ruledtabular}\n\\end{table}\n\\end{verbatim}\n\nUnder \\revtex~3, tables automatically break across pages. \\revtex~4\nprovides some of this functionality. However, this requires adding the\ntable a float placement option of [H] (meaning put the table\n``here'') to the \\envb{table} command.\n\nLong tables are more robustly handled by using the\n\\classname{longtable.sty} package included with the standard \\LaTeXe\\\ndistribution (put \\verb+\\usepackage{longtable}+ in the preamble). This\npackage gives precise control over the layout of the table. \\revtex~4\ngoes out of its way to provide patches so that the \\env{longtable}\nenvironment will work within a two-column format. A new\n\\env{longtable*} environment is also provided for long tables that are\ntoo wide for a narrow column. (Note that the \\env{table*} and\n\\env{longtable*} environments should always be used rather than\nattempting to use the \\env{widetext} environment.)\n\nTo create tables with columns of numbers aligned on decimal points,\nload the standard \\LaTeXe\\ \\classname{dcolumn} package and use the\n\\verb+d+ column specifier. The content of each cell in the column is\nimplicitly in math mode: Use of math delimiters (\\verb+$+) is unnecessary\nin a \\verb+d+ column.\n\nFootnotes within a table can be specified with the\n\\cmd{\\footnote} command (see Sec.~\\ref{sec:foot}). \n\n\\section{Font selection}\n\nThe largest difference between \\revtex~3 and \\revtex~4 with respect to\nfonts is that \\revtex~4 allows one use the \\LaTeXe\\ font commands such\nas \\cmd{\\textit}, \\cmd{\\texttt}, \\cmd{\\textbf} etc. These commands\nshould be used in place of the basic \\TeX\/\\LaTeX\\ 2.09 font commands\nsuch as \\cmd{\\it}, \\cmd{\\tt}, \\cmd{\\bf}, etc. The new font commands\nbetter handle subtleties such as italic correction and scaling in\nsuper- and subscripts.\n\n\\section{Math and Symbols}\n\n\\revtex~4 depends more heavily on packages from the standard \\LaTeXe\\\ndistribution and AMS-\\LaTeX\\ than \\revtex~3 did. Thus, \\revtex~4 users\nshould make sure their \\LaTeXe\\ distributions are up to date and they\nshould install AMS-\\LaTeX\\ 2.0 as well. In general, if any fine control of\nequation layout, special math symbols, or other specialized math\nconstructs are needed, users should look to the \\classname{amsmath}\npackage (see the AMS-\\LaTeX\\ documentation).\n\n\\revtex~4 provides a small number of additional diacritics, symbols,\nand bold parentheses. Table~\\ref{tab:revsymb} summarizes this.\n\n\\begin{table}\n\\caption{Special \\revtex~4 symbols, accents, and boldfaced parentheses \ndefined in \\file{revsymb.sty}}\n\\label{tab:revsymb}\n\\begin{ruledtabular}\n\\begin{tabular}{ll|ll}\n\\cmd\\lambdabar & $\\lambdabar$ &\\cmd\\openone & $\\openone$\\\\\n\\cmd\\altsuccsim & $\\altsuccsim$ & \\cmd\\altprecsim & $\\altprecsim$ \\\\\n\\cmd\\alt & $\\alt$ & \\cmd\\agt & $\\agt$ \\\\\n\\cmd\\tensor\\ x & $\\tensor x$ & \\cmd\\overstar\\ x & $\\overstar x$ \\\\\n\\cmd\\loarrow\\ x & $\\loarrow x$ & \\cmd\\roarrow\\ x & $\\roarrow x$ \\\\\n\\cmd\\biglb\\ ( \\cmd\\bigrb ) & $\\biglb( \\bigrb)$ &\n\\cmd\\Biglb\\ ( \\cmd\\Bigrb )& $\\Biglb( \\Bigrb)$ \\\\\n& & \\\\\n\\cmd\\bigglb\\ ( \\cmd\\biggrb ) & $\\bigglb( \\biggrb)$ &\n\\cmd\\Bigglb\\ ( \\cmd\\Biggrb\\ ) & $\\Bigglb( \\Biggrb)$ \\\\\n\\end{tabular}\n\\end{ruledtabular}\n\\end{table}\n\nHere is a partial list of the more notable changes between \\revtex~3\nand \\revtex~4 math:\n\\begin{itemize}\n\\item Bold math characters should now be handle via the standard\n\\LaTeXe\\ \\classname{bm} package (use \\cmd{\\bm} instead of \\cmd{\\bbox}).\n\\cmd{\\bm} will handle Greek letters and other symbols.\n\n\\item Use the class options \\classoption{amsmath},\n\\classoption{amsfonts} and \\classoption{amssymb} to get even more math\nfonts and symbols. \\cmd{\\mathfrak} and \\cmd{\\mathbb} will, for instance, give\nFraktur and Blackboard Bold symbols.\n\n\\item Use the \\classoption{fleqn} class option for making equation\nflush left or right. \\cmd{\\FL} and \\cmd{\\FR} are no longer provided.\n\n\\item In place of \\cmd{\\eqnum}, load the \\classname{amsmath} package\n[\\verb+\\usepackage{amsmath}+] and use \\cmd{\\tag}.\n\n\\item In place of \\cmd{\\case}, use \\cmd{\\textstyle}\\cmd{\\frac}.\n\n\\item In place of the \\env{mathletters} environment, load the\n\\classname{amsmath} package and use \\env{subequations} environment.\n\n\\item In place of \\cmd{\\slantfrac}, use \\cmd{\\frac}.\n\n\\item The macros \\cmd{\\corresponds}, \\cmd{\\overdots}, and\n\\cmd{\\overcirc} have been removed. See Table~\\ref{tab:obsolete}.\n\n\\end{itemize}\n\n\\section{Obsolete \\revtex~3.1 commands}\n\nTable~\\ref{tab:obsolete} summarizes more differences between \\revtex~4\nand \\revtex~3, particularly which \\revtex~3 commands are now obsolete.\n\n\\begin{table*}\n\\caption{Differences between \\revtex~3.1 and \\revtex~4\nmarkup}\\label{tab:diff31}\n\\label{tab:obsolete}\n\\begin{ruledtabular}\n\\begin{tabular}{lp{330pt}}\n\\textbf{\\revtex~3.1 command}&\\textbf{\\revtex~4 replacement}\n\\lrstrut\\\\\n\\cmd\\documentstyle\\oarg{options}\\aarg{\\classname{revtex}}&\\cmd\\documentclass\\oarg{options}\\aarg{\\classname{revtex4}}\n\\\\\noption \\classoption{manuscript}& \\classoption{preprint}\n\\\\\n\\cmd\\tighten\\ preamble command & \\classoption{tightenlines} class option\n\\\\\n\\cmd\\draft\\ preamble command & \\classoption{draft} class option\n\\\\\n\\cmd\\author & \\cmd\\author\\marg{name} may appear\nmultiple times; each signifies a new author name.\\\\\n & \\cmd\\collaboration\\marg{name}:\nCollaboration name (should appear after last \\cmd\\author)\\\\\n & \\cmd\\homepage\\marg{URL}: URL for preceding author\\\\\n & \\cmd\\email\\marg{email}: email\naddress for preceding author\\\\\n & \\cmd{\\altaffiliation}: alternate\naffiliation for preceding \\cmd\\author\\\\\n\\cmd\\thanks & \\cmd\\thanks, but use only for\ninformation not covered by \\cmd{\\email}, \\cmd{\\homepage}, or \\cmd{\\altaffilitiation}\\\\\n\\cmd\\and & obsolete, remove this command\\\\\n\\cmd\\address & \\cmd\\affiliation\\marg{institution}\\ gives the affiliation for the group of authors above\\\\\n & \\cmd\\affiliation\\oarg{note} lets you specify a footnote to this institution\\\\\n & \\cmd\\noaffiliation\\ signifies that the above authors have no affiliation\\\\\n\n\\cmd\\preprint & \\cmd\\preprint\\marg{number} can appear multiple times, and must precede \\cmd\\maketitle\\\\\n\\cmd\\pacs & \\cmd\\pacs\\ must precede \\cmd\\maketitle\\\\\n\\env{abstract} environment & \\env{abstract} environment must precede \\cmd\\maketitle\\\\\n\\cmd\\wideabs & obsolete, remove this command\\\\\n\\cmd\\maketitle & \\cmd\\maketitle\\ must follow\n\\emph{all} front matter data commands\\\\\n\\cmd\\narrowtext & obsolete, remove this command\\\\\n\\cmd\\mediumtext & obsolete, remove this command\\\\\n\\cmd\\widetext & obsolete, replace with \\env{widetext} environment\\\\\n\\cmd\\FL & obsolete, remove this command\\\\\n\\cmd\\FR & obsolete, remove this command\\\\\n\\cmd\\eqnum & replace with \\cmd\\tag, load \\classname{amsmath}\\\\\n\\env{mathletters} & replace with \\env{subequations}, load\n\\classname{amsmath}\\\\\n\\env{tabular} environment & No longer puts in doubled-rules. Enclose \\env{tabular} in \\env{ruledtabular} to get old behavior.\\\\\n\\env{quasitable} environment & obsolete, \\env{tabular} environment no longer\nputs in rules\\\\\n\\env{references} environment & replace with \\env{thebibliography}\\verb+{}+\\\\\n\\cmd\\case & replace with \\cmd\\textstyle\\cmd\\frac\\\\\n\\cmd\\slantfrac & replace with \\cmd\\frac\\\\\n\\cmd\\tablenote & replace with \\cmd\\footnote\\\\\n\\cmd\\tablenotemark & replace with \\cmd\\footnotemark\\\\\n\\cmd\\tablenotetext & replace with \\cmd\\footnotetext\\lrstrut\\\\\n\\cmd\\overcirc & Use standard \\LaTeXe\\ \\cmd\\mathring\\ \\\\\n\\cmd\\overdots & Use \\cmd\\dddot\\ with \\classoption{amsmath}\\\\\n\\cmd\\corresponds & Use \\cmd\\triangleq\\ with \\classoption{amssymb}\\\\\n\\classoption{epsf} class option & \\verb+\\usepackage{epsfig}+\\\\\n\\end{tabular}\n\\end{ruledtabular}\n\\end{table*}\n\n\n\\section{Converting a \\revtex~3.1 Document to \\revtex~4}\\label{sec:conv31}%\n\n\\revtex~3 documents can be converted to \\revtex~4 rather\nstraightforwardly. The following checklist covers most of the major\nsteps involved.\n\n\\begin{itemize}\n\\item Change \\cmd\\documentstyle\\verb+{revtex}+ to\n\\cmd\\documentclass\\verb+{revtex4}+, and run the document under\n\\LaTeXe\\ instead of \\LaTeX2.09.\n\n\\item\nReplace the \\cmd\\draft\\ command with the \\classoption{draft} class option.\n\n\\item\nReplace the \\cmd\\tighten\\ command with the \\classoption{tightenlines}\nclass option.\n\n\\item\nFor each \\cmd\\author\\ command, split the multiple authors into\nindividual \\cmd\\author\\ commands. Remove any instances of \\cmd\\and.\n\n\\item For superscript-style associations between authors and\naffiliations, remove explicit superscripts and use the\n\\classoption{superscriptaddress} class option.\n\n\\item\nUse \\cmd\\affiliation\\ instead of \\cmd\\address.\n\n\\item\nPut \\cmd\\maketitle\\ after the \\env{abstract} environment and any\n\\cmd\\pacs\\ commands.\n\n\\item If double-ruled table borders are desired, enclose \\env{tabular}\nenviroments in \\env{ruledtabular} environments.\n\n\\item\nConvert long tables to \\env{longtable}, and load the\n\\classname{longtable} package. Alternatively, give the \\env{table}\nan [H] float placement parameter so that the table will break automatically.\n\n\\item\nReplace any instances of the \\cmd\\widetext\\ and \\cmd\\narrowtext\\\ncommands with the \\env{widetext} environment.\nUsually, the \\envb{widetext} statement will replace the \\cmd\\widetext\\\ncommand, and the \\enve{widetext} statement replaces the matching\n\\cmd\\narrowtext\\ command.\n\nNote in this connection that due to a curious feature of \\LaTeX\\\nitself, \\revtex~4 having a \\env{widetext} environment means that it\nalso has a definition for the \\cmd\\widetext\\ command, even though the\nlatter cammand is not intended to be used in your document.\nTherefore, it is particularly important to remove\nall \\cmd\\widetext\\ commands when converting to \\revtex~4.\n\n\\item\nRemove all obsolete commands: \\cmd\\FL, \\cmd\\FR, \\cmd\\narrowtext, and\n\\cmd\\mediumtext\\ (see Table~\\ref{tab:diff31}).\n\n\\item\nReplace \\cmd\\case\\ with \\cmd\\frac. If a fraction needs to be set\nin text style despite being in a display equation, use the\nconstruction \\cmd\\textstyle\\cmd\\frac. Note that \\cmd\\frac\\ does not\nsupport the syntax \\cmd\\case\\verb+1\/2+.\n\n\\item\nReplace \\cmd\\slantfrac\\ with \\cmd\\frac.\n\n\\item\nChange \\cmd\\frak\\ to \\cmd\\mathfrak\\marg{char}\\index{Fraktur} and\n\\cmd\\Bbb\\ to \\cmd\\mathbb\\marg{char}\\index{Blackboard Bold}, and invoke\none of the class options \\classoption{amsfonts} or\n\\classoption{amssymb}.\n\n\\item\nReplace environment \\env{mathletters} with environment\n\\env{subequations} and load the \\classname{amsmath} package.\n\n\\item\nReplace \\cmd\\eqnum\\ with \\cmd\\tag\\ and load the \\classname{amsmath} package.\n\n\\item\nReplace \\cmd\\bbox\\ with \\cmd\\bm\\ and load the \\classname{bm} package.\n\n\\item\nIf using the \\cmd\\text\\ command, load the \\classname{amsmath} package.\n\n\\item\nIf using the \\verb+d+ column specifier in \\env{tabular} environments,\nload the \\classname{dcolumn} package. Under \\classname{dcolumn}, the\ncontent of each \\verb+d+ column cell is implicitly in math mode:\nremove any \\verb+$+ math delimiters appearing in cells in a \\verb+d+\ncolumn.\n\n\\item\nReplace \\cmd\\tablenote\\ with \\cmd\\footnote, \\cmd\\tablenotemark\\ with\n\\cmd\\footnotemark, and \\cmd\\tablenotetext\\ with \\cmd\\footnotetext.\n\n\\item\nReplace \\envb{references} with \\envb{thebibliography}\\verb+{}+;\n\\enve{references} with \\enve{thebibliography}.\n\\end{itemize}\n\\end{document}\n\n\\section{}+, \\verb+\\subsection{}+,\n\\verb+\\subsubsection{}+ & Start a new section or\nsubsection.\\\\\n\\verb+\\section*{}+ & Start a new section without a number.\\\\\n\\verb+","meta":{"redpajama_set_name":"RedPajamaArXiv"}} {"text":"\\section{Introduction}\n\nRecently, the LIGO-Virgo\\footnote{\\url{https:\/\/www.ligo.caltech.edu\/}} collaboration detected six sources of gravitational waves (GWs), five from merging BH-BH binaries \\citep{abbott16a,abbott16b,abbott17,abbott17a,abbott17b} and one from merging NS-NS binaries \\citep{abbott17c}. With ongoing improvements to LIGO and upcoming instruments such as \\textit{LISA}\\footnote{\\url{http:\/\/www.et-gw.eu}} and the Einstein Telescope\\footnote{\\url{https:\/\/lisa.nasa.gov}}, hundreds of BH-BH, BH-NS and NS-NS binary sources may be detected within a few years. Thus the modeling of the formation and evolution of BH and NS binaries is crucial for interpreting the signals from all the GW sources we expect to observe.\n\nThe origins of BH and NS mergers are actively debated. Several scenarios have been proposed, such as isolated binary evolution in the galactic field \\citep{bel16b}, gas-assisted mergers \\citep{bart17,sto17,tag18}, triple systems in the field \\citep{ant17,sil17,ll18} and dynamically formed in dense clusters \\citep{wen03,anm14}, mergers of binaries in galactic nuclei \\citep{mill2009,ole2009,antoper12,prod15,ant16,van16,chen17,petr17,chen18,fern18,hamer18,hoan18,rand18} and other dense stellar systems \\citep{askar17,baner18,cho18,frak18,rod18,rodlo18,sam18}. Each model predicts different rates (generally of the order of $\\sim\\ \\mathrm{few}$ Gpc$^{-3}$ yr$^{-1}$) and can in principle be distinguished from other channels using the observed mass, eccentricity, spin and redshift distributions \\citep[see e.g.][]{olea16}. For instance, dynamically assembled mergers are expected to have a non-negligible probability of appearing eccentric when observed \\citep{antoper12,samas18,zevin18}.\n\nMost of the literature pertaining to dynamically-induced mergers focuses on BH and NS binaries forming in globular clusters, while only a few studies have paid attention to the formation of compact-object binaries in the vicinity of a super massive black hole (SMBH) and in nuclear star clusters \\citep[e.g.][]{hoan18,leigh16,leigh18}. The pioneering work by \\citet{antfab2010} and \\citet{antoper12} showed that SMBHs can induce Lidov-Kozai (LK) oscillations on BH-BH and NS-NS binaries orbiting in its vicinity \\citep{lid62,koz62}, thus enhancing the probability of merging compact binaries. In this scenario, the eccentricity of the BH\/NS binary reaches large values \\citep[for a review on LK mechanisms see][]{nao16}, then GW emission drives the binary to merge. While \\citet{antoper12} adopted a secular treatment for the equations of motion at the quadruple level of approximation, \\citet{hoan18} considered soft binaries and the importance of expansion up to the octuple order. These calculations adopt the secular approximation to study triples, which must satisfy hierarchical conditions \\citep{nao16}. In some cases, the inner binary may undergo rapid oscillations in the angular momentum and eccentricity, thus the secular theory is not anymore an adequate description of the three-body equations of motion \\citep*{antoper12,antognini14,anm14,luo16,ll18,grish18}. For these cases, direct precise $N$-body simulations, including regularization schemes and Post-Newtonian (PN) terms, are required to follow accurately the orbits of the objects up to the final merger \\citep{grish18,fraglei18b,fraglei18}. Recently, \\citet{van16} used $N$-body simulations of small ($\\sim 300$-$4000$ stars) clusters surrounding $10^3$-$10^4\\msun$ black holes to study the effect of LK oscillations in dense environments, while \\citet{agu18} used few-body simulations to check the merger rates of BH-BH binaries delivered by infalling star clusters at typical distances of $\\sim$ few pc.\n\nIn this paper, we revisit the SMBH-induced mergers of compact binaries orbiting in its vicinity. We consider a three-body system consisting of an inner binary comprised of a BH-BH\/NS-NS\/BH-NS binary, and an outer binary comprised of the SMBH and the centre of mass of the inner binary. Figure~\\ref{fig:threebody} depicts the system we study in the present paper. We denote the mass of the SMBH as $\\msmbh$ and the mass of the objects in the inner binary as $m_1$ and $m_2$, while the semimajor axis and eccentricity of the inner orbit are $\\ain$ and $\\ein$, respectively, and for the outer orbit, these are $\\aout$ and $\\eout$, respectively. While previous papers mainly adopted a secular approximation for the equations of motion, with a few direct N-body integrations in comparison to secular evolution \\citep[e.g.][]{antoper12,hoan18}, here, we make the first systematic and statistical study of BH-BH, NS-NS and BH-NS mergers in the proximity of an SMBH by means of direct high-precision $N$-body simulations, including Post-Newtonian (PN) terms up to order PN2.5. Moreover, we consider how different masses of the SMBH affect the mergers of compact binaries, and adopt a mass spectrum for the BHs, while also studying different spatial distributions for the merging binaries. Finally, we discuss observational diagnostics that can help discriminate this compact object merger channel from other ones. \n\nThe paper is organized as follows. In Section~\\ref{sect:bhnsnuclei}, we discuss the properties and dynamics of BHs and NSs in galactic nuclei. In Section~\\ref{sect:secular}, we discuss the state-of-the-art secular approximations currently being used in the literature. In Section~\\ref{sect:bhnsmergers}, we present our numerical methods to determine the rate of BH-BH, NS-NS and BH-NS mergers in galactic nuclei, for which we discuss the results. In Section~\\ref{sect:timedistrates}, we discuss the predicted rate of compact object mergers in galactic nuclei and compare our results to secular approximations, while, in Section~\\ref{sect:implications}, we discuss the observational signatures of these events. Finally, in Section~\\ref{sect:conc}, we discuss the implications of our findings and draw our conclusions.\n\n\\begin{figure} \n\\centering\n\\includegraphics[scale=0.45]{threebody.pdf}\n\\caption{The three-body system studied in the present work. We denote the mass of the SMBH as $\\msmbh$, and the masses of the binary components as $m_1$ and $m_2$. The semimajor axis and eccentricity of the outer orbit are $\\aout$ and $\\eout$, respectively, while for the inner orbit they are $\\ain$ and $\\ein$, respectively.}\n\\label{fig:threebody}\n\\end{figure}\n\n\\section{Black hole and neutron star binaries in galactic nuclei}\n\\label{sect:bhnsnuclei}\n\nThe evidence in favour of the presence of close binaries composed of compact objects (COs) such as white dwarfs (WDs), and especially NSs and BHs, in dense stellar environments is rapidly growing. Recently, \\citet{hailey17} reported observations of a dozen quiescent X-ray binaries that form a central density cusp within $\\sim$ 1 parsec of Sagittarius A$^*$. The authors argue that the emission spectra they observe are inconsistent with a population of accreting WDs, suggesting that the X-ray binaries must contain mostly NSs and BHs. However, the relative numbers of these two types of COs is an open question. Six other X-ray transients are known to be present in the inner parsec of the Galactic Centre, which are also strongly indicative of binaries containing COs \\citep[e.g.][]{muno05,hailey17}. Given the very dense stellar environments in galactic nuclei combined with the presence of a central SMBH, these CO binaries can undergo several fates. They can be hardened to shorter orbital periods, either by direct scattering interactions with single stars \\citep{per09b,leigh18}, or by LK oscillations due to the SMBH combined with gravitational wave (GW) emission acting at pericentre \\citep[e.g.][]{antoper12,prod15}.\n\nIn their Table 1, \\citet{generosoz18} combine the reported statistics from the literature to provide estimates for the numbers of BHs and NSs in the Galactic Centre. In short, the number of NS X-ray Binaries (XRBs) per stellar mass in the Galactic Centre is roughly three orders of magnitude higher than in the field, and comparable to the number expected to be in globular clusters. Similarly, the number of BH XRBs per stellar mass is roughly three orders of magnitude higher than in the field, and roughly an order of magnitude higher than in any known globular cluster \\citep[e.g.][]{strader12,leigh14,leigh16}. \n\nThese large numbers of NS and BH binaries must come from somewhere. Yet, little is known about binaries in our Galactic Centre. A likely explanation is that they are the remnants of massive O\/B stars, since hundreds of these are known to be present in the inner $\\sim 1$ pc of Sagittarius A*. In the Solar neighborhood, the massive-star binary fraction is very high ($\\gtrsim 70\\%$) and the most massive binaries have semi-major axes of up to a few AU \\citep{sana12}. This alone is suggestive of a high rate of NS and BH formation in nuclear star clusters \\citep[e.g.][]{Levin03,genzel08}. Interestingly, the discovery of even a single magnetar within $\\lesssim 0.1$ pc of Sgr A* would also argue in favour of a high rate of NS formation, given their short active lifetimes \\citep{mori13}. \n\nThis population of CO binaries is continuously depleted through dynamical interactions with other stars and COs (evaporation) and GW mergers. A few mechanisms may replenish the CO population. The binaries may come from outside the central region near the SMBH, which thus serves as a continuous source term \\citep{hop09,alex17}. In this scenario, CO binaries form far from the innermost region around the SMBH and gradually migrate on a 2-body relaxation timescale,\n\\begin{eqnarray}\nT_{2b}&=&1.6\\times 10^{10}\\mathrm{yr}\\ \\left(\\frac{\\sigma}{300\\ \\mathrm{km s}^{-1}}\\right)^3\\left(\\frac{m}{\\msun}\\right)^{-1}\\times\\nonumber\\\\\n&\\times&\\left(\\frac{\\rho}{2.1\\times 10^{6}\\msun\\ \\mathrm{pc}^{-3}}\\right)^{-1}\\left(\\frac{\\ln \\Lambda}{15}\\right)^{-1} \\ ,\n\\label{eqn:t2b}\n\\end{eqnarray}\ntowards smaller distances, where they become active in the Lidov-Kozai regime. Here, $\\rho$ and $\\sigma$ are the 1-D density and velocity dispersion in the Galactic Centre, respectively, $\\ln \\Lambda$ is the Coulomb logarithm and $m$ is the average stellar mass. On the other hand, our Galactic Centre contains a large population of young massive O-type stars, many of which have been observed to reside in a stellar disk. Likely, most of them were born \\textit{in-situ} as a consequence of the fragmentation of a gaseous disk formed from an infalling gaseous clump \\citep{genz10}. An important question is which process can make their orbit, which are not observed closer than $\\sim 0.05$ pc, approach the SMBH, where efficient Lidov-Kozai oscillations take place. Both planet-like migration in the gaseous disc \\citep{baru11} and disc instability \\citep{madi09} have been proposed to make the \\textit{in situ} binaries migrate, but to which innermost distance with respect to the SMBH is not known exactly. Other mechanisms include triple\/quadruple disruptions, where a triple\/quadruple is disrupted and the inner binary is left orbiting the SMBH \\citep{per09,Gin11,fgu18,fgi18}, and infalls of star clusters \\citep*{antmer,fck17}.\n\nOutside our own Milky Way, \\citet{secunda18} recently showed that CO binaries can form efficiently in Active Galactic Nucleus (AGN) disks. The COs migrate in the disk due to differential torques exerted by the gas, moving toward migration traps, where the torques actually cancel \\citep[e.g.][]{bellovary16}. The COs drift toward the trap and get stuck there, waiting for the next CO to migrate toward it. Once close enough, the two COs can undergo a strong interaction and end up forming a bound binary due to the dissipative effects of the gas. COs can also accrete from the disk gas in this scenario, possibly growing substantially and in some cases even forming an IMBH \\citep{mckernan12,mckernan14}. \n\n\\begin{table*}\n\\caption{Models: name, SMBH mass ($M_\\mathrm{SMBH}$), binary type, slope of the BH mass function ($\\beta$), slope of the outer semi-major axis distribution ($\\alpha$), $\\ain$ distribution, $\\ein$ distribution, merger fraction ($f_{\\rm merge}$).}\n\\centering\n\\begin{tabular}{lccccccc}\n\\hline\nName &\t$M_\\mathrm{SMBH}$ (M$_\\odot$) & Binary Type & $\\beta$ & $\\alpha$ & $f(\\ain)$ & $f(\\ein)$ & $f_{\\rm merge}$ \\\\\n\\hline\\hline\nMW\t& $4\\times 10^6$ & BH-BH & $1$ & $2$ & \\citet{hoan18} & uniform & $0.045$ \\\\\nMW\t& $4\\times 10^6$ & BH-BH & $1$ & $2$ & \\citet{antoper12} & thermal & $0.261$ \\\\\nMW\t& $4\\times 10^6$ & BH-BH & $2$ & $2$ & \\citet{hoan18} & uniform & $0.05$ \\\\\nMW\t& $4\\times 10^6$ & BH-BH & $3$ & $2$ & \\citet{hoan18} & uniform & $0.036$ \\\\\nMW\t& $4\\times 10^6$ & BH-BH & $4$ & $2$ & \\citet{hoan18} & uniform & $0.041$ \\\\\nMW\t& $4\\times 10^6$ & BH-BH & $1$ & $0$ & \\citet{hoan18} & uniform & $0.035$ \\\\\nMW\t& $4\\times 10^6$ & BH-BH & $1$ & $3$ & \\citet{hoan18} & uniform & $0.051$ \\\\\nMW\t& $4\\times 10^6$ & BH-NS & $1$ & $2$ & \\citet{hoan18} & uniform & $0.026$ \\\\\nMW\t& $4\\times 10^6$ & BH-NS & $2$ & $2$ & \\citet{hoan18} & uniform & $0.029$ \\\\\nMW\t& $4\\times 10^6$ & BH-NS & $3$ & $2$ & \\citet{hoan18} & uniform & $0.028$ \\\\\nMW\t& $4\\times 10^6$ & NS-NS & $-$ & $1.5$ & \\citet{hoan18} & uniform & $0.032$ \\\\\nMW\t& $4\\times 10^6$ & NS-NS & $-$ & $2$ & \\citet{hoan18} & uniform & $0.028$ \\\\\nMW\t& $4\\times 10^6$ & NS-NS & $-$ & $2$ & \\citet{antoper12} & thermal & $0.079$ \\\\\n\\hline\nGN\t& $1\\times 10^8$ & BH-BH & $1$ & $2$ & \\citet{hoan18} & uniform & $0.072$ \\\\\nGN\t& $1\\times 10^8$ & BH-BH & $1$ & $2$ & \\citet{antoper12} & thermal & $0.258$ \\\\\nGN\t& $1\\times 10^8$ & BH-NS & $1$ & $2$ & \\citet{hoan18} & uniform & $0.054$ \\\\\nGN\t& $1\\times 10^8$ & NS-NS & $-$ & $2$ & \\citet{hoan18} & uniform & $0.044$ \\\\\nGN\t& $1\\times 10^8$ & NS-NS & $-$ & $2$ & \\citet{antoper12} & thermal & $0.133$ \\\\\n\\hline\nGN2\t& $1\\times 10^9$ & BH-BH & $1$ & $2$ & \\citet{hoan18} & uniform & $0.087$ \\\\\nGN2\t& $1\\times 10^9$ & BH-BH & $1$ & $2$ & \\citet{antoper12} & thermal & $0.3$ \\\\\nGN2\t& $1\\times 10^9$ & BH-NS & $1$ & $2$ & \\citet{hoan18} & uniform & $0.061$ \\\\\nGN2\t& $1\\times 10^9$ & NS-NS & $-$ & $2$ & \\citet{hoan18} & uniform & $0.048$ \\\\\nGN2\t& $1\\times 10^9$ & NS-NS & $-$ & $2$ & \\citet{antoper12} & thermal & $0.055$ \\\\\n\\hline\n\\end{tabular}\n\\label{tab:models}\n\\end{table*}\n\n\\subsection{Outer semi-major axis and eccentricity}\n \nThe numbers and spatial profiles of BHs and NSs are poorly known in galactic nuclei. In general, stars tend to form a power-law density cusp around an SMBH. The classical result by \\citet{bahcall76} shows that a population of equal-mass objects forms a power-law density cusp around an SMBH, $n(r)\\propto r^{-\\alpha}$, where $\\alpha=7\/4$. For multi-mass distributions, lighter and heavier objects develop shallower and steeper cusps, respectively \\citep{hopm06,Fre06,ale09,pre10,aharon16,baumg18}, while source terms may make the cusp steeper as well \\citep{aharon15,frasar18}. Recent observations of the Milky Way's centre showed that the slope of the cusp appears to be shallower \\citep[$\\alpha\\sim 5\/4$;][]{gall18,scho18}. As BHs and NSs are heavier than average stars, they are expected to relax into steeper cusps, with BHs relaxing into steeper cusps than NSs \\citep{Fre06,hopal06}. If not all BHs have the same mass, but follow a mass distribution, only the more massive ones would have steeper slopes, while low-mass BHs should follow shallower slopes \\citep{aharon16}.\n\nIn the present study, we assume that the BH and NS number densities follow a cusp with $\\alpha=2$. We also study the effects of the cusp slope, by considering a steeper cusp ($\\alpha=3$) and a uniform density profile ($\\alpha=0$) for BHs, and a shallower cusp ($\\alpha=1.5$) for NSs. For the maximum outer semi-major axis, we take $a_{\\rm out}^{M}=0.1$ pc following \\citet{hoan18}, which approximately corresponds to the value at which the eccentric Lidov-Kozai timescale is equal to the timescale over which accumulated fly-bys from single stars tend to unbind the binary (see Eqs.~\\ref{eqn:binevap}-\\ref{t_oct}). However, the ratio of these two timescales generally depends on the binary and cusp properties since wide binaries could be affected by the LK mechanism at larger distances. We also consider one model where we take $a_{\\rm out}^{M}=0.5$ pc, to assess how our results depend on this parameter. As discussed, \\textit{in-situ} formation occurs in our Galactic Centre at larger distances and some mechanism that delivers such CO binaries closer to the SMBH has to be invoked. Finally, we sample the outer orbital eccentricity from a thermal distribution \\citep{jeans1919}.\n\n\\subsection{Inner semi-major axis and eccentricity}\n\nThe inner binary (BH-BH\/NS-NS\/BH-NS) semi-major axis and eccentricity are not well known. Different models predict different distributions. Moreover, the dense environments characteristic of galactic nuclei should cause both distributions to diffuse over time, thus changing the relative distributions. \\citet{hop09} made the only attempt to model binaries very close to the SMBH, even though this pioneering study accounted only for the 2-body relaxation process. Other relaxation processes, such as resonant relaxation \\citep{rauch96}, may affect the distribution as well \\citep{hamer18}. For what concerns eccentricity, though it mostly depends on the natal-kick and the common-envelope phase, it also depends on the scattering of the CO binaries by local COs and stars. As a consequence, \\textit{in-situ} formation would probably favour circular binaries, while the migration scenario would most likely prefer a thermal distribution, as a consequence of the energy exchange through many dynamical encounters of the CO binaries with other stars and COs in galactic nuclei.\n\n\\citet[][see Fig.~1]{antoper12} used the results of \\citet{belc04} for the initial distribution of BH-BH binaries orbits, while, for NSs, they used the observed pulsar binary population as found in the ATNF pulsar catalog \\footnote{http:\/\/www.atnf.csiro.au\/research\/pulsar\/psrcat} \\citep{manc05}. We note that their distributions referred to isolated binaries and used a simplified approach for to account for the softening and binary-destruction due to the crowded environments (following the approach in \\citep{per07}). Inner eccentricities were sampled from a uniform distribution. Recently, \\citet{hoan18} drew the inner semi-major axes from a log-uniform distribution in the range $0.1$-$50$ AU, somewhat consistent with the observed distribution from \\citet{sana12}, which favors short period binaries, and the inner eccentricities from a uniform distribution \\citep{ragh10}. Note the caveat that this distribution corresponds to massive MS binaries, and not to CO binaries which already evolved and change their configurations. \n\n\\subsection{Masses}\nThe mass distribution of BHs is unknown, even in isolation. Moreover, in galactic nuclei, irrespective of the original mass function, mass-segregation can make the effective BH mass function even steeper in the inner-most regions \\citep{aharon16}. This would in turn also affect the outer orbit distribution, since more massive BHs would have steeper slopes \\citep{aharon16}.\n\nIn our models, we sample the masses of the BHs from\n\\begin{equation}\n\\frac{dN}{dm} \\propto M^{-\\beta}\\ ,\n\\label{eqn:bhmassfunc}\n\\end{equation}\nin the mass range $5\\msun$-$100\\msun$\\footnote{Note that pulsational pair instability may limit the maximum mass to $\\sim 50\\msun$ \\citep{bel2016}.}. To check how the results depend on the slope of the BH mass function, we run models with $\\beta=1$, $2$, $3$, $4$ for both the BH-BH and BH-NS binaries \\citep{olea16}. For NSs, we fix the mass to $1.3\\msun$ \\citep*[e.g.][]{fpb18}.\n\n\\subsection{Inclinations and relevant angles}\nWe draw the initial mutual inclination $i_0$ between the inner and outer orbit from an isotropic distribution (i.e. uniform in $\\cos i$). The other relevant angles, such as the arguments of pericentre, nodes and mean anomalies, are drawn randomly.\n\n\\subsection{Timescales in galactic nuclei}\nIn the dense stellar environment of a galactic nucleus, several dynamical processes other than 2-body relaxation (Eq.~\\ref{eqn:t2b}) can take place and affect the evolution of the stellar and compact object populations. On smaller timescales than $T_{2b}$, resonant relaxation \\citep{rauch96,kocs15} randomizes the direction and magnitude (hence eccentricity) of the outer orbit on a typical timescale\n\\begin{equation}\nT_{\\rm RR}=9.2\\times 10^{8}\\ \\mathrm{yr}\\ \\left(\\frac{\\msmbh}{4\\times 10^6 \\msun}\\right)^{1\/2}\\left(\\frac{a_{out}}{0.1\\ \\mathrm{pc}}\\right)^{3\/2}\\left(\\frac{m}{\\msun}\\right)^{-1}\\ .\n\\label{eqn:trr}\n\\end{equation}\nOn even shorter timescales, vector resonant relaxation changes the direction (hence the relative inclination) of the outer orbit angular momentum on a typical timescale\n\\begin{eqnarray}\nT_{\\rm VRR}&=&7.6\\times 10^{6}\\ \\mathrm{yr}\\ \\left(\\frac{\\msmbh}{4\\times 10^6 \\msun}\\right)^{1\/2}\\times\\nonumber\\\\\n&\\times & \\left(\\frac{a_{\\rm out}}{0.1\\ \\mathrm{pc}}\\right)^{3\/2}\\left(\\frac{m}{\\msun}\\right)^{-1}\\left(\\frac{N}{6000}\\right)^{-1\/2}\\ ,\n\\label{eqn:tvrr}\n\\end{eqnarray}\nwhere $N$ is the number of stars within $a_{\\rm out}$. In the context of Kozai-Lidov oscillations, vector resonant relaxation plays a role, since it may affect the initial inclination of the inner and outer orbit of the CO binary on timescales comparable to or even shorter than the Kozai-Lidov timescale \\citep{hamer18}.\n\nFinally, binaries may evaporate due to dynamical interactions with field stars in the dense environment of a galactic nucleus when \n\\begin{equation}\n\\frac{E_b}{(m_1+m_2)\\sigma^2}\\lesssim 1\\ ,\n\\end{equation}\nwhere $E_b$ is the binary internal orbital energy and $\\sigma$ is the velocity dispersion. This happens on an evaporation timescale \\citep{binntrem87}\n\\begin{eqnarray}\nT_{\\rm EV}&=&3.2\\times 10^{7}\\ \\mathrm{yr} \\left(\\frac{m_1+m_2}{2\\msun}\\right)\\left(\\frac{\\sigma}{300\\ \\mathrm{km s}^{-1}}\\right)\\left(\\frac{m}{\\msun}\\right)^{-1}\\times\\nonumber\\\\\n&\\times&\\left(\\frac{a_{\\rm in}}{1\\ \\mathrm{AU}}\\right)^{-1}\\left(\\frac{\\rho}{2.1\\times 10^{6}\\msun\\ \\mathrm{pc}^{-3}}\\right)^{-1}\\left(\\frac{\\ln \\Lambda}{15}\\right)^{-1}\\ .\n\\label{eqn:binevap}\n\\end{eqnarray}\n\n\\section{Secular averaging techniques}\n\\label{sect:secular}\nThe merger time of an isolated binary of component masses $m_1$, $m_2$, semimajor axis $a$ and eccentricity $e$ emitting GWs is \\citep{pet64}\n\\begin{equation}\nT_{{\\rm GW}}(a,e)=\\frac{5}{256}\\frac{c^{5}a^{4}}{G^{3}m_{1}m_{2}(m_{1}+m_{2})}(1-e^{2})^{7\/2}. \\label{eq:t_merge}\n\\end{equation}\nFor a triple system made up of an inner binary that is orbited by an outer companion, the inner eccentricity can be pumped by the tidal potential of a distant body via the Lidov-Kozai (LK) mechanism \\citep{lid62,koz62}. The LK oscillations occur on a secular timescale \\citep{antognini15}\n\\begin{equation}\nt_{\\rm sec} = \\frac{8}{15\\pi}\\frac{m_{{\\rm tot}}}{m_{{\\rm out}}}\\frac{P_{{\\rm out}}^{2}}{P_{{\\rm in}}}(1-\\eoutsq)^{3\/2}\\ ,\n\\label{t_sec}\n\\end{equation}\nwhere $m_{\\rm out} = \\msmbh$ and $m_{\\rm tot} = \\msmbh + m_{\\rm bin} \\approx \\msmbh $, $P_{{\\rm in}}$ and $P_{{\\rm out}}$ are the orbital periods of the inner and outer binary, respectively. The large values attained by the inner eccentricity make the overall merger time of the inner binary shorter since it efficiently dissipates energy when $e \\sim e_{\\rm max}$ \\citep[e.g., see][]{antoper12}. The LK-induced merger time is \\citep{antoper12,rand18, ll18}\n\\begin{equation}\nT_{\\rm GW}^{\\rm LK}(a,e_{\\rm max}) \\approx T_{\\rm GW}(a,e_{\\rm max})\/\\sqrt{1-e_{\\rm max}^2}\\ \\label{eq:T_LKGW} .\n\\end{equation}\nThe maximal eccentricity is a function mostly of the initial mutual inclination, $i_0$, and is usually evaluated by the secular approximation, which relies on double-averaging of both the inner and the outer orbits \\citep{rand18}. In the leading, quadrupole order, the system is integrable and has been widely studied \\citep[see recent review by][and references therein]{nao16}, where coupled oscillations between the eccentricity and inclination of the inner binary are excited for sufficiently large initial mutual inclinations. The inner binary eccentricity approaches almost unity as $i_0$ approaches $\\sim 90$ deg.\n\nWhen the outer orbit is eccentric and the inner binary has an extreme mass ratio, octupole-level perturbations turn the system from integrable to chaotic \\citep{ln14}, and can potentially induce extreme orbital eccentricities, orbital flips and even direct collisions \\citep{Katz11,ln11}. The strength of the octupole perturbation is encapsulated in the octupole parameter as\n\\begin{equation}\n\\epsilon_{\\rm oct} \\equiv \\frac{m_1 - m_2}{m_1 + m_2}\\frac{\\ain}{\\aout}\\frac{\\eout}{1 -\\eoutsq}. \\label{eps_oct}\n\\end{equation}\nGenerally, increasing $\\epsilon_{\\rm oct}$ will increase the parameter space corresponding to orbital flips and very large eccentricities. The typical timescale for an orbital flip is \\citep{antognini15}\n\\begin{equation}\nt_{\\rm flip} = \\frac{8}{\\pi}\\sqrt{\\frac{10}{\\epsilon_{\\rm oct}}} t_{\\rm sec} \\label{t_oct}\\,.\n\\end{equation}\n\nThe secular approximation assumes that the triple system is hierarchical, namely that $P_{\\rm in} \/ P_{\\rm out} \\propto (\\ain \/ \\aout)^{2\/3} \\ll 1$, thus ignoring short-term variations ($t \\sim P_{\\rm out }$) of the osculating elements, whose typical strength can be parametrized by the so-called 'single-averaging parameter' \\citep{luo16}\n\\begin{equation}\n\\epsilon_{{\\rm SA}}\\equiv\\left(\\frac{\\ain}{\\aout (1-\\eoutsq)} \\right)^{3\/2}\\left(\\frac{\\msmbh}{m_{\\rm bin}}\\right)^{1\/2}=\\frac{P_{{\\rm out}}}{2\\pi\\tau_{{\\rm sec}}}.\\label{eq:epssa}\n\\end{equation}\nAlso, \\citet{luo16} point out that in addition to the fluctuating terms, additional secular evolution can take place. Consequently, the resulting fate of the system could be different, since additional extra apsidal and nodal precession changes the structure of the LK resonance. \\citet{grish17} showed that extra apsidal precession shifts the critical inclination for the LK resonance and affects the Hill stability limit of irregular satellites.\n\nRecently, \\citet{grish18} used \\citet{luo16}'s result to find an analytic formula for the maximal eccentricity that can be reached due to LK oscillations\n\\begin{align}\ne_{\\rm max} & = \\bar{e}_{\\rm max} + \\delta e\\ ,\\nonumber \\\\\n\\bar{e}_{\\rm max} & = \\sqrt{1-\\frac{5}{3}\\cos^{2}i_{0}\\frac{1+\\frac{9}{8}\\epsilon_{{\\rm SA}}\\cos i_{0}}{1-\\frac{9}{8}\\epsilon_{{\\rm SA}}\\cos i_{0}}}\\ , \\nonumber \\\\\n\\delta e & =\\frac{135}{128}\\bar{e}_{\\rm max}^{\\rm SA}\\epsilon_{\\rm SA}\\left[ \\frac{16}{9}\\sqrt{\\frac{3}{5}}\\sqrt{1-\\bar{e}_{\\rm max}^{2}}+\\epsilon_{{\\rm SA}}-2\\epsilon_{{\\rm SA}}\\bar{e}_{\\rm max}^{2}\\right]\\ \\label{eq:emax_corr}.\n\\end{align}\nIf the inner binary is too compact or the tertiary is too far away, the maximal eccentricity could be quenched, e.g. by GR (general relativistic) precession, and the average maximal eccentricity is given by solving \\citep{grish18}\n\\begin{align}\n\\bar{A}(1-\\bar{e}_{{\\rm max}}^{2}) & =8\\frac{\\epsilon_{{\\rm GR}}}{\\bar{e}_{{\\rm max}}^{2}}\\sqrt{1-\\bar{e}_{\\rm max}^{2}} + 15\\bar{j}_{z}^{2}\\left(1+\\frac{9}{8}\\epsilon_{{\\rm SA}}\\bar{j}_{z}\\right)\\nonumber \\\\\n\\bar{A}(\\bar{j}_{z},\\bar{e}_{{\\rm max}}) & \\equiv9-\\epsilon_{{\\rm SA}}\\frac{81}{8}\\bar{j}_{z}+8\\frac{\\epsilon_{{\\rm GR}}}{\\bar{e}_{{\\rm max}}^{2}}\\,, \\label{eq:jmin_e0}\n\\end{align}\nwhere $\\bar{j}_{z} = \\sqrt{(1-e_0^2)\\cos i_0}$ is the initial normalized angular momentum of the inner binary and\n\\begin{equation}\n\\epsilon_{{\\rm GR}}\\equiv\\frac{3m_{{\\rm bin}}(1-e_{{\\rm out}}^{2})^{3\/2}}{m_{{\\rm out}}}\\left(\\frac{a_{{\\rm out}}}{a}\\right)^{3}\\frac{G m_{\\rm bin}}{ac^2}\\label{eq:epsgr}\n\\end{equation}\nmeasures the ratio between the apsidal precession rates induced by Lidov-Kozai and GR perturbations. The above formulae are valid in the limit $\\epsilon_{\\rm oct}=0$. \n\nIn some configurations the inner binary may undergo rapid oscillations in the angular momentum and eccentricity, thus the secular theory is no longer an adequate description of the three-body equations of motion \\citep{antoper12,antognini14}. For instance, this can happen when the typical time-scale for the angular momentum of the inner orbit to change by of order itself becomes comparable to (or even shorter than) the outer or inner orbital periods \\citep{anm14}. Thus, in this case, the secular approximation can fail to predict both the correct maximum eccentricity and merger time. Although computationally expensive (in particular in the case of a third very massive companion as in this paper), direct $N$-body simulations including Post-Newtonian (PN) terms represent the most reliable option for accurately studying the effects of the tertiary companion in reducing the GW merger time of the inner binary.\n\n\\section{N-Body Simulations: black hole and neutron star mergers in galactic nuclei}\n\\label{sect:bhnsmergers}\n\n\\begin{figure} \n\\centering\n\\includegraphics[scale=0.525]{inclainbhbh.pdf}\n\\includegraphics[scale=0.525]{inclmtotbhbh.pdf}\n\\caption{Inclination as a function of the initial semi-major axis $\\ain$ and the total mass in merged BH-BH binaries for all models with $f(\\ain)$ from \\citet{hoan18}. Most BH binaries that merge have initial inclinations $\\sim 90^\\circ$, where the enhancement in the maximum eccentricity is expected to be larger due to Lidov-Kozai oscillations.}\n\\label{fig:incl}\n\\end{figure}\n\n\\begin{figure*} \n\\centering\n\\begin{minipage}{20.5cm}\n\\hspace{0.5cm}\n\\includegraphics[scale=0.5]{ainbhbh.pdf}\n\\hspace{1.5cm}\n\\includegraphics[scale=0.5]{aoutbhbh.pdf}\n\\end{minipage}\n\\begin{minipage}{20.5cm}\n\\hspace{0.5cm}\n\\includegraphics[scale=0.5]{ainbhns.pdf}\n\\hspace{1.5cm}\n\\includegraphics[scale=0.5]{aoutbhns.pdf}\n\\end{minipage}\n\\begin{minipage}{20.5cm}\n\\hspace{0.5cm}\n\\includegraphics[scale=0.5]{ainnsns.pdf}\n\\hspace{1.5cm}\n\\includegraphics[scale=0.5]{aoutnsns.pdf}\n\\end{minipage}\n\\caption{Cumulative distributions of the inner semi-major axis $\\ain$ (left) and of the outer semi-major axis $\\aout$ (right) for BH-BH (top), BH-NS (centre), NS-NS (bottom) binaries that merge in all models with $f(\\ain)$ from \\citet{hoan18}.}\n\\label{fig:ainaout}\n\\end{figure*}\n\n\\begin{figure*} \n\\centering\n\\begin{minipage}{20.5cm}\n\\includegraphics[scale=0.55]{mtotbhbh.pdf}\n\\hspace{1cm}\n\\includegraphics[scale=0.55]{mtotbhns.pdf}\n\\end{minipage}\n\\caption{Distribution of the total ($m_1+m_2$) BH-BH mass (left) and primary BH mass ($m_1$) in BH-NS binaries (right) for all models with $f(\\ain)$ from \\citet{hoan18}. As expected, the main parameter that affects the mass distribution is the slope $\\beta$ of the BH mass function.}\n\\label{fig:masstot}\n\\end{figure*}\n\nIn this section, we use $N$-body simulations to study the fate of BH-BH, NS-NS and BH-NS binaries in galactic nuclei that host an SMBH. We consider three different SMBH masses, i.e. $\\msmbh=4\\times 10^6\\msun$ for a Milky-Way-like nucleus (Models MW), $\\msmbh=10^8\\msun$ for a M31-like nucleus (Models GN) and $\\msmbh=10^9\\msun$ for more massive host galaxies (Models GN2). For the inner semi-major axis and eccentricities, we follow the prescriptions given by \\citet{hoan18}, but also run some models with the sampling suggested by \\citet{antoper12} to check how the initial conditions affect the final rates. Given the set of initial parameters as described in Sect.~\\ref{sect:bhnsnuclei}, we draw the main parameters of the three-body system and require that the inner binary does not cross the Roche limit of the SMBH at its orbital pericentre distance\n\\begin{equation}\n\\frac{\\aout}{\\ain}> \\eta \\frac{1+\\ein}{(1-\\eout)}\\left(\\frac{3\\msmbh}{m_1+m_2}\\right)^{1\/3}\\,.\n\\label{eqn:hills}\n\\end{equation}\nFollowing \\citet{antoper12}, we set $\\eta=4$ since at shorter distances the inner binary is unstable. We then integrate the triple SMBH-CO-CO differential equations of motion\n\\begin{equation}\n{\\ddot{\\textbf{r}}}_i=-G\\sum\\limits_{j\\ne i}\\frac{m_j(\\textbf{r}_i-\\textbf{r}_j)}{\\left|\\textbf{r}_i-\\textbf{r}_j\\right|^3}\\ ,\n\\end{equation}\nwith $i=1$,$2$,$3$. The integrations are performed using the \\textsc{archain} code \\citep{mik06,mik08}. This code is fully regularized and is able to model the evolution of objects of arbitrary mass ratios and eccentricities with extreme accuracy, even over long periods of time. We include PN corrections up to order PN2.5. \n\nFor each set of parameters in Tab.~\\ref{tab:models}, we run $\\sim 1500$ simulations up to a maximum integration time of $T=1$ Myr, for a total of $\\sim 35000$ simulations. From the computational point of view, this limit represents a good compromise between the numerical effort (large mass ratios and GW effects slow down the code) and the size of the statistical sample we want to take into account. From the physical point of view, we note that our total integration time is smaller than the typical timescale for vector resonant relaxation to operate \\citep[$\\sim$ few Myr, see Eq.~\\ref{eqn:tvrr};][]{rauch96,kocs15}, which reorients the binary centre-of-mass orbital plane with respect to the SMBH, thus affecting the relative inclination of the inner and outer orbits and the relative LK dynamics, rendering the $3$-body approximation insufficient \\citep{hamer18}. Furthermore, in-plane precession induced by the nuclear cluster potential and departure from spherical symmetry of the galactic nucleus would make the CO center of mass orbit precess even faster than vector resonant relaxation alone in a MW-like nucleus \\citep{petr17}. Finally, we also note that our total integration time is smaller than the typical evaporation time of the CO binaries in galactic nuclei (see Eq.~\\ref{eqn:binevap}). Taken all together, these considerations justify our choice of maximum integration time. Nevertheless, we also take into account in our rate calculations the binaries that are not affected by the LK cycles within $1$~Myr and merge by emission of GWs alone on longer timescales, without the assistance of LK oscillations\\footnote{Note that these binaries could however be significantly perturbed by dynamical interactions, even before evaporating \\citep{leigh16}}. Thus, our estimations correspond to a lower limit. We consider in total $23$ different models, which take into account different COs in the inner binary (BH-BH, BH-NS and NS-NS), different masses of the SMBHs, different slopes of the BH\/NS mass functions, different spatial distributions of the CO binaries, and different inner semi-major axis and eccentricity distributions. Table~\\ref{tab:models} summarizes all the models considered in this work.\n\nIn our simulations the CO binary has three possible fates: (i) the CO binary can survive on an orbit perturbed with respect to the initial one; (ii) the CO binary can be tidally broken apart by differential forces exerted by the SMBH, and its components will either be captured by the SMBH or ejected from the galactic nucleus; (iii) the CO binary merges producing an GW merger event. We distinguish among these possible outcomes by computing the mechanical energy of the CO binary. If the relative energy remains negative, we consider the binary survived (case (i)), otherwise we consider the binary unbound (case (ii)). Finally, if the CO binary merges, which occurs if the relative radii of the two COs overlap directly, we have a GW merger event (case (iii)).\n\n\\begin{table}\n\\caption{BH masses already detected via GW emission.}\n\\centering\n\\begin{tabular}{lcc}\n\\hline\nName &\t$m_1$\t(M$_\\odot$) &\t$m_2$\t(M$_\\odot$) \\\\\n\\hline\\hline\nGW150914 & $36.0^{+5.0}_{-4.0}$ & $29.0^{+4.0}_{-4.0}$ \\\\\nGW151226 & $14.2^{+8.3}_{-3.7}$ & $7.5^{+2.3}_{-2.3}$ \\\\\nGW170104 & $31.2^{+8.4}_{-6.0}$ & $19.4^{+5.3}_{-5.9}$ \\\\\nGW170608 & $12.0^{+7.0}_{-2.0}$ & $7.0^{+2.0}_{-2.0}$ \\\\\nGW170814 & $30.5^{+5.7}_{-3.0}$ & $25.3^{+2.8}_{-4.2}$ \\\\\nLVT151012 & $23.0^{+18.0}_{-6.0}$ & $13.0^{+4.0}_{-5.0}$ \\\\\n\\hline\n\\end{tabular}\n\\label{tab:bhligo}\n\\end{table}\n\n\\subsection{Inclination distribution}\n\nWe illustrate in Fig.~\\ref{fig:incl} the inclination as a function of semi-major axis $\\ain$ and total mass in merged BH-BH binaries in our simulations for all the models with $f(\\ain)$ from \\citet{hoan18}. Most of the BH-BH binaries that merge have initial inclinations $\\sim 90^\\circ$, where the enhancement of the maximum eccentricity is expected to be larger due to LK oscillations. As discussed in Sect.~\\ref{sect:secular}, the exact LK window angle depends on the physical quantities of the system \\citep{grish17,grish18}, and the final distribution of surviving systems lacks highly inclined binaries \\citep{fraglei18}. The lack of highly inclined systems shows the importance of the LK mechanism, since BH-BH binaries that successfully undergo a merger event originally orbit in a plane highly inclined with respect to the outer orbital plane \\citep{fraglei18}. In these binaries, the LK mechanism influences the dynamics of the system and induces oscillations both in eccentricity and inclination, whenever not suppressed by GR precession. Figure~\\ref{fig:incl} also shows that some systems that merge have inclinations far from $\\sim 90^\\circ$, in particular when the total binary mass is large and the inner semi-major axis is small. Also, these systems typically have relatively high initial inner eccentricities. \n\n\\begin{figure*} \n\\centering\n\\begin{minipage}{20.5cm}\n\\hspace{0.5cm}\n\\includegraphics[scale=0.50]{data_mwb1a2.pdf}\n\\hspace{1.5cm}\n\\includegraphics[scale=0.50]{data_mwb2a2.pdf}\n\\end{minipage}\n\\begin{minipage}{20.5cm}\n\\hspace{0.5cm}\n\\includegraphics[scale=0.50]{data_mwb3a2.pdf}\n\\hspace{1.5cm}\n\\includegraphics[scale=0.50]{data_mwb1a3.pdf}\n\\end{minipage}\n\\begin{minipage}{20.5cm}\n\\hspace{0.5cm}\n\\includegraphics[scale=0.50]{data_gnb1a2.pdf}\n\\hspace{1.5cm}\n\\includegraphics[scale=0.50]{data_gn2b1a2.pdf}\n\\end{minipage}\n\\caption{Density maps of the masses of the BH-BH binaries that merge in our models (with $f(\\ain)$ from \\citet{hoan18}), along with a comparison to the observed BH masses detected via GW emission \\citep[see Tab.~\\ref{tab:bhligo};][]{abbott16a,abbott16b,abbott17,abbott17a,abbott17b}.}\n\\label{fig:massbhdata}\n\\end{figure*}\n\n\\subsection{Inner and outer orbital parameter distributions}\n\nWe present in Fig.~\\ref{fig:ainaout} the cumulative distribution of $\\ain$ (left panel) for BH-BH (top), BH-NS (centre), NS-NS (bottom) binaries for all models with $f(\\ain)$ from \\citet{hoan18}. Both the SMBH mass and the slope $\\alpha$ of the CO binary spatial distribution around the SMBH significantly affect the inner and outer semi-major axes of merging binaries, as a consequence of their Hill stability \\citep{grish17}. Larger SMBH masses imply that, on average, smaller values of $\\ain$ are needed to avoid tidal disruption of the CO binaries after only a few orbits about the SMBH. For the same reason, large SMBH masses typically produce mergers at larger distances from the SMBH. Obviously, steep binary distributions (i.e., large $\\alpha$'s) imply that the simulated CO binaries will, on average, be closer to the SMBH when they merge. Hence, smaller inner semi-major axes are needed to avoid tidal dissociation by the SMBH. Finally, the total mass of the CO binary plays some role in shaping the final outer orbital semi-major axis distribution. CO binaries with smaller total masses ($m_1+m_2$) typically merge at larger distances from the SMBH, since their binding energy is easily overcome by the gravitational pull of the SMBH, which tends to break the binaries at smaller distances. For BHs, this translates into steeper mass functions typically producing mergers at farther distances from the SMBH.\n\n\\subsection{Mass distribution}\n\nFigure~\\ref{fig:masstot} shows the distribution of the total ($m_1+m_2$) BH-BH mass (left) and the primary BH mass ($m_1$) in BH-NS binaries (right) that merge in all models with $f(\\ain)$ from \\citet{hoan18}. The resulting mass distribution is barely affected by the slope of the binary spatial distribution around the SMBH, with a roughly constant shape in the range $\\sim 25\\msun- 125\\msun$ and a tail extending up to $\\sim 180\\msun$ for BH-BH binaries. Also, the mass of the central SMBH does not significantly affect the mass distribution. As expected, the parameter that governs the resulting shape of the mass distribution is the slope $\\beta$ of the BH mass function: the shallower the BH mass function, the larger the typical total mass of merging BH-BH binaries. In the case $\\beta=1$, we find that $\\sim 95\\%$ of the mergers have $m_1+m_2\\lesssim 150\\msun$, while roughly all the mergers have total masses $\\lesssim 100\\msun$, $\\lesssim 50\\msun$, and $\\lesssim 25\\msun$ for $\\beta=2$, $\\beta=3$, and $\\beta=4$, respectively. Similar results also hold for BH-NS binaries.\n\nThe slope of the BH mass function is unknown. We can use the results of our simulations along with the BH-BH merger events observed by LIGO \\citep[see Tab.~\\ref{tab:bhligo};][]{abbott16a,abbott16b,abbott17,abbott17a,abbott17b} to constrain the BH mass function, assuming these mergers took place in a galactic nucleus. We show in Fig.~\\ref{fig:massbhdata} density maps for the masses of the two merging BHs ($m_1>m_2$), along with data from the LIGO-observed BH merger events. It is clear that a steep mass function ($\\beta>1$) seems to be disfavored by the current data, which suggest a shallower BH mass function. However, we note that BH mass measurements via GW observations are biased towards more massive BHs, since these are more easily observed by LIGO. We also note that, although the mass distribution is only slightly affected by the SMBH mass, the data seem to prefer more massive nuclei than the Milky-Way. Note, however, that mass-segregation processes, that can give rise to much steeper effective mass-functions of BHs in galactic nuclei \\citep{aharon16}, only operate in small-SMBH nuclei where relaxation (and mass-segregation) times are short.\n\n\\subsection{Eccentricity}\n\nFor the systems that merge in our simulations, we compute a proxy for the GW frequency of the merging binaries. This is taken to be the frequency corresponding to the harmonic that gives the maximal emission of GWs \\citep{wen03}\n\\begin{equation} \nf_{\\rm GW}=\\frac{\\sqrt{G(m_1+m_2)}}{\\pi}\\frac{(1+e_{\\rm in})^{1.1954}}{[\\ain(1-e_{\\rm in}^2)]^{1.5}}\\ .\n\\end{equation}\nFigure~\\ref{fig:eccligo} reports the distribution of eccentricities at the moment the binaries enter the LIGO frequency band ($10$ Hz) for BH-BH mergers in a Milky Way-like nucleus, for different values of $\\beta$ and $\\alpha$. The distributions have a double peak at $e_{\\rm 10Hz}\\sim 10^{-2}$ and $e_{\\rm 10Hz}\\sim 1$. In our model where we take $a_{\\rm out}^{M}=0.5$ pc, we find a similar distribution of eccentricities. Binaries merging in galactic nuclei typically have larger eccentricities than those formed through most other channels, particularly mergers in isolated binary evolution and in SBHBs ejected from star clusters. However, mergers that follow from the GW capture scenario in clusters \\citep{zevin18} and galactic nuclei \\citep{gondan2018,rass2019}, from resonant binary-single scattering in clusters \\citep{sam18}, from hierarchical triples \\citep{ant17,fragk2019,flp2019,fralo2019}, and from BH binaries orbiting intermediate-mass black holes in star clusters \\citep{fragbr2019} also present a similar peak at high eccentricities. We find that typically $\\sim 20$--$30\\%$ of binaries have $e\\gtrsim 0.1$ in the LIGO band \\citep{gond2019}. In our runs, we also find that some of the CO binaries that merge do not merge due to LK oscillations, but instead merge by emission of GWs on timescales longer than $1$ Myr and with eccentricities at $10$ Hz much smaller than the typical eccentricity reported in Figure~\\ref{fig:eccligo}. Their relative fraction is typically $\\sim 20\\%$-$50\\%$ of the total mergers we find in our simulations. When their contribution is taken into account, the fraction of binaries that enter the LIGO band with $e\\gtrsim 0.1$ decreases to $\\sim 10\\%$-$20\\%$. This fraction is still larger than previously estimated values ($\\sim 1$\\%) found in the literature \\citep{antoper12,van16,rand18}, probably due to the different integration schemes adopted. In a secular approach, the averaged equations of motion could smear out the peak at high eccentricities and lower the number of binaries entering the LIGO band with very high eccentricities.\n\nFinally, the high eccentricities we find in those binaries that merge may imply that a fraction of these binaries could emit their maximum power at higher frequencies, possibly in the range of LISA.\n\n\\begin{figure}\n\\begin{centering}\n\\includegraphics[scale=0.5]{ecc.pdf}\n\\par\\end{centering}\n\\caption{Distribution of eccentricities at the moment the binaries enter the LIGO frequency band ($10$ Hz) for BH-BH mergers in a Milky Way-like nucleus, for different values of $\\beta$ and $\\alpha$. The vertical line shows the minimum $e_{\\rm 10Hz}=0.081$ where LIGO\/VIRGO\/KAGRA network may distinguish eccentric sources from circular sources \\citep{gond2019}. A significant fraction of binaries have a significant eccentricity in the LIGO band.}\n\\label{fig:eccligo}\n\\end{figure}\n\n\\section{Merger time distributions and rates}\n\\label{sect:timedistrates}\n\n\\begin{figure*} \n\\centering\n\\begin{minipage}{20.5cm}\n\\hspace{0.5cm}\n\\includegraphics[scale=0.5]{ratebhbh.pdf}\n\\hspace{1.5cm}\n\\includegraphics[scale=0.5]{tgwtfinbhbh.pdf}\n\\end{minipage}\n\\begin{minipage}{20.5cm}\n\\hspace{0.5cm}\n\\includegraphics[scale=0.5]{ratebhns.pdf}\n\\hspace{1.5cm}\n\\includegraphics[scale=0.5]{tgwtfinbhns.pdf}\n\\end{minipage}\n\\begin{minipage}{20.5cm}\n\\hspace{0.5cm}\n\\includegraphics[scale=0.5]{ratensns.pdf}\n\\hspace{1.5cm}\n\\includegraphics[scale=0.5]{tgwtfinnsns.pdf}\n\\end{minipage}\n\\caption{Left panel: cumulative merger time ($\\tmerg$) distribution for BH-BH (top), BH-NS (centre) and NS-NS (bottom) binaries, for all models with $f(\\ain)$ from \\citet{hoan18}. Right panel: $\\tmerg$ as a function of the nominal \\citet{pet64} GW merger time-scale $T_{GW}$, for the same models as shown in the left panel.}\n\\label{fig:tmerg}\n\\end{figure*}\n\n\\begin{figure}\n\\includegraphics[width=8cm]{figure_8_revised.png}\n\\caption{\\label{fig:tdist} Cumulative distribution of merger times for merged orbits. Red: \\textsc{archain} N-body simulation. The analytic merger estimate for the \\textsc{archain} initial conditions is from Eq. \\ref{eq:T_LKGW} is shown in blue (secular approximation) and green (GPF18 correction, see text). Cyan: \\textsc{SecuLab} secular simulation. The distribution of initial conditions is the same but the sample is larger ($N=10^4$).}\n\\end{figure}\n\nAs discussed, the SMBH plays a fundamental role in reducing the merger timescale from the nominal value in Eq.~(\\ref{eq:t_merge}) \\citep{antoper12,hoan18,fraglei18b}. In Fig.~\\ref{fig:tmerg}, we present for all models the cumulative distribution of merger times ($\\tmerg$; left panel) for BH-BH (top), BH-NS (centre) and NS-NS (bottom) binaries. The merger time distribution is nearly independent of our assumptions for the BH mass function slope $\\beta$ and the CO binary spatial distribution slope $\\alpha$. It depends only on the SMBH mass. Larger SMBH masses imply shorter merger times due to more intense perturbations from the SMBH. In the right panel of Fig.~\\ref{fig:tmerg}, we show $\\tmerg$ as a function of the nominal \\citep{pet64} GW merger time-scale $T_{\\rm GW}$, for all binaries that merge in our simulations. Due to oscillations in the orbital elements, the CO binaries merge much faster than predicted by Eq.~\\ref{eq:t_merge}, by several orders of magnitude.\n\n\\subsection{Comparing secular techniques and N-body simulations}\n\\label{subsect:comparison}\n\nIn order to compare the distribution of merger times with different prescriptions, we take the initial conditions and calculate the merger time from Eq. (\\ref{eq:T_LKGW}). We find the maximal eccentricity by solving Eq.~(\\ref{eq:jmin_e0}) \\citep{grish18}. For the secular case, we use $\\epsilon_{\\rm SA}=0$, which also implies $\\delta e=0$ from Eq. (\\ref{eq:emax_corr}), while for the corrected case, $\\epsilon_{\\rm SA} > 0 $ is set from the system's initial conditions. If the merger time is shorter than the secular LK timescale that is required to reach the maximal eccentricity, we use the secular LK time. We then filter out the merger times longer than $1\\ \\rm Myr$ and compare to the simulated merger times. \n\nWe use the initial conditions of the MW NS-NS case with $\\alpha=2$ to calculate $T_{\\rm GW}^{\\rm LK}$ as prescribed above. In addition, in order to compare with \\textsc{archain}, we draw $N=10^4$ initial conditions from the same distribution and use a secular code to evolve them up to $T=1\\ \\rm Myr$ and record their merger times. For the secular code, we use \\textsc{SecuLab}\\footnote{https:\/\/github.com\/eugeneg88\/SecuLab}, a publicly available code that solves the secular equations of motion up to octupole order with additional secular 2.5PN terms. \n\nTable \\ref{tab:tmerge_nsns} shows the expected merger fractions from the direct N-body and the secular codes, respectively, together with different methods of evaluating the merger time. $T_{{\\rm GW}}^{{\\rm LK}}$ is evaluated once using the secular approximation (sec), and the corrected eccentricity (\\citealp{grish18}; GPF18). The merger rate is given by the fraction of initial conditions that result in $T_{{\\rm GW}}^{{\\rm LK}}\\le 1\\ \\rm Myr$. We see that $T_{{\\rm GW}}^{{\\rm LK}}$ slightly overestimates the merger rate in the secular regime, but underpredicts it in the N-body regime.\n\nOverall, the number of events from both simulations exceeds their expected analytic estimate. For the secular case, the ratio between the merger rate obtained numerically from \\textsc{SecuLab} and analytically from Eq. (\\ref{eq:T_LKGW}) is $0.004\/0.0028=1.43$. Similarly, for the N-body case, this ratio is $0.028\/0.01=2.8$. The ratio of merger rates for both simulations is about $0.028\/0.004 = 7$. The possible origins of these discrepancies are discussed below.\n\n\\begin{table}\n\\caption{Fraction of merger for the MW NS-NS case. First and second row are the fractions from \\textsc{SecuLab} and \\textsc{archain}, respectively. Third and forth row are the expected mergers from evaluating Eq. \\ref{eq:T_LKGW} with different maximal eccentricity evaluations (see text).}\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|}\n\\hline \n & SecuLab & ARCHAIN & $T_{{\\rm GW}}^{{\\rm LK}}$ sec & $T_{{\\rm GW}}^{{\\rm LK}}$ GPF18 \\tabularnewline\n\\hline \n\\hline \n$f_{{\\rm merge}}$ & $0.004$ & $0.028$ & $0.0028$ & $0.01$\\tabularnewline\n\\hline \n\\end{tabular}\n\\par\\end{center}\n\\label{tab:tmerge_nsns}\n\\end{table}\n\n\\begin{table}\n\\caption{$D$ and $p$ values for double-sided Kolmogorov-Smirnov (KS) tests of the cumulative distributions of the merger times.}\n\\begin{center}\n\\begin{tabular}{|c|c|c|}\n\\hline \n$(D,p)$ values & SecuLab & ARCHAIN\\tabularnewline\n\\hline \n\\hline \n$T_{{\\rm GW}}^{{\\rm LK}}$ sec & $0.4,0.23$ & $0.29,0.58$\\tabularnewline\n\\hline \n$T_{{\\rm GW}}^{{\\rm LK}}$ GPF18 & $0.2,0.52$ & $0.19,0.5$\\tabularnewline\n\\hline \n\\end{tabular}\n\\par\\end{center}\n\\label{tab:kstests}\n\\end{table}\n\nFigure \\ref{fig:tdist} shows the cumulative distribution of the merger times for the MW NS-NS case, where both secular and direct N-body simulations were used. Overall, the total number of mergers predicted by the corrected GPF18 model is larger, since larger eccentricities are involved, which is compatible with the numerical results. The secular model is less accurate with decreasing merger timescales; this is because the eccentricities involved are extreme, therefore deviations from the secular regime are more severe. \n\n\nOn longer merger timescales, $t_{\\rm merge} \\gtrsim 10^4 \\ \\rm yr$, if the typical LK timescales are short enough the maximal eccentricity attained, $e_{\\rm max}$, is usually larger than the predicted one from 1PN theory alone (see their Fig. 5 of \\citealp{grish18}). Therefore, the mergers occur of faster than expected and the CDF is underestimated. On the other hand, for short merger times, the maximal eccentricity of the GPF18 model unbound, and the merger occurs on the secular timescale. If, however, $t_{\\rm LK} \\sqrt{1 - e_{\\rm max}^2} \\lesssim P_{\\rm in}$, the GPF18 model also breaks down and cannot describe the system \\citep{ant17}, while the value of $e_{\\rm max}$ is stochastic. This is because the fraction of time the inner orbit spends near $e_{\\rm max}$ (i.e. $\\sim t_{\\rm LK} \\sqrt{1 - e_{\\rm max}^2}$) is too short for the inner orbit to complete one revolution and reach pericentre. Thus it takes longer time (at least a few secular times) to merge and the CDF is overestimated.\n\nIn order to get a better sense of the corrected prescription, we preform two sided Kolmogorov-Smirnov (KS) tests comparing the cumulative distribution function (CDF) from the simulated results versus the secular merger time distributions. Table \\ref{tab:kstests} shows the resulting D values and p values upon comparing the $N$-body and secular simulations with the secular and the corrected GPF18 model. \n\nFor the KS statistics, the D values are better for the GPF18 model, since the distance between the simulated and GPF18 CDFs is smaller. The p values are comparable for all models. A possible statistical artifact could be the low number of predicted mergers for the secular case. This suggesting that neither of the models is comparable with the simulated distributions.\n\nTo summarize, the simulated distribution cannot be fully described by the GPF18 model in a statistical sense, but the overall trend of larger merger fractions on shorter merger timescales is consistent with the simulations. Thus, further improvements in the analytic understanding both in the GPF18 model and in the \\cite{pet64} formulae are highly desired and deserve future work.\n\n\\subsection{Merger Rates}\n\\label{subsect:rates}\n\n\\begin{table*}\n\\caption{Rates of CO binary mergers (in $\\ \\mathrm{Gpc}^{-3}\\ \\mathrm{yr}^{-1}$) as a function of the SMBH mass.}\n\\centering\n\\begin{tabular}{lcccc}\n\\hline\n$\\msmbh$ ($\\msun$) & $\\Gamma_{\\rm BH-BH}$ & $\\Gamma_{\\rm BH-NS}$ & $\\Gamma_{\\rm NS-NS}$ & $\\Gamma^{\\rm SF}_{\\rm NS-NS}$\\\\\n\\hline\\hline\n$4\\times 10^6$ & $0.52$ & $0.10$ & $1.71\\times 10^{-3}$ & $0.16$ \\\\\n$10^8$ \t\t\t& $0.24$ & $0.08$ & $1.27\\times 10^{-3}$ & $0.11$ \\\\\n$10^9$ \t\t\t& $0.17$ & $0.06$ & $0.41\\times 10^{-3}$ & $0.04$ \\\\\n\\hline\n\\end{tabular}\n\\label{tab:rates}\n\\end{table*}\n\nAlthough we explore only a limited number of SMBH masses, we note that the range is relatively representative of what is expected for galactic nuclei. With the results of our simulations in hand, we can derive the expected merger rates of BH-BH, BH-NS and NS-NS binaries. With this, we can infer the dependence of the rate on the distribution of SMBH masses in the nearby Universe. \n\nFollowing \\citet{hamer18}, we calculate the merger rate for CO binaries as\n\\begin{equation}\n\\Gamma(\\msmbh)=n_{\\rm gal} f_{\\rm SMBH} \\Gamma^{\\rm sup}_{\\rm CO} f_{\\rm bin} f_{\\rm merge}\\ ,\n\\end{equation}\nwhere $n_{\\rm gal}$ is the galaxy density, $f_{\\rm SMBH}\\approx 0.5$ is the fraction of galaxies containing an SMBH \\citep{anto15a,anto15b}, $\\Gamma^{\\rm sup}_{\\rm CO}$ is the compact object supply rate, $f_{\\rm bin}$ is the fraction\nof stars forming compact object binaries, and $f_{\\rm merge}$ is the fraction of mergers we find in our simulations. For galaxies, we assume that the SMBH number density scales as $\\Phi(\\msmbh)\\propto 1\/\\msmbh$ \\citep{aller02}, hence the integrated number density of galaxies scales as\n\\begin{equation}\nn_{\\rm gal}\\propto \\int \\Phi(\\msmbh) d\\msmbh \\propto \\log(\\msmbh)\\ .\n\\end{equation}\nAs in \\citet{hamer18}, we neglect the weak dependence on the SMBH mass and fix $n_{\\rm gal}=0.02$ Mpc$^{-3}$ \\citep*{cons05}. \n\nThe fraction of CO binaries in the GC strongly depends on the assumptions regarding their origins. Several possibilities have been discussed in \\citet{antoper12}; here we focus on two: ex-situ and in-situ origins. In the ex-situ scenario, stars form outside the nuclear cluster and then diffuse inwards. In the in-situ formation scenario, stars are formed in-situ close to the SMBH. We use simplified assumptions to estimate the supply rate in both cases. For a relaxed nuclear cluster, Fokker-Planck, Monte-Carlo and N-body simulations suggest that the fractions of BHs and NSs in the central 0.1 pc are of the order of $\\gamma_{\\rm CO}=0.06,\\,0.01$ for BHs and NSs, respectively (the higher BH fractions are due to mass-segregation), assuming the background stellar population has a continuous star-formation rate \\citep{hopm06}. Following \\cite{antoper12} we take initial binary fractions of $f_{\\rm bin}=0.1,\\,0.07$ for BHs and NSs, respectively. For the compact object formation rate, we assume that the compact objects are supplied to the galactic nucleus by $2$-body relaxation and mass segregation\n\\begin{equation}\n\\Gamma^{\\rm sup}_{\\rm CO}=\\frac{\\gamma_{\\rm CO} N_* (0.1\\ \\mathrm{pc})}{t_{\\rm seg}(0.1\\ \\mathrm{pc})}\\ \\propto \\msmbh^{(3-\\beta)\/\\beta}\\ ,\n\\end{equation}\nwhere $\\gamma_{\\rm CO}$ is the fractional number of compact objects; $t_{\\rm seg} = T_{\\rm 2b}(m_{\\rm bin}\/{\\rm M_\\odot})$ is the timescale for mass segregation for binaries with mass $m_{\\rm bin}$; and we assume $\\msmbh\\propto \\sigma^4$ \\citep{merr01}.\n\nNormalizing the rates to the Milky Way's Galactic Centre\n\\begin{equation}\n\\Gamma^{\\rm sup}_{\\rm BH}=2.5\\times10^{-6} \\left(\\frac{4\\times 10^6\\msun}{\\msmbh} \\right)^{1\/4} \\mathrm{yr}^{-1}\\ .\n\\end{equation}\n\\begin{equation}\n\\Gamma^{\\rm sup}_{\\rm NS}=2.3\\times10^{-8} \\left(\\frac{4\\times 10^6\\msun}{\\msmbh} \\right)^{1\/4} \\mathrm{yr}^{-1}\\ .\n\\end{equation}\nThe final expression for our rate becomes\n\\begin{equation}\n\\Gamma_{\\rm BH}(\\msmbh)= 3.5f_{\\rm merge}\\ \\mathrm{Gpc}^{-3}\\ \\mathrm{yr}^{-1} \\times\\left(\\frac{4\\times 10^6\\msun}{\\msmbh} \\right)^{1\/4} ,\n\\label{eqn:ratebh}\n\\end{equation}\n\\begin{equation}\n\\Gamma_{\\rm NS}(\\msmbh)= 3.2\\times 10^{-2} f_{\\rm merge}\\ \\mathrm{Gpc}^{-3}\\ \\mathrm{yr}^{-1} \\times\\left(\\frac{4\\times 10^6\\msun}{\\msmbh} \\right)^{1\/4} ,\n\\end{equation}\nwhich is weakly dependent on the SMBH mass, and where the merger fraction $f_{\\rm merge}$, which is typically a few up to a few tens of percents for the various models we considered, can be found in Tab.~\\ref{tab:models}. We note that in $f_{\\rm merge}$ we have also included the CO binaries that would merge by emission of GWs in timescales $>1$ Myr, without the assistance of LK oscillations. These are typically a few percent of the total mergers in the case of BH-NS and NS-NS binaries, while $\\sim 20$-$50\\%$ in the case of BH-BH binaries.\n\nNote that the evaporation time of NS binaries (see Eq.~\\ref{eqn:binevap}) could become comparable to the segregation time, and therefore the rates of NS-NS mergers could be even lower, if supplied from outside the central region of the nuclear cluster. If more massive stellar-BHs exist, the most massive ones will dominate the inner regions due to strong mass-segregation \\citep{alexander09,aharon16}, and can be resupplied into the inner regions faster, enhancing the rates by up to a factor of a few. The larger numbers and faster supply will therefore bias the mass-function of merging BHs through this process to higher masses. Also, the relaxation and mass-segregation times in non-cuspy nuclear clusters, or when no nuclear cluster exists (e.g. for SMBHs more massive than $\\sim10^8\\,$ M$_\\odot$), could be so long that the stellar density around the SMBHs is low. As a consequence, the resupply of stars close to the SMBH cannot be efficiently attained through 2-body relaxation processes, but is more likely to depend on the gas-inflow and in-situ star formation close to the SMBH \\citep{ant13,ant14}.\n\nThe star-formation rate close to non-resolved regions around SMBHs is difficult to estimate theoretically. Here, we try to use an empirical estimate based on our own resolved Galactic Centre \\citep[see e.g.][]{bart09}. Approximately $\\sim 200$ O-stars (likely to later form stellar black holes) are observed and inferred to have formed over the last 10 Myrs in the young stellar disk close ($\\sim 0.05-0.5$ pc) to the SMBH. The number of lower-mass B-stars in the same environment suggests that similar continuous star-formation has not occurred over the last 100 Myr. Based on these observations we may consider an in-situ formation rate of BHs of $\\sim 200\/10^8=2\\times10^{-6}$ yr$^{-1}$, i.e. comparable to the estimated supply rate of $2.5\\times 10^{-6}$ from mass-segregation of BHs from outside the central regions. The comparable formation rate of NSs, however, would increase their resupply rates to the same level as BHs, i.e. much higher than the resupply from NSs migrating in from the outside ($\\sim 2\\times 10^{-6}$ yr$^{-1}$) and thereby\n\\begin{equation}\n\\Gamma_{\\rm NS}^{\\rm SF}(\\msmbh)= 2.8 f_{\\rm merge}\\ \\mathrm{Gpc}^{-3}\\ \\mathrm{yr}^{-1} \\times\\left(\\frac{4\\times 10^6\\msun}{\\msmbh} \\right)^{1\/4}\\ .\n\\end{equation}\n\nTable~\\ref{tab:rates} reports the resulting rates as a function of the SMBH mass. Upon using the semi-major axis and eccentricity distributions following the prescriptions of \\citet{antoper12}, we typically get a merger fraction $\\sim 2-5$ times larger than in the case of adopting the initial conditions from \\citet{hoan18}. This is probably related to the fact that the semi-major axes of CO binaries are typically smaller in the former case. BH-NS binaries should have mass segregation times similar to BH-BH binaries, hence we use Eq.~\\ref{eqn:ratebh}. For all SMBH masses considered in this study, the rates are in the range $\\sim 0.17$-$0.52 \\ \\mathrm{Gpc}^{-3}\\ \\mathrm{yr}^{-1}$, $\\sim 0.06$-$0.10 \\ \\mathrm{Gpc}^{-3}\\ \\mathrm{yr}^{-1}$ and $\\sim 0.41$-$1.71\\times 10^{-3} \\ \\mathrm{Gpc}^{-3}\\ \\mathrm{yr}^{-1}$ for BH-BH, BH-NS and NS-NS binaries, respectively. In the star-formation channel, the NS-NS rate may be as high as $\\sim 0.04$-$0.16 \\ \\mathrm{Gpc}^{-3}\\ \\mathrm{yr}^{-1}$. We note that the merger rate is a decreasing function of the SMBH mass, even though the relative fraction of merger events is typically larger for more massive SMBHs (see Tab.~\\ref{tab:models}). On the other hand, more massive SMBHs imply longer relaxation times, that contribute to a reduction in the merger fraction and make the relative rates smaller. Finally, we also run one model where we take $a_{\\rm out}^{M}=0.5$ pc, to check how the results depend on this parameter; wide binaries can be affected by LK cycles at distances larger than $\\sim 0.1$ pc. In this case, we find that typically our merger fraction $f_{\\rm merge}$ is reduced by a factor of $\\sim 2$--$3$\\footnote{Note that these binaries could however be perturbed by dynamical interactions \\citep[see e.g.][]{leigh16}}.\n\n\\subsection{Comparison of merger rates to previous studies} \nWe find rates comparable with though somewhat different from the ones of \\citet{antoper12} and those of \\citet{petr17} for spherical clusters (the latter finds ten times higher rates for the case of a non-spherical nuclear cluster - not modelled, or compared with in our work), but lower with respect to other works that explored the role of the SMBH in reducing the merger timescale of binaries due to Lidov-Kozai oscillations \\citep{hoan18}. Other merger channels typically predict larger rates. Typical values for globular clusters are $\\sim 2-10$ Gpc$^{-3}$ yr$^{-1}$ \\citep{askar17,frak18,rod18} and for nuclear star clusters are $\\sim 1-15$ Gpc$^{-3}$ yr$^{-1}$ \\citep{ant16}. For reference, the BH-BH merger rate inferred by LIGO is $\\sim 12-213$ Gpc$^{-3}$ \\citep{abbott17}.\n\nBefore directly comparing different estimates, we emphasize that any rate estimate is highly uncertain and should be considered as an order-of-magnitude estimate, since it depends on the specific assumptions regarding the star-formation history in galactic nuclei and the supply rate of compact objects at various distances from the SMBH, which remains poorly constrained in the literature.\n\nWe note that previous studies that used a similar approach made use of much larger supply rates, $\\sim 10$-$40$ times higher than considered here. The differences arise for several reasons, in particular from the different assumptions on the star-formation rate in the Galactic Centre and nuclear star cluster. \\citet{petr17}, \\citet{hamer18} and \\citet{hoan18} considered a resupply rate of BHs of $10^{-5}-10^{-4}$ yr$^{-1}$ from star-formation, where the latter rate is derived assuming a top-heavy initial mass function from \\citep{maness07}. However, this gives rise to several difficulties: (1) This rate is based on the formation rates derived by \\citet{lockmann09a} and \\citet{lockmann09b} for both NSs and BHs lumped together, while the formation rate for NSs is actually $\\sim 8$ times higher than that of BHs for regular (e.g. Kroupa or Miller-Scalo) IMFs, and becomes comparable only for top-heavy IMFs; one can therefore not discuss the number of BHs taking these numbers at face value, but consider the division between BHs and NSs (this issue was accounted for by \\citet{petr17}, \\citet{hamer18}, but not in \\citet{hoan18}), and its dependence on the assumed IMF. (2) The higher rate estimates are based on a top-heavy IMF, which in-turn is based on results from \\citet{maness07}. However, this top-heavy IMF was only derived from observations of old low-mass stars and the results are therefore highly problematic for the use in this context, as they are based on a large extrapolation from the low-mass regime up to that of NS and BH progenitors not probed at all by \\citet{maness07}. Also note that low-mass stars could already dynamically evolve through mass-segregation and their observed distribution in the GC does not necessarily reflect the actual IMF \\citep[e.g. see][]{aharon15}. Moreover, direct observations of {\\emph{massive stars}} in the GC today, though suggestive of a somewhat top-heavy IMF \\citep{lu13}, find a much steeper power-law of $-1.7\\pm0.2$ compared with $\\sim-0.8$ in the Maness et al. study of low-mass stars (which also supersedes the shallow power-law results from from Bartko et al. 2010 regarding massive stars), compared with $-2.3$ for Salpeter or Kroupa IMFs in the relevant mass-range. (3) Even more important, all of these estimates considered star formation throughout the nuclear cluster, rather than the innermost regions, where the induced mergers actually take place (especially in the cases considered by \\cite{hoan18}), i.e. the COs would have to migrate inwards over long timescales, and one should then refer to the ex-situ resupply rate discussed above. Indeed, a recent paper by Zhang et al. (2019) exploring the long-term evolution of binaries in the cluster shows that the long-term softening and disruption of binaries due to stars and the SMBH effectively quench the contribution of secular evolution induced mergers, consistent with the points raised above in the case of ex-situ supply. The overall numbers of COs derived when assuming in-situ formation in these other papers are therefore at least 10 times higher than actually expected in the central parts, and, in fact, are at least 10 times higher than what one can infer from the observed young massive stars in the GC \\citep{bart09,bart10}, or those inferred from X-ray sources \\citep{muno05}.\n\nWe also briefly note that the binary fraction of BHs and NSs in nuclear clusters is not well known, and the different studies make somewhat different assumptions and definitions in their calculations; for example, we calculate the fraction of BHs\/NSs given the fraction of massive progenitors in the population and then multiply by the binary fraction, making use of past theoretical and observational studies of BH\/NSs, while \\cite{petr17} and \\cite{hamer18} calculated the total binary fraction, including the dependence on the mass-function derived from stellar evolution. The differences on this point, however, are of order a factor of 1-2, and do not amount to a significant difference. Finally, our models originally considered binaries up to 0.1 pc where most of the current star-formation is observed (the young stellar disk). As mentioned above the merger fractions we find at larger separations up to 0.5 pc are smaller, and therefore, in our models these can contribute at most a comparable number of sources. This potential factor of two in the rate could then also help reconcile some of the differences in comparisons with the previous models which effectively considered star-formation throughout the central pc.\n\nIn summary, we believe the overall higher rate estimates near SMBHs by a factor of a few up to $\\sim 10s$ in some of the previous studies mostly arise from the different assumptions on the star-formation rate in the Galactic Centre and nuclear star cluster. If one assumes that it formed only through in-situ star formation and that star formation occurred as observed today (at $\\lesssim 0.1$ pc) throughout the last $\\sim 10$ Gyr, then the resupply rate considered in previous studies should be valid. For the Milky Way, however, these assumptions seem not to be consistent both with the star-formation history over the last $100$ Myrs given the observations of OB stars and X-ray sources (as discussed above), and with the inferred long-term star-formation history over the lifetime of the Galaxy (Nogueras-Lara et al., in prep; R. Sch{\\\"o}del, private communication), suggesting most of the stars in the Galactic Centre formed very early ($\\gtrsim 8$ Gyrs), and at most a few percents formed in the last Gyr. Nevertheless, the overall star-formation history in our Galaxy remains poorly constrained, and one can not exclude that a more vigorous and continuous star-formation could occur in the nuclei of other galaxies, in which case a higher rate estimates may apply, thus a higher merger rate of COs.\n\n\\section{Electromagnetic counterparts and observational signatures of SMBH-induced CO mergers}\n\\label{sect:implications}\n\nAs we noted previously (cfr Sec.~\\ref{sect:bhnsmergers}) the very high (close to unity) eccentricity, with which the GW signal enters the LIGO band in the scenario explored potentially proviude an important\nobservational diagnostic of CO mergers induced by LK oscillations. In the following, we discuss further\nobservational diagnostics of this merger channel in relation to\npossible electromagnetic (EM) counterparts to the mergers.\nIn particular, mergers of compact object binaries are expected to be associated with a strong release of electromagnetic radiation, if the right conditions arise to power an energetic outflow.\n\nIn the case of a NS-NS merger, tidal disruption during the inspiral phase leaves behind an accretion torus surrounding the merged object (either a NS or a BH), unless the two NSs have identical masses \\citep{Shibata2006,Rezzolla2010,Giacomazzo2013,Hotokezaka2013,Kiuchi2014,Ruiz2016,Radice2016}. An energetic engine can be driven by rapid accretion onto to the remnant object and\/or by dipole radiation losses if the remnant is an hypermassive or stable NS \\citep{Giacomazzo2013b,Ciolfi2017}. Growth and collimation of magnetic fields during the merger, as well as neutrino losses, are then believed to\npower a relativistic outflow. Dissipation within the expanding flow, and later interaction of the flow with the interstellar medium, gives rise to radiation that spans a wide window in the electromagnetic spectrum, from high-energy $\\gamma$-rays down to the radio. This basic scenario has been observationally confirmed with the recent event GW170817\/GRB170817A\n\\citep{Abbott2017a}.\n\nMergers of BH-NS binaries (always resulting in a BH as the resulting compact remnant) are expected to be accompanied by the formation of an hyperaccreting disk only if the mass ratio between the BH and NS does not exceed the value $\\sim3-5$, with the precise value depending on the equation of state of the NS and the BH spin \\citep{Pannarale2011,Foucart2012,Foucart2018}. For larger mass ratios, the tidal disruption radius of the NS is smaller than the radius of the innermost stable circular orbit, and no disk\nwill form, resulting in a direct plunge into the BH (see e.g. \\citealt{Bartos2013} for a review). If a rapidly accreting disk forms, then the resulting EM phenomenology is expected to be similar to that of the NS-NS case, at least in so far as the bulk properties are concerned. For the initial conditions explored in this work, $\\sim 10$-$20$\\% of mergers is expected to have mass ratios $\\lesssim 5$, and hence possibly giving rise to an accretion-powered EM counterpart.\n\nIn the case of a BH-BH binary merger, there is no tidally disrupted material which can readily supply the accretion power for a relativistic outflow\\footnote{Note however that alternative scenarios, involving pure electromagnetic energy, have also been invoked as alternatives to accretion to produce energy \n\\citep{Zhang2016,Liebling2016}.}. However, following the tentative detection of a $\\gamma$-ray counterpart by the \\textit{Fermi} satellite to the event GW150914 \\citep{Connaughton2016}, several\tideas were proposed for\tproviding the merged BH with a baryonic remannt to accrete from \\citep{Perna2016,Loeb2016, Woosley2016,Murase2016, Stone2017, Bartos2017,Kimura2017, Janiuk2017,DeMink2017}. Within the context of this study, the scenario proposed by \\citet{Bartos2017} is of particular relevance; they note that BH-BH binaries merging within an AGN disk can accrete a significant amount of gas from the disk, well above\tthe Eddington rate, and\tpossibly give rise to high-energy EM emission.\n\nElectromagnetic counterparts to binary mergers provide crucial information on the production mechanism of the binaries, since they can potentially allow a much better localization compared to GWs\nalone. A distinctive signature of binary mergers enhanced by LK oscillations in the vicinity of SMBHs is their relatively short merger timescale compared to that of other formation channels. For example, the classical channel of isolated binary evolution predicts merger times $\\sim $~100~Myr-15~Gyr \\citep{Belczynski2006}. The short lifetimes of the binaries, coupled with their production in the galactic centers, lead to correspondingly short distances traveled prior to mergers. We find that these distances are typically $\\lesssim 0.1$~pc, which makes these merger events practically occurring within the close nuclear region. This constitutes a major\tdifference with respect to the\tstandard isolated binary\tevolution scenario \\citep{Perna2002,Belczynski2006,Oshau2017,Perna2018}: whether it is a small or a large galaxy, the bulk of the merger events occurs at projected distances (from the galaxy center) $\\ga 100$~pc\\footnote{Note that, even if the isolated binary evolution scenario does predict a fraction of tight binaries with sub-Myr lifetimes \\citep{Belczynski2006}, and even ultra-short merger times \\citep{Michaely2018}, but the merger sites are still dominated by large scales since isolated binaries are born throughout the galactic disk.}. Localization via EM counterparts hence becomes an especially useful discriminant.\n\nShort GRBs associated with NS-NS mergers, and BH-NS mergers with a small enough mass ratio to allow tidal disruption, are expected to be followed by broadband radiation called afterglow, resulting from the\ndissipation of a relativistic shock propagating in the interstellar medium \\citep*{Sari1998}. The maximum flux intensity (at any wavelength) is given by\n\\begin{equation}\nF_{\\nu,{\\rm max}}=\n110\\;{n_1}^{1\/2}{\\xi_B}^{1\/2}\\;E_{52}D_{28}^{-2}\\;(1+z)\\;{\\rm mJy}\\;,\n\\label{eq:Fnu2} \\end{equation}\nwhere $E_{52}$ is the explosion energy in units of $10^{52}$~erg, $D_{28}$ the luminosity distance in units of $10^{28}$~cm, $z$ is the redshift, $n_1$ the number density of the interstellar medium in cm$^{-3}$, and it is assumed that the magnetic field energy density in the shock rest frame is a fraction $\\xi_B$ of the equipartition value.\n\nThe broadband spectrum evolves with time, and we compute it numerically using the formalism of \\citet{Sari1998}. For an energy $E=10^{50}$~erg as typical of short GRBs, standard assumptions for the shock parameters and medium ambient density $\\sim $ a few cm$^{-3}$ (more typical of the inner regions of a galaxy), the afterglow luminosity in some representative bands (X-rays and radio) at some typical observation times is found to be $L_{[2-10]{\\rm kev}}\\sim 5\\times 10^{45}$~erg~s$^{-1}$ at $t_{\\rm obs}=1$~hr and $L_{5GHz}\\sim 6\\times 10^{30}$~erg~s$^{-1}$~Hz$^{-1}$ at $t_{\\rm obs}=7$~days.\nIn the X-rays, a representative flux threshold is the {\\em Swift}\/XRT flux sensitivity of $F_{\\rm lim}=2.5\\times 10^{-13}$~erg~s$^{-1}$~cm$^{-2}$, while in the radio, a 1hr integration with the VLA leads to a flux threshold for detection of $F_{\\rm lim}\\approx 50\\mu$Jy. In both these bands, detection would\t\nbe possible up to a redshift of $\\sim~2$, considerably larger than the LIGO horizon\\footnote{It should however be noted that in the X-rays, when the shock is still moving at relativistic speeds, relativistic beaming of the emission will lead to a reduced luminosity for jets which are not observed on-axis. The fraction of on-axis jets is expected to be on the order of $1\/20$ by taking the jet size of $\\sim 16$~deg inferred for short GRBs \\citep{Fong2015}.}. The detection distances are larger than in the isolated binary evolution scenario, for which the large traveled distances lead to a sizable fraction of mergers to occur in low-density environments, where the afterglow luminosity is considerably dimmer.\n\nAdditionally note that, independently of the post-merger EM signal, a fraction on the order\tof a few $\\times 10^{-3}$ of the GW sources is expected to be accompanied by an SN-type precursor \\citep{Michaely2018}. This is\tdue to the fact\tthat the distribution of the delay time between the\nlast SN\texplosion and the binary merger\thas a non-negligible tail of ultra-short times, on the order of 1-100~yr (see also \\citep{Dominik2012}).\n\nDetection of an EM counterpart to a GW event generally allows a redshift measurement. The redshift distribution of the channel studied here would be one that follows the star-formation rate, since the merger times are shorter or at most comparable to the lifetimes of the most massive stars (as a reference, the lifetime of a $\\sim 100M_\\odot$ star is about 1 Myr). This would hence constitute another observational diagnostics.\n\nThe relatively easier prospects for detecting EM counterparts from CO mergers in galactic nuclei makes this channel especially useful for extraction of astrophysical and cosmological information from combined\nGW\/EM detections. This includes, among other, measurements of the Hubble constant, new tests of the Lorentz invariance, constraints on the speed of GWs, probes of the physics of mergers and jet formation, constraints on the\tequation of state of neutron stars \\citep[see][and references therein]{Abbott2017a}.\n\n\\section{Discussion and summary}\n\\label{sect:conc}\n\nIn this paper, we have revisited the SMBH-induced mergers of compact binaries orbiting within its sphere of influence. While previous studies in the literature adopted the secular approximation for the equations of motion \\citep{antoper12,hamer18,hoan18}, here we have performed an extensive statistical study of BH-BH, NS-NS and BH-NS binary mergers by means of $\\sim 35000$ direct high-precision regularized $N$-body simulations, including Post-Newtonian (PN) terms up to order PN2.5. \n\nWe have shown that the secular approach breaks down for systems with mild and extreme hierarchies. We used the recent corrections to the maximal eccentricity $e_{\\rm max}$ and merger times in the quasi-secular regime \\citep[][GPF18]{grish18} and tested it against N-body population synthesis integrations. The total number of mergers is under-predicted by a factor of $\\sim 6-10$ in the secular approach, and by a factor of $\\sim 2-3$ by the corrected GPF18 model. The CDF of merger times fail to fit either of the distributions, although the D-value distance between the simulated CDF and the GPF18 model is closer than for the secular approach. The difference can be attributed to the original underestimate of $e_{\\rm max}$ in the latter model, which leads to slower and less frequent mergers. \n\nIn our numerical simulations, we have considered different SMBH masses, different slopes for the BH mass function and the binary spatial distributions, and different CO binary semi-major axis and eccentricity distributions. We find that the majority of binary mergers happen when the mutual inclination of the binary orbit and its center of mass orbit around the SMBH is $i_0 \\sim 90^\\circ$, as a consequence of the Lidov-Kozai mechanism. We have also shown that the distributions of the inner and outer semi-major axes of the merging binaries depend mainly on the mass of the SMBH and on the slope $\\alpha$ of the binary spatial distribution around the SMBH. On the other hand, the shape of the resulting CO mass distributions depend on the slope $\\beta$ of the BH mass function. BH mergers observed by LIGO seem to favour $\\beta\\sim 1$, if those mergers were to happen around SMBHs. We have also discussed that the fraction of binaries that enter the LIGO band with $e\\gtrsim 0.1$ is $\\sim 10\\%$-$20\\%$, larger than previous values found in the literature \\citep{antoper12,van16,rand18}, due to the different integration schemes adopted.\n\nWe have also calculated the resulting rates as a function of the SMBH mass. We find that the merger rates are a decreasing function of the SMBH mass and are in the ranges $\\sim 0.17$-$0.52 \\ \\mathrm{Gpc}^{-3}\\ \\mathrm{yr}^{-1}$, $\\sim 0.06$-$0.10 \\ \\mathrm{Gpc}^{-3}\\ \\mathrm{yr}^{-1}$ and $\\sim 0.41$-$1.71\\times 10^{-3} \\ \\mathrm{Gpc}^{-3}\\ \\mathrm{yr}^{-1}$ for BH-BH, BH-NS and NS-NS binaries, respectively. In the star-formation channel, the NS-NS rate may be as high as $\\sim 0.04$-$0.16 \\ \\mathrm{Gpc}^{-3}\\ \\mathrm{yr}^{-1}$. We find rates consistent with \\citet{antoper12}, but lower with respect to previous works \\citep{petr17,hamer18,hoan18}, which may have overestimated the amount of CO binaries supplied to galactic nuclei through star-formation.\n\nWe have also discussed the possible EM counterparts of these events. Due to their locations, these mergers may have higher probabilities of being detected also via their EM counterparts, hence making these CO mergers especially valuable for cosmological and astrophysical purposes.\n\nFinally, we note that we have adopted $1$ Myr for the maximum integration time in our simulations, since this limit sets a good compromise between the computational effort and the size of the statistical sample we generate. This choice is further justified by noting that the typical timescale for vector resonant relaxation to operate is $\\sim 1$-$10$ Myr, over which the mutual orbital inclination is reoriented by interactions with other background objects, and renders the 3-body approximation insufficient \\citep{hamer18}. Also, we have neglected the possible precession of the CO binaries' motion induced by continual weak interactions with other stars and COs in the stellar cusp surrounding the SMBH \\citep{alex17}. A comprehensive $N$-body study over long integration timescales that includes both the SMBH-induced Lidov-Kozai oscillations and the detailed effects of the background stars surrounding the CO binaries deserves consideration in future work.\n\n\\section*{Acknowledgements}\n\nGF is supported by the Foreign Postdoctoral Fellowship Program of the Israel Academy of Sciences and Humanities. GF also acknowledges support from an Arskin postdoctoral fellowship at the Hebrew University of Jerusalem. EG acknowledges support from the Technion Irwin and Joan Jacobs Excellence Fellowship for outstanding graduate students. EG and HBP acknowledge support by Israel Science Foundation I-CORE grant 1829\/12. NL and RP acknowledge support by NSF award AST-1616157. GF thanks Seppo Mikkola for helpful discussions on the use of the code \\textsc{archain}. Simulations were run on the \\textit{Astric} cluster at the Hebrew University of Jerusalem. The Center for Computational Astrophysics at the Flatiron Institute is supported by the Simons Foundation.\n\n\\bibliographystyle{mn2e}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}