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+{"text":"\\section{}\n\n\\subsubsection{Introduction}\nThe idea of fault-tolerant topological quantum computation is premised on both the existence of non-Abelian anyons and our ability to manipulate them~\\cite{Nayak2008}. Fractional quantum Hall (FQH) states are the best-established examples of topological phases with FQH states at filling fractions $\\nu=5\/2$ and $7\/2$ being arguably the strongest candidates for non-Abelian phases. They are predicted to have non-Abelian charge-e\/4 excitations if their ground states are in either the Pfaffian \/ Moore--Read ~\\cite{Moore1991}, anti-Pfaffian~\\cite{Lee2007a,Levin2007a} or particle-hole-symmetric Pfaffian ('PH-Pfaffian') ~\\cite{Bonderson2013b,Chen2014,Son2015,Zucker2016} universality class. In addition to their electrical charge, these excitations also carry the non-Abelian topological charge of Ising anyons ~\\cite{Nayak1996c,Nayak2008}, which can be understood as the presence of a Majorana zero mode~\\cite{Read2000}. A combined quantum state of a pair of such anyons (known as a \\emph{fusion channel}) can be viewed as an Abelian charge-e\/2 excitation (which is a ``conventional'' Laughlin quasiparticle), either with or without a neutral fermion~\\cite{Milovanovic1996} -- see Figure~\\ref{fig:braiding}. The presence or absence of the neutral mode determines the fermion parity of the state.  The nature of these excitations\u2014both their charge and statistics\u2014can be probed by interferometry experiments~\\cite{Fradkin1998,Stern2006a,Bonderson2006a,Bonderson2008a,Bishara2009a,Halperin2011a}.  The non-Abelian properties of the $e\/4$ quasiparticles should manifest themselves in the even-odd effect, whereby the interference between two different paths for an $e\/4$ quasiparticle is switched on or off whenever the difference between the paths encircles, respectively, an even or odd number of $e\/4$ quasiparticles, see Figure~\\ref{fig:braiding}. Meantime, the Abelian $e\/2$ quasiparticle should show interference regardless of the number of encircled quasiparticles, with a pattern similar to other Laughlin quasiparticles. In a realistic Fabry-P\\'{e}rot interferometer, the interference pattern should consist of oscillations due to all types of charged quasiparticles present in the system. It is the goal of this study to experimentally determine the full set of observed oscillation frequencies at $\\nu=5\/2$ and $7\/2$ and to compare the observed frequencies with theoretical predictions based on the braiding properties of the $e\/2$ and $e\/4$ quasiparticles.\n\nPrevious interferometry experiments at $\\nu=5\/2$~\\cite{Willett2007,Willett2009a,Willett2010a,Willett2013a,Willett2013b} have observed resistance oscillations consistent with charge $e\/4$ and charge $e\/2$  excitations displaying, respectively, non-Abelian braiding and Abelian braiding statistics.  Meanwhile, noise~\\cite{Dolev2008a}, tunneling~\\cite{Radu2008a}, and charge sensing measurements~\\cite{Venkatachalam2011} at $\\nu=5\/2$ have all found signatures of $e\/4$ quasiparticles but no indication of  $e\/2$ quasiparticles. These measurements did not probe the braiding statistics of the excitations, only their electrical charge.  By contrast, interferometry measurements can provide information about both the charge and the braiding statistics of the quasiparticles. A recent measurement of the thermal Hall conductivity~\\cite{Banerjee2018} is an indirect probe of the topological order of the bulk and, therefore, an indirect measure, at best, of the presence of quasiparticles with non-Abelian braiding statistics in the bulk.\n\nThe key novel feature of the study presented here is the new class of ultra-high mobility AlGaAs heterostructures in which the Al alloy layers are purified to the extreme, promoting stronger electron-electron correlations, and thus more robust quantum Hall states than previously attained. This improvement in the material purification also results in a substantially larger amplitude of resistance oscillations observed in new interferometer devices. Specifically, in addition to providing more solid evidence for the non-Abelian nature of the $\\nu=5\/2$ state, we report the first experimental evidence in support of the similar nature of the $\\nu=7\/2$ state. Furthermore, oscillations consistent with the even-odd effect associated with transport by non-Abelian charge $e\/4$ quaiparticles are stable in the time scales of hours or even days.  When an instability occurs, it takes the form of a $\\pi$ phase shift consistent with the change of the fusion channel of non-Abelian anyons, thus providing further evidence for the non-Abelian nature of the state. Meantime, the associated time scales strengthen the case for using such FQH systems as a platform for topological quantum computation.\n\nBoth heterostructure and interferometer designs used in this study allow us to address another potential issue that has been plaguing earlier interference studies. Specifically, resistance oscillations of a mesoscopic quantum Hall island can be due to some combination of the Aharonov\u2013Bohm (AB) and Coulomb blockade effects. The latter are expected to dominate in smaller devices~\\cite{Bhattacharyya2019}, and some early results~\\cite{Mcclure2012a,Camino2005b,Camino2007a} are consistent with this. However, here we use special heterostructures that contain additional conducting layers able to screen the long-range Coulomb interactions (see Section~\\ref{sec:S2} of Supplementary Materials). This layering promotes AB oscillations even in relatively small quantum Hall interferometers at $\\nu=5\/2$ and $7\/3$~\\cite{Willett2007,Willett2009a, Willett2010a,Willett2013a,Willett2013b}, yet retains the high quality needed to observe $\\nu=5\/2$ and $\\nu=7\/2$ states. The case for AB oscillations aided by parallel conductors is supported by recent measurements~\\cite{Nakamura2019} at $\\nu=1\/3$.  Our measurements here also employ large aperture interferometers that contribute to suppression of Coulomb blockade effects, thus working in conjunction with the heterostructure layering to promote AB.\n\nThe manuscript is arranged as follows. First we present an extensive review of quantum Hall interferometry, emphasizing the expected periodicities of the AB oscillations for both Abelian and non-Abelian quasiparticles expected at $\\nu=5\/2$ and $7\/2$.  The consequences of the non-Abelian fusion manifesting itself as the fermion parity are discussed (and displayed schematically in Figure~\\ref{fig:braiding}). Next we turn to the description of the experimental methods used to probe these predictions.  We then present our principal experimental results \u2013 the stable interference oscillations observed at $\\nu=5\/2$ and $7\/2$,  also demonstrating occasional $\\pi$ phase shifts.  We show the full power spectra of these AB oscillations and identify spectral peak positions observed at these filling fractions. We also present the data in support of stability of the non-Abelian fusion, which can be deduced from the oscillations attributable to charge $e\/4$ quasiparticles, with changes in fusion channel manifesting themselves as directly observed $\\pi$ phase shifts. In addition we demonstrate the ability to control a specific component of the interference spectrum attributable to different braiding processes. Taken together, these data significantly strengthen the case for both the non-Abelian nature of the FQH states at $\\nu=5\/2$ and $7\/2$ (while providing the first experimental evidence for the latter) and for their potential applications for quantum information processing.\n\n\n\\subsubsection{Introduction to fractional quantum Hall interferometry}\n\\label{sec:intro}\n\n\\begin{figure}[htb]\n\\centering\n    \\includegraphics[width=0.9\\columnwidth]{figure01.pdf}\n    \\caption{Fermion parity and the non-Abelian even--odd effect.\\\\\n(a) The fusion of two non-Abelian e\/4  quasiparticles has two possible outcomes, one with and one without a neutral fermionic mode.  The fermion parity is a quantum number associated with an even number of such quasiparticles. It is changed when one constituent e\/4  quasiparticle is braided by an external  e\/4 quasiparticle.\\\\\n(b) When $e\/4$ quasiparticles backscatter from the lower to the upper edge of the Hall bar at the two constrictions, the outcome depends dramatically on the parity of $e\/4$ quasiparticles inside the interferometer. If their number is even, quasiparticles propagating along two possible paths interfere akin to the familiar double slit interference. If this number is odd, the interference disappears since with each tunneling quasiparticle the fermion parity is either switched or not, depending on which path is taken. The final quantum states are therefore orthogonal to one another, which precludes interference.\n}\n    \\label{fig:braiding}\n\\end{figure}\n\nFabry-P\\'{e}rot interferometry in the two-dimensional electron gas (2DEG) in the quantum Hall regime is due to interference between two different paths by which electrical current can flow from source to drain along edge states and across constrictions (see Figure~\\ref{fig:interferometer}). The interference pattern is thus determined by the total phase difference accumulated along the two paths, which in turn consists of both the Aharonov--Bohm phase, determined by the charge of the propagating quasiparticles and the enclosed flux, and the statistical contribution, determined by the statistics of the quasiparticles and the number and type of the quasiparticles enclosed between the two paths. This overall phase difference can be changed experimentally by changing either the enclosed flux or the number of quasiparticles within the interferometer loop. As is discussed in more detail in Section~\\ref{sec:S1} of Supplementary Materials, for Abelian quasiparticles this change is given by\n\\begin{equation}\n\\Delta \\gamma_{e^*}  =2 \\pi\\left(\\frac{\\Delta\\Phi}{\\Phi_0} \\right)\\left(\\frac{e^*}{e}\\right)+2\\theta_{e^*}\\Delta N_{e^*}\n\\label{eq:AB phase}\n\\end{equation}\t\nIn this expression, $\\Delta\\Phi$ is the change in the encircled flux, $\\Phi_0=hc\/e \\approx 41\\,\\text{G}\\,\\upmu \\text{m}^2$ is the flux quantum, and $\\Delta N_{e^*}$ is the change in number of the enclosed quasiparticles of charge $e^*$. Their braiding statistics is described by statistical angle $\\theta_{e^*}$ \u2013 a phase acquired by the wavefunction upon counter-clockwise exchange of two identical quasiparticles. If the interfering quasiparticles are non-Abelian, the effect may be more pronounced as both the phase and the amplitude of the interference term can depend on the number of quasiparticles inside the interferometer~\\cite{Bonderson2006b}, leading e.g. to the predicted even-odd effect for $e\/4$ quasiparticles in non-Abelian $\\nu=5\/2$ or $7\/2$ QH states.\n\nAccording to Eq.~(\\ref{eq:AB phase}), two parameters can potentially be varied in interferometric studies: the encircled flux and the number of enclosed quasiparticles. While varying them independently may seem experimentally hard, the variation of different combinations of them is achieved by: (i) varying the side gate ($V_s$) at fixed magnetic field~\\cite{Willett2009a,Willett2010a,Willett2013a}, and (ii) varying the magnetic field at fixed gate voltage~\\cite{Willett2013b}.  The active area $A$ in an interferometer -- the area encircled by the current paths -- is defined by surface gates. Varying the applied voltages on these gates changes this area; consequently, it changes both the enclosed flux and the number of $e\/4$ quasiparticles randomly localized within the area. This method has shown signatures of both $e\/2$ and $e\/4$ quasiparticles and also demonstrated a pattern of oscillations consistent with the non-Abelian nature of the latter, specifically the aforementioned even-odd effect~\\cite{Stern2006a,Bonderson2006a} whereby the Aharonov--Bohm oscillations associated with electrical transport by $e\/4$ quasiparticles is only observed when an even number of $e\/4$ quasiparticles is localized in the interferometer loop while Aharonov--Bohm oscillations associated with electrical transport by $e\/2$ quasiparticles is always observed. When the magnetic field is varied with fixed gate voltage, the enclosed magnetic flux number and the enclosed quasiparticle number change in tandem (see Section~\\ref{sec:S1} of Supplementary Materials). Specifically, in the simplest model that assumes the independence of the active area of the interferometer from the magnetic field (thus discounting the possibility of so-called Coulomb domination ~\\cite{Halperin2011a,Keyserlingk2015}), the change in the number of bulk quasiparticles in response to the change in flux $\\Delta\\Phi$ is given by $\\Delta N_{e^*}=-(\\Delta\\Phi\/\\Phi_0 )(\\nu e\/e^* )$, resulting in\n\\begin{equation}\n\\Delta \\gamma_{e^*}=\\left(\\frac{\\Delta\\Phi}{\\Phi_0}\\right)\\left[2 \\pi\\left(\\frac{e^*}{e}\\right)-2\\theta_{e^*}  \\left(\\frac{\\nu e}{e^*}\\right)\\right].\n\\label{eq:AB_phase2}\n\\end{equation}\n\n\nThe putative non-Abelian nature of $e\/4$ quasiparticles should manifest itself in specific small-period oscillations centered around $5f_0$ for the $\\nu=5\/2$ state (and  $7f_0$ for the $\\nu=7\/2$ state), with $f_0= 1\/\\Phi_0$ being the oscillation frequency corresponding to the period of one flux quantum.  This is a consequence of the combination of the even-odd effect (illustrated schematically in Figure~\\ref{fig:braiding}; see also schematic for fermion parity) and the systematic variation of the $e\/4$ quasiparticle number as the magnetic field is changed ($B$-field sweep): the resistance oscillates with period corresponding to the flux needed to increase the quasiparticle number by two, namely the period of  $2\\Phi_0 e^*\/\\nu e=\\Phi_0\/5$ at $\\nu=5\/2$ and $\\Phi_0\/7$ at $\\nu=7\/2$. Such high-frequency peaks should be an unmistakable signature of non-Abelian statistics and they were first reported in the earlier study~\\cite{Willett2013b} at 5\/2 filling. However, a more complicated picture emerges when all quasiparticle types are considered.  Specifically, if both $e\/4$ and $e\/2$ excitations are present, one should observe oscillations due to all permutations of interfering and enclosed quasiparticles. A straightforward generalization of Eqs.~(\\ref{eq:AB phase})--(\\ref{eq:AB_phase2}) for the Abelian phase acquired by interfering quasiparticles of type $a$ encircling bulk quasiparticles of type $b$, which is given by:\n\\begin{multline}\n\\Delta \\gamma_{ab}=2 \\pi\\left(\\frac{\\Delta\\Phi}{\\Phi_0} \\right)\\left(\\frac{e_a}{e}\\right)+2\\theta_{ab}\\Delta N_b\n\\\\\n=\\left(\\frac{\\Delta\\Phi}{\\Phi_0} \\right)\\left[2 \\pi\\left(\\frac{e_a}{e}\\right)-2\\theta_{ab} \\left(\\frac{\\nu e}{e_b}\\right)\\right].\n\\label{eq:interferenece_general}\n\\end{multline}\nDirect application of this expression results in the oscillation periods of $\\Phi_0$ for $e\/4$ quasiparticles interfering around $e\/2$ quasiparticles and $\\Phi_0\/2$ for the $e\/2$ quasiparticles interfering around either $e\/4$ or $e\/2$ quasiparticles.  Finally, the interference of $e\/4$ around $e\/4$ quasiparticles would na\u00efvely result in the period of $\\Phi_0$ for the Moore--Read Pfaffian state and $2\\Phi_0\/3$ for the anti-Pfaffian state (see Section~\\ref{sec:S1} of Supplementary Materials for more detail). However, this is not the case since in both states the $e\/4$ excitations are actually non-Abelian. Therefore the interference turns on and off with each shift $\\Delta N_{e\/4}=\\pm 1$, resulting in the aforementioned small period of $\\Delta\\Phi=\\Phi_0\/5$. When the number of bulk $e\/4$ excitations is even, the interference is not simply governed by $\\theta_{e\/4}$; it also depends on the fusion channel of the enclosed quasiparticles. There are three basic possibilities: (i) the fusion channel, which determines the fermion parity, is fixed by the energetics and remains largely stable during the magnetic field sweep across the $\\nu=5\/2$ plateau, (ii) the fusion channel is random but its autocorrelation time is longer or comparable to the time it takes to change the flux by one flux quantum, and (iii) the fusion channel fluctuates rapidly on the time scale of changing the flux by $\\Delta\\Phi=\\Phi_0$. Focusing on the first scenario, let us assume that the net fusion channel of the bulk quasiparticles is always trivial. Physically this means that from the point of view of interference, the bulk is equivalent to a collection of $e\/2$ Laughlin quasiparticles, which would result in the aforementioned Abelian factor in the interference pattern, with period of $\\Phi_0$ irrespective of the exact nature of the $\\nu=5\/2$ state. The net result would be a convolution of non-Abelian $5f_0$  and Abelian $f_0$ oscillations, resulting in spectral peaks at $4f_0$ and $6f_0$. Were the fusion channel to contain a fermion instead, the bulk quasiparticles would to be in a different fermion parity state and the overall phase of Abelian oscillations would shift by $\\pi$ with no change in the oscillation period. In the second scenario, the fluctuations in the fusion channel -- fluctuations in the fermion parity -- would scramble the $f_0$ component (due to random $\\pi$ phase shifts throughout the magnetic field sweep) thus eliminating the beats, resulting in a single spectral peak at $5f_0$. Finally, in the third scenario, the interference of charge-$e\/4$ excitations around other $e\/4$ excitations would be eliminated entirely: their interference is suppressed for odd numbers of enclosed $e\/4$ quasiparticles by their non-Abelian nature and for even numbers by rapid phase fluctuations. Therefore, observations of high-frequency spectral peak(s) both in the previous~\\cite{Willett2013b} and present studies can be interpreted not only as a confirmation of the non-Abelian nature of the $\\nu=5\/2$ state but also as a validation of the results of the earlier side-gate studies as the main conceptual criticism of those was rooted in doubts about the fusion channel stability~\\cite{Rosenow2008a}. We should also note that the first and second scenarios may coexist within the sweep across the entire $\\nu=5\/2$  plateau: one could envision e.g. a situation whereby the parity is stable near the middle of the plateau while becoming progressively less stable closer to its margins, where the concentration of the bulk quasiparticles becomes larger and hence their typical distance to the edge smaller. The latter would enhance tunneling of neutral fermions between the edge and the localized quasiparticles, scrambling the well-defined fermion parity in the bulk~\\cite{Rosenow2008a}. In such a case one would find oscillation peaks at $4f_0$ and $6f_0$ near the middle of the plateau and $5f_0$ closer to its flanks.\n\nThis construct can also be applied to  $\\nu=7\/2$, where the charge $e\/4$ quasiparticle is similarly expected to obey non-Abelian statistics.  Upon magnetic field sweep the non-Abelian even-odd effect will manifest itself through the resistance oscillations with period corresponding to the magnetic field increment needed to change the number of $e\/4$ quasiparticles by two, which in terms of flux corresponds to the period of $2\\Phi_0 e^*\/(\\nu e)=\\Phi_0\/7$.  In the measured power spectrum of resistance interference oscillations this is a peak at frequency $7f_0$ . The corresponding frequency $7f_0$ in the Fourier spectrum may be then modulated by the frequency of $1.5f_0$ corresponding to $e\/4$ quasiparticles interfering around $e\/2$ quasiparticles, according to Eq.~(\\ref{eq:interferenece_general}). Therefore the expected spectral features can occur either at $7f_0$ or at $7 f_0 \\pm 1.5 f_0$, depending upon the fermion parity stability and the Fourier transform window.  Finally, the spectrum could also display a peak corresponding to $e\/2$ interfering around e\/2, and from Eq. (2) this frequency would be $3f_0$.  The $7\/2$ spectrum could then be comprised of peaks at either $1.5f_0$, $5.5f_0$ and  $8.5f_0$ or  at $1.5f_0$  and $7f_0$ (or, perhaps, at all of those); and in addition there might be a spectral peak at $3f_0$ as well.\nIn summary, the observed resistance oscillations should be a combination of interference patterns resulting from charge $e\/4$ Ising anyon encircling another $e\/4$ Ising anyon; a charge $e\/2$ Abelian anyon encircling another charge $e\/2$ Abelian anyon as well as both types of anyons encircling the other kind. In addition, the phase of the charge $e\/4$ quasiparticle interference should depend on the parity of neutral fermions inside the loop. At 5\/2 filling the first type of process leads to a resistance that oscillates with magnetic flux with frequency $5f_0$ whereas at $7\/2$ the corresponding frequency is $7f_0$. The $e\/2-e\/2$ interference in these states should result in oscillations with frequency $2f_0$ and $3f_0$ respectively; we will show later that the amplitude of these oscillations can be tuned independently of $e\/4$ interference oscillations, effectively allowing for them to be turned on or off.  Finally, the interference of $e\/4$ quasiparticles around Laughlin $e\/2$ quasiparticles produces oscillations with frequency $f_0$ at $\\nu=5\/2$ and $1.5f_0$ at $\\nu=7\/2$; its convolution with the first type of process results in oscillation frequencies $5f_0\\pm f_0$ and $7f_0\\pm 1.5f_0$ respectively. The stability of the fermion parity should determine whether a measured high-frequency spectral peak is split in this manner or remains centered at $5f_0$ or $7f_0$ (with an additional possibility of both scenarios occurring within the same plateau). Irrespective of those details, an appearance of such high-frequency spectral features in the vicinity of $5f_0$ at $\\nu=5\/2$ and $7f_0$ at $\\nu=7\/2$ should be an unmistakable signature of non-Abelian nature of those states.\n\n\n\\subsubsection{Methods}\n\\label{sec:methods}\nSeveral essential unique experimental methods contribute to the results in this study and are outlined here, followed by description of interference device operation needed to understand the results.  Specific heterostructure designs and growth features are crucially important: breakthrough improvement in the purity of Al in the GaAs\/AlGaAs heterostructure quantum wells, consequently increasing the electron correlation effects, and placement and electron population of conducting layers parallel to the principal quantum well to suppress Coulomb blockade.\nAluminum purity in our heterostructures was improved by first assessing the oxygen impurity levels in the heterostructure AlGaAs layers~\\cite{Chung2018}, then developing methods of offline Al-effusion furnace bakes to reduce these charged impurities.  These bakes ultimately resulted in AlGaAs barrier material used in the heterostructures of this study with about eight times fewer impurities than previous material~\\cite{Chung2018a}.  In these extreme high purity materials, and in previously grown heterostructures~\\cite{Willett2007,Willett2009a,Willett2010a,Willett2013a,Willett2013b}, a unique multiple conduction layering structure was employed for materials used in our interference measurements. Parallel to the principal quantum well both above and below, poorly conducting layers are grown that suppresses Coulomb blockade or domination of the interferometer's laterally confined electron layer in the principal well. These parallel layers are populated differentially by the doping layers, and illumination of the samples at low temperatures contributes further to that population: such illumination of the samples is distinctly unique to our method versus all others~\\cite{Dolev2008a,Radu2008a,Venkatachalam2011,Banerjee2018,Bhattacharyya2019,Mcclure2012a,Camino2005b,Camino2007a, Nakamura2019}.\nDifferent illumination, cool-down, and gating histories for a given sample can produce different electron populations in the layers. Each such history is numbered and in the results sections is referred to as the preparation number for each sample.  See Section~\\ref{sec:S2} of Supplementary Materials for details on heterostructure construction and illumination and on the Al purification method for this study. Also see Section~\\ref{sec:S2} for details of the preparation histories applied to samples used in our measurements.\n\n\\begin{figure}[thb]\n\\centering\n    \\includegraphics[width=\\columnwidth]{figure02a.pdf}\n    \\caption{\n   \n   \nSchematic and electron-micrograph images of interference devices.  The interferometer is defined by surface gates operated at voltages $V_b$ and $V_s$ that fully deplete the electron population below them.   Currents propagate along the edges of the sample and the gates: contacts diffused into the heterostructure away from the device are used to measure resistance across the device longitudinally, $R_L$, and in Hall configuration across the device $R_D$. Edge current can backscatter at one of the two constrictions resulting in interference between the two paths. Device images show one device with no structure inside area $A$ (top -- device type a) and another one with a top gate central dot (device type b). In each device the lithographic separation $d_g$ of gates defining the constrictions is $1\\upmu$m. The actual tunneling distance between the edge currents $d$ is controlled by the gate voltage $V_b$.\n}\n    \\label{fig:interferometer}\n\\end{figure}\n\nAn interference device is shown schematically in Figure~\\ref{fig:interferometer}.   The top gates are charged to a negative voltage sufficient to fully deplete the underlying electron layer.  At high magnetic fields as prescribed for 5\/2 filling factor (filling factor $\\nu$ being the ratio of the electron areal density to the magnetic flux density), the currents carrying the excitations of the fractional quantum Hall state will travel along the edge of these depleted areas, surrounding an area of the bulk filling factor $\\nu$. The important principal physical property of the interferometer device is two separated locations where these edge currents are brought in proximity (marked 1 and 2 in the Figure).  At these points backscattering from one edge to the other can occur, and with this backscattering two different current paths are established that can interfere, as shown by the dashed lines in the schematic.  The one path encircles the area marked A in the schematic, and changes in the magnetic flux number within area A or changes in the particle number within area A will cause phase accumulation for that path (the Aharonov--Bohm and statistical phase contributions). Interference of that path and the one not entering the area A produce oscillations in the resistance measured across the interference device.  The voltages on the top gates can be adjusted to promote backscattering (gates marked $V_b$) and to change the enclosed area A (gates marked $V_s$).  The separation of the backscattering top gates, distance marked  $d_g$ in Figure~\\ref{fig:interferometer}, is sufficiently large that for nominal voltages on $V_b$ the backscattering is weak, an important feature to maintain the 5\/2 fractional Hall state contiguously from outside to inside the active area A of the interferometer.  Note also that area A is ultimately the area which is enclosed by the edge states in the quantized Hall systems, and not the lithographic area.  The location of the edge states is determined electrostatically and can be modified by the applied gate voltages.  Although the lithographic area is several square microns, the active area A is typically less than one square micron. In one interferometer device type a small dot is placed centrally in the area A and is accessed by an air-bridge that extends over one of the side gates marked $V_s$ in Figure~\\ref{fig:interferometer}. Although such type b devices (i.e. those with a top central gate) are present in several of the samples used in this study, the central gate was kept grounded for all the measurements and preparations for the purposes of obtaining the data presented in this paper.  Also shown are electron micrographs of the interferometers.\n\nResistance and resistance oscillations are measured using low noise lock-in amplifier techniques.   A constant current (typically 2nA) is driven through the 2D electron system underlying the interferometer top gate structure, and the voltage, and so resistance, is determined with a four-terminal measurement.  The voltage drop along the same edge of the 2D electron system and across the device gives the longitudinal resistance $R_L$; across the device and across the two edges of the 2D system gives diagonal resistance $R_D$.  Similar measurements performed away from the interferometer device yield $R_{xx}$ and $R_{xy}$ respectively.\n\n\\begin{figure}[hbt]\n\\centering\n    \\includegraphics[width=0.85\\columnwidth]{Figure3.pdf}\n    \\caption{\n    (a) Longitudinal resistance $R_L$ measured across the interferometer between filling factors $\\nu=3$ and $\\nu=4$ in an ultra-high mobility heterostructure.  Integer and fractional quantum Hall states as well as phase-separated (nematic) states  are labeled. Temperature $\\sim20$mK, sample 6, preparation 16, device type b.\\\\\n    (b) A blow-up of $R_L(B)$ near the filling factor $7\/2$ demonstrating large-amplitude interference oscillations. Trace sets show oscillations near $\\nu=7\/2$ for  both up- and down- sweeps of magnetic field $B$.  The set of six sweeps shown here covers a time of 8.75 hours, 87.5 minutes for each directional sweep. The data demonstrate both a high level of reproducibility and stability for the time and magnetic field ranges over which the data were taken. See also Figures~\\ref{fig:S5-2-2}, \\ref{fig:add_osc1}, \\ref{fig:add_osc2}.\\\\\n    (c) A Fourier transform of these oscillations (red) overlayed with the Fourier transform of oscillations observed at $\\nu=3$ (blue). Vertical green lines mark the frequency $f_0$ of the integer spectral peak and its multiple $7f_0$, the expected frequency of the non-Abelian even-odd effect at $\\nu=7\/2$.\n}\n    \\label{fig:Hall_trace}\n\\end{figure}\n\nAn example of longitudinal resistance $R_L$ across an interferometer in an ultra-high mobility heterostructure characterized by improved Al purity is shown in Figure~\\ref{fig:Hall_trace}(a). (Also see Section~\\ref{sec:S4} of Supplementary Materials.) In comparison to heterostructures without improved Al purity (see Figure~\\ref{fig:power_spectra} and Figure~\\ref{fig:S5-2-1}), this resistance trace shows sharper resistance features throughout this filling factor range. A blow-up of the $R_L$ trace near $\\nu=7\/2$ reveals a set of reproducible oscillations shown in Figure~\\ref{fig:Hall_trace}(b). Their Fourier transform reveals a prominent peak at a frequency roughly seven times that of the main spectral peak observed at an integer filling fraction, as shown in Figure~\\ref{fig:Hall_trace}(c). A similar set of data for $\\nu=5\/2$ is shown in Figure~\\ref{fig:5_halves_interf} in the Results section below. The analysis of these oscillations, their spectra and their attribution to the non-Abelian even-odd effect is the main focus of this paper.\n\nThe interference oscillations in the measured resistance are analyzed by applying fast Fourier transforms (FFT) to the data.  Because the oscillations are observed near the minima of resistance of quantum Hall states, the corresponding background \u2013 the shape of the minimum - is subtracted before the FFT is applied.  The subtracted background is determined equivalently by either a polynomial fit or a running large element smoothing of the minimum.\nBy following this procedure at both integer ($\\nu=4$) and fractional ($\\nu=16\/5$) filling, we can test the validity of our approach and, specifically, confirm the expectation that our interferometers operate in the Aharonov--Bohm and not Coulomb-dominated regime, thus justifying the key assumption used in deriving Eq.~(\\ref{eq:AB_phase2}). Specifically, at the $\\nu=16\/5$ FQH state one expects a Laughlin state with $e^*=e\/5$ and $2\\theta_{e^*}=2\\pi\/5$.  Consequently, Eq.~(\\ref{eq:AB_phase2}) predicts the phase accumulation of $-6\\pi$ per additional flux quantum resulting in the expected AB periodicity of $\\Delta\\Phi = \\Phi_0\/3$, which is what we observe experimentally -- see Figure~\\ref{fig:integer_vs_fractional_peak_positions}.\n\\begin{figure}[htb]\n\\centering\n    \\includegraphics[width=0.9\\columnwidth]{integer-fifth-comparison.pdf}\n    \\caption{A comparison of magnetic field sweeps and their Fourier transforms at $\\nu=4$ and $\\nu=16\/5$ in the same sample (sample~6, preparation~25, device type~b, $T \\sim 20$mK). A dominant spectral feature observed at $\\nu=16\/5$ is located at approximately three times the frequency of the integer peak, as expected from Eq.~(\\ref{eq:AB_phase2}).}\n    \\label{fig:integer_vs_fractional_peak_positions}\n\\end{figure}\n\nFurther evidence of the Aharonov--Bohm nature of the observed interference is provided by the ``pajama plot'' allowing one to trace the lines of constant phase in the $B-V_s$ plane. For the Aharonov--Bohm oscillations, one expects these lines to have a negative slope whereas Coulomb domination should result in a positive slope~\\cite{Halperin2011a}; the data shown in Figure~\\ref{fig:pajama_plot} for $\\nu=3$ are clearly consistent with the former. (Figures~\\ref{fig:integer_vs_fractional_peak_positions} and \\ref{fig:pajama_plot} show data for different preparations of the same sample.)  Also see Section~\\ref{sec:S4} of Supplementary Materials for more details on the Aharonov--Bohm nature of the interference observed at integer filling factors in our samples.\n\\begin{figure}[bht]\n\\centering\n    \\includegraphics[width=\\columnwidth]{integercolorplot.pdf}\n    \\caption{A ``pajama plot'' showing the resistance oscillations at $\\nu=3$ as a function of both magnetic field $B$ and side gate voltage $V_s$ consistent with the Aharonov--Bohm as opposed to Coulomb dominated nature of observed oscillations (sample~6, preparation~3, device type~b).}\n    \\label{fig:pajama_plot}\n\\end{figure}\n\n\nFinally, we would like to reiterate the importance of two fundamental improvements between the new high Al purity shielded wells heterostructures and the previously used shielded well samples: the high Al purity samples can display a ten-fold increase in the amplitude of the interference oscillations at $\\nu=7\/2$ and $5\/2$ compared to those previously observed in shielded-well samples, and, furthermore, the high Al purity materials also demonstrate sharper definition of the fractional states and the reentrant phases.  This later point is shown comparing the $R_L$ data in Figures~\\ref{fig:Hall_trace}(a) and \\ref{fig:5_halves_interf} versus that of Figure~\\ref{fig:power_spectra}(a) (comparison is also made in supplemental Figure~\\ref{fig:S5-2-1}). Figure~\\ref{fig:power_spectra}(a) shows only continuous evolution in $R_L$ from the re-entrant phases to, for instance, the 5\/2 minimum, in stark contrast to the abrupt changes in $R_L$ in sweeping the magnetic field from the re-entrant phases to $\\nu=5\/2$ shown in Figure~\\ref{fig:5_halves_interf}. In Figure~\\ref{fig:Hall_trace}(a) the transitions from re-entrant phases to $\\nu=7\/2$ are also distinct.  It is posited but not proven that this relative sharpening of the re-entrant features, less mixing with the target $\\nu=5\/2$ and $\\nu=7\/2$ states, may contribute to the larger amplitude of the oscillations at those FQH states.\n\n\n\n\n\\subsubsection{Results}\n\\label{sec:results}\n  \t\nWe now turn to our main findings, beginning with the analysis of the interference oscillations for the first time observed at $\\nu=7\/2$. Figure~\\ref{fig:Hall_trace}(a) depicts a representative quantum Hall trace between integer fillings $\\nu=3$ and $4$ observed in a higher purity sample. It shows a well-defined feature at filling fraction = $7\/2$.  Zooming into the corresponding resistance minima reveals the oscillating behavior of the resistance $R_L$ as a function of the magnetic field. As can be seen in Figure~\\ref{fig:Hall_trace}(b), the observed oscillations are both prominent, with the typical amplitude of order $20\\Upomega$, and remarkably reproducible \u2013 the figure presents the results of six different magnetic field sweeps separated from one another by hours. We should also note the generic nature of these oscillations: they are seen in different preparations and within a significant range of gate voltage settings -- see Figures~\\ref{fig:add_osc1} and \\ref{fig:add_osc2} of Supplementary Materials. The most significant feature of these oscillations is their frequency. Indeed, as we will demonstrate, it is inconsistent with any Abelian AB interference but is consistent with the non-Abelian even-odd effect.\n\nIn order to analyze the origin of these oscillations, we must first establish the reference frequency for our interferometer.  Because we do not know the precise active area of the interferometer, we rely on the oscillation spectra measured at integer filling factors to extract the reference frequency $f_0$ -- see e.g Figures~\\ref{fig:Hall_trace}(c) and~\\ref{fig:integer_vs_fractional_peak_positions} or Figure~\\ref{fig:S5-2-1} of Supplementary Materials.  Since only electrons can interfere in the integer QH regime, the oscillation period observed as the magnetic field is swept across the integer plateau corresponds to the change of one flux quantum through the active area of the interferometer.\nThis procedure has been repeated for each sample, each cool-down, and each gate voltage setting (see also Figure~\\ref{fig:power_spectra}, which exemplifies this procedure).  Equipped with the knowledge of the electron AB oscillation frequency $f_0$ we proceed to examine the oscillation spectra at filling fractions $7\/2$ and $5\/2$.\n\n\\begin{figure}[htb]\n\\centering\n    \\includegraphics[width=0.95\\columnwidth]{figure04.pdf}\n    \\caption{Interference oscillations at $\\nu=7\/2$ and their power spectra.\\\\\n(a) Transport between filling factors $\\nu=3$ and $\\nu=4$.\\\\\n(b)\\&(c) Successive blow-ups of $R_L(B)$ around the minimum at $\\nu=7\/2$, demonstrating sets of interference oscillations (sample~6, preparation~8, device type~b, $T \\sim 20$mK).\\\\\n(d) The Fourier spectrum of the sweep in panel (c) with the FFT window marked by the red vertical lines. $f_0$ marks the position of the FFT peak at $\\nu=3$ for this sample and preparation (see Figure~\\ref{fig:S5-2-1}).\n\\\\\n(e) An FFT using a larger range of $B$-field centered around $\\nu=7\/2$ can potentially express the full complement of $e\/4$  and $e\/2$ braids as shown for $\\nu=5\/2$ in Figure~\\ref{fig:power_spectra}. The FFT window corresponding to $\\nu = 3.5\\pm 0.03$ is marked by red vertical lines panel (b). The vertical lines marked $1.5f_0$, $3f_0$, $5.5f_0$, $7f_0$, and $8.5f_0$  are the respective multiples of $f_0$ identified at the integer filling, which correspond to expected features at $\\nu=7\/2$ (see text for details). A broad spectral feature is centered around $7f_0$, with a sharp \\emph{minimum} developing at that frequency.\nAdditional $\\nu=7\/2$ spectra are shown in Figure~\\ref{fig:interf_model}(b) and in Section~\\ref{sec:S5-4} of Supplementary Materials, all demonstrating a large spectral feature centered at their respective $7f_0$ frequencies, as expected for non-Abelian $e\/4$ at $\\nu=7\/2$.\n}\n    \\label{fig:7_halves_interf}\n\\end{figure}\n\nHaving established the reference frequency $f_0$, we focus our attention to the interference signal at $\\nu=7\/2$ shown in Figure~\\ref{fig:Hall_trace}. A crucial parameter in  the analysis of  the resistance oscillations is the width of the Fourier transform window. Figure~\\ref{fig:7_halves_interf} shows a series of successive zooms into the resistance minimum at this filling and the corresponding Fourier transforms. The measurements presented here have been done using a different preparation of the same sample that was used for the measurements in Figure~\\ref{fig:Hall_trace}. Similarly to what has been shown in Figure~\\ref{fig:Hall_trace}, using the narrower FFT window (marked in the last of these zooms, Figure~\\ref{fig:7_halves_interf}(c)) results in a prominent spectral peak seen in Figure~\\ref{fig:7_halves_interf}(d) at $\\sim 91\\pm4\\, \\text{kG}^{-1}$, the frequency of easily identifiable reproducible oscillations shown in Figure~\\ref{fig:7_halves_interf}(c), as well as~\\ref{fig:Hall_trace}(b). The reference frequency $f_0$ has been determined to be $\\sim 12.5\\pm0.5\\, \\text{kG}^{-1}$ in this sample, so the observed spectral peak corresponds to $7f_0$ within reasonable precision \u2013 the very frequency expected for the even-odd effect in this quantum Hall state. A larger Fourier transform window bounded by vertical red bars in Figure~\\ref{fig:7_halves_interf}(b) reveals lower-frequency features in the Fourier spectrum as can be seen in Figure~\\ref{fig:7_halves_interf}(e). It also points towards more complex structure of the spectral peak centered around $7f_0$, which will be discussed later.\n\nOne of the remarkable features of the observed oscillations attributable to the non-Abelian $e\/4$ quasiparticles is their temporal stability. Generally, there are two ways of inferring the fermion parity stability: 1) direct observation of the phase stability of the $5f_0$ and $7f_0$ frequency oscillations respectively at $\\nu=5\/2$ and $7\/2$, and 2) indirect inference via the presence or absence of spectral properties, split peaks at  $(5\\pm 1)f_0$ at $\\nu=5\/2$ and  $(7\\pm 1.5)f_0$ at $\\nu=7\/2$; see Section~\\ref{sec:intro} and also Section~\\ref{sec:S1} of Supplementary Materials). While there is some indication of such splitting in Figure~\\ref{fig:7_halves_interf}(e), it is far from conclusive. In general, such indirect inference is complicated; in order to achieve the required resolution, it requires a sufficiently large magnetic field range that should be employed in the FFT, which at the same time makes the data more susceptible to low-frequency noise -- see Figure~\\ref{fig:7_halves_interf}(d) versus \\ref{fig:7_halves_interf}(e).\n\nNevertheless this does not preclude us from studying the fermion parity stability in the temporal domain by repeated sweeps within a limited magnetic field interval.\nTheir reproducibility includes their phase which remains stable over several hours. What is even more remarkable is that when the phase of the oscillations fluctuates, it happens predominantly through phase jumping by $\\pi$, as can be seen in Figure~\\ref{fig:fermionic_parity}(a). This is exactly what one would expect in the case of fluctuating fermion parity inside the interferometer: the phase of the $e\/4$ interference is shifted by $\\pi$ every time the overall fusion channel of quasiparticles inside the interferometer \u2013 their combined fermion parity \u2013 changes. Such a $\\pi$ jump can be seen in Figure~\\ref{fig:fermionic_parity}(b) representing three different magnetic field sweeps taken over eight hours, each individual trace takes about 100 min.   Two of them are perfectly in phase while the third one displays a $\\pi$ phase jump whose location is indicated by an arrow. Similar behavior is observed at $\\nu=5\/2$: runs of reproducible oscillations are punctuated by occasional $\\pi$ shifts, the predominant instability \u2013 see Figure~\\ref{fig:5_halves_interf} and Section~\\ref{sec:S5-6} of Supplementary Materials, Figures~\\ref{fig:S5-6-1} and \\ref{fig:S5-6-2}.\n\\begin{figure}[htb]\n\\centering\n    \\includegraphics[width=0.95\\columnwidth]{figure06.pdf}\n    \\caption{$\\pi$ phase shifts in high-frequency oscillations at $\\nu=7\/2$:\\\\\n    (a)  Magnetic field sweeps taken near $\\nu=7\/2$ as in Figure~\\ref{fig:7_halves_interf} from the same sample but of a different preparation (sample~6, preparation~19, device type~b, $T \\sim 20$mK).  Both are down-sweeps.  The two traces of $R_L$ oscillations are out of phase by $\\pi$. The $\\pi$ phase jump seemingly occurred away from the depicted oscillations showing the full extent of the sweep. Such a phase jump  is consistent with a change in fermion parity.\\\\\n    Panel (b) shows three consecutive $B$-field up traces (same sample and preparation, but with $B$-sweeps separated by days from (a)).  Traces 1 and 2 (bottom of panel (b)) overlap, consistent with an unchanging fermion parity.  In trace~3 (top of panel (b), shown alongside trace~2 repeated for comparison) a $\\pi$ phase jump has occurred in the time intervening trace~2 and 3 as this phase difference is apparent from the lowest shown $B$-field up to about 48.85kG (marked by an arrow), at which point the phase reverts to that of trace~2.  The envelopes of the oscillations are not shifted, which is consistent with the fermion parity change rather than a change of the encircled charge. (See also Supplementary Materials, Section~\\ref{sec:S5-6}.)\n}\n    \\label{fig:fermionic_parity}\n\\end{figure}\n\nThis direct observation of the oscillation stability gives us the means of establishing the temporal stability, the magnetic field dependence of that stability by examining different magnetic field intervals, and influence of a host of other parameters, such as different gating configurations.  The ability to directly observe the phase stability stands as a new platform for studying fermion parity. The fact that we can actually see the rare phase jumps and assign their characteristic timescale is significant: this not only further validates our theoretical model, it may also have profound consequences for the future of such systems in quantum computation. After all, the fermion parity is supposed to encode the state of a topological qubit; its remarkable stability is a key to its potential utility.\n\n\n\nGiven the potential importance of the $\\pi$ phase jumps, it is essential to ask what experimental parameters can control their incidence and prevalence. One parameter conceivably influencing these $\\pi$ phase jumps \u2013 the density of encircled  $e\/4$ quasiparticles \u2013 can be estimated from the collected data. By simply counting of the $7f_0$ oscillation periods from the center of the  $7\/2$ FQH plateau, and using  $f_0$ to determine area A (see Figure~\\ref{fig:power_spectra}), we estimate their density to be $\\sim 170 \\upmu\\text{m}^{-2}$ towards the edge of the plateau. This puts their separation at $\\sim 0.07\\upmu\\text{m} = 70\\text{nm}$ for the maximum density, which corresponds to about 5 magnetic lengths at these magnetic fields.  (For more detail on the calculations see Supplementary Materials, Section~\\ref{sec:S5-6}.) This suggests that the oscillations occur over a range of reasonable separations between  $e\/4$ quasiparticles. Yet it does not provide any indication of a critical density that should trigger phase jumps. A more systematic study of these phase jumps is presently underway.\n\n\\begin{figure}[htb]\n\\includegraphics[width=\\columnwidth]{figS5-6-1}\n\\caption{Interference oscillations and their spectrum at $\\nu=5\/2$.\\\\\nThe top left panel shows the trace of $R_L$ from filling factor $\\nu=3$ to 2;\nits blow-up in the vicinity of $\\nu=5\/2$ (circled area) is shown in the bottom panel, with the corresponding  FFT spectrum  shown above in the right panel. $f_0$ is frequency of oscillations observed at integer filling in the same sample.  \\\\\nThe lower panel shows three $B$-field sweeps in a single direction, demonstrating significant overlap of the resistance oscillations.  The frequency of rapid oscillations is $\\sim 5f_0$ shown in the power spectrum.  Each trace is 60 minutes, so total data collection time is 6 hours (the opposite sweep direction is not shown).\\\\\nSample~6, preparation~18, device type~b, $T\\sim 20mK$.\n}\n\\label{fig:5_halves_interf}\n\\end{figure}\nWe now turn our attention to the interference oscillations observed at $\\nu=5\/2$.\nThe interference oscillations observed at this filling fraction in the devices with high Al purification strongly resemble those at $\\nu=7\/2$.\nAs shown in Figure~\\ref{fig:5_halves_interf}, these oscillations are remarkably stable and occur at five times the reference frequency $f_0$, which is once again indicative of the non-Abelian even--odd effect. Notice that another prominent spectral feature is located near $f_0$ itself.\n\\begin{figure}[htb]\n\\centering\n    \\includegraphics[width=\\columnwidth]{figure10.pdf}\n    \\caption{Extracting power spectra in $B$-field sweeps and interference spectra at $\\nu=5\/2$:\\\\\n    (a) Transport ($R_L$) through an interference device shown in Figure~\\ref{fig:interferometer}, bottom micrograph (device type~b), between filling factors $\\nu=2$ and $3$ in a high mobility heterostructure without additional Al purification.  The power spectrum is extracted from a $B$-field sweep as follows:  the background resistance near $\\nu=3$ is subtracted, after which a fast Fourier transform (FFT) is applied to the residual oscillations, producing the power spectrum that shows a principal peak at roughly $6\\,\\text{kG}^{-1}$, the frequency identified as $f_0$ whose value is  closely matched at other integral filling factors.  See also Fig.~\\ref{fig:S5-2-1}. Temperature $\\sim 20$mK, sample~2, preparation~2, device type~b.\\\\\n    (b) Power spectrum of $R_L(B)$ near $\\nu=5\/2$.  The FFT window is bracketed in panel (a).  After background subtraction and application of the FFT, four dominant peaks located near $f_0$, $2f_0$, $4f_0$ and $6f_0$ can be discerned. Here $f_0$ is the frequency of the Aharonov--Bohm oscillations in the integer quantum Hall state (panel (a)), and marked in panels (a) and (b) by the solid red vertical line. Solid green lines in panel (b) indicate multiples of that frequency with the shaded area around them indicating the potential error margins stemming from the error of $\\pm 1\/8\\;\\text{kG}^{-1}$ in determining $f_0$. Meantime the dashed lines indicate multiples of the actual frequency of the first peak in panel (b).}\n    \\label{fig:power_spectra}\n\\end{figure}\n\nBefore we examine the spectra in more detail, it worth reiterating that the highest predicted oscillation frequency for Abelian AB interference at $\\nu=5\/2$ is $2f_0$; any significant spectral weight at higher frequencies (specifically, within the range between $4f_0$ and $6f_0$ for $\\nu=5\/2$) that is not a result of noise is an indicator of a non-Abelian  nature of the state. Specifically, it should be interpreted as the evidence of the even-odd effect, which is a consequence of the quasiparticles' statistics and not charge; were the $e\/4$ quasiparticles Abelian instead, their interference would result in oscillations with frequency $f_0$ at $\\nu=5\/2$ and $1.5f_0$ at $\\nu=7\/2$.\n\nWhile we empirically attribute the substantially more prominent high-frequency oscillations associated with the even--odd effect to the improved Al purity of the newer heterostructures, we emphasize that samples without this high Al purity can still demonstrate high frequency oscillations at $\\nu=5\/2$ as reported in the earlier work~\\cite{Willett2013b}.  The focus of those studies was only on the oscillations associated with the even--odd effect; meantime recent improvements in those materials and devices (without the Al purity change) have provided higher quality samples with better defined FQH states of interest. This in turn allows for larger Fourier transform windows, letting us focus on the finer details of the oscillation spectra, including their low-frequency features, than was possible in the previous studies. Results of such study of the full spectral frequencies follow here.\n\nA representative Fourier spectrum of the oscillations at $\\nu=5\/2$ is shown in Figure~\\ref{fig:power_spectra}(b) along with the possible interpretation of its most prominent features.  The oscillations in $R_L$ around filling fraction $\\nu=3$ are used to determine the integer Aharonov--Bohm interference frequency $f_0$; details of the FFT analysis and magnetic field window size used for each FFT in results are described in Section~\\ref{sec:S3} of Supplementary Materials. The observed\nspectral peaks are located near  $f_0$, $2f_0$, $4f_0$  and $6f_0$, with $f_0$ being the aforementioned reference frequency. (See Section~\\ref{sec:S5-7a} of Supplementary Materials for more details on peak identification.) In order to verify this identification of the observed  peaks, in Figure~\\ref{fig:sup_peak_fit} we plot their actual locations vs. integer factor $m$ that we assign to these peaks in five different samples, with two different preparations for one of the samples -- 6 data sets altogether. Each set of observed peak frequencies at $\\nu=5\/2$ has been rescaled by the coefficient $\\tilde{f_0}$ obtained from linear fit for each sample's frequency data.  We can then compare $\\tilde{f_0}$ obtained from linear fit to frequency $f_0$ corresponding to spectral peaks measured at $\\nu=3$ for each sample\/prepapration; the results of this comparison are shown in  Table~\\ref{tab:sup_f_0}.\n\n\\begin{figure}[thb]\n\\centering\n    \\includegraphics[width=\\columnwidth]{peaks_samples.pdf}\n    \\caption{Frequencies corresponding to the most prominent spectral peaks at $\\nu=5\/2$ in 6 different data sets from different samples and preparations.\n    Frequencies are normalized by ${\\tilde f}_0$, which is the parameter obtained by applying linear fit to each observed set of peak frequencies (shown in circles) with the assumption that they correspond to integer multiples of this fundamental frequency. Open squares correspond to the frequencies obtained by taking the average value of the two frequencies that mark the width at half-maximum for each peak, normalized by the same coefficients. The data set identification is as follows: 1 - sample~2, preparation~2; 2 - same sample\/preparation four days later; 3 - sample~2, preparation~1; 4 - sample~5, preparation~2; 5 - sample~3, preparation~1; 6 - sample~4, preparation~1. Note that two of the samples did not exhibit peaks at or near $2f_0$ -- cf. traces in Fig.~\\ref{fig:2f0_peak}(a,b) corresponding to data sets 3 \\& 2 shown here. $T \\sim 20$mK.}\n    \\label{fig:sup_peak_fit}\n\\end{figure}\n\n\nThe peak locations shown in Figure~\\ref{fig:sup_peak_fit}  correspond to the highest measured values of spectral peaks. In Figures~\\ref{fig:peak_identification} and \\ref{fig:sup_peak_identification} in Supplementary Materials we present a comparison between two peak identification processes, one identifying peak locations by their highest value and the other using the average of the two points corresponding to their half-values.  The observed features are consistent with the expectations for both Abelian and non-Abelian interference processes due to charge $e\/4$ and charge $e\/2$ quasiparticles. (We remark that the peak identification procedure used here is the same as was illustrated earlier in Figure~\\ref{fig:integer_vs_fractional_peak_positions} showing the comparison of actual traces and their Fourier transforms for $\\nu=4$ and $\\nu=16\/5$, further strengthening our confidence in identification of observed spectral features.)\n\\begin{table}[bht]\n  \\begin{tabular}{| c | c | c |}\n        \\hline\n    $\\tilde{f_0}\\;\\;(\\text{kG}^{-1})$ & ${f_0}\\;\\;(\\text{kG}^{-1})$ & sample\/preparation \\\\\n      \\hline\\hline\n    6.2 & $6$ &  sample 2, prep 2\\\\\n    5.8 & $6\\frac{1}{2}$ &  sample 2, prep 2, later\\\\\n    6.3 & 7 & sample 2, prep 1\\\\\n    10.0 & $9\\frac{1}{4}$ & sample 5, prep 2\\\\\n    4.5 & $4\\frac{3}{4}$ &  sample 3, prep 1\\\\\n    4.7 & $4\\frac{1}{2}$ & sample 4, prep 1\\\\\n    \\hline\n  \\end{tabular}\n  \\caption{Comparison of frequencies $\\tilde{f_0}$ obtained from linear fit of the $\\nu=5\/2$ spectral data to ${f_0}$ observed at $\\nu=3$ for each sample\/preparation.}\n  \\label{tab:sup_f_0}\n\\end{table}\n\nIn the power spectrum presented in Figure~\\ref{fig:power_spectra}(b) the amplitudes of these four peaks drop progressively with the frequency, a common property of these spectra in our study, but with exceptions (see Supplementary Materials, Figure~\\ref{fig:S5-3-1}).  The base frequency, $f_0$, depends on the size of the interferometer, as seen by comparing the power spectra from different devices, but the ratios of the power spectrum positions remain close to  1:2:4:6.  (Note that the base frequency $f_0$ can be measured at the different integer filling factors, a property of AB oscillations, and this is demonstrated in Supplementary Materials, Figure~\\ref{fig:S4-1} and \\ref{fig:S4-2a}.)  The spectral features shown in Figure~\\ref{fig:power_spectra}(b) are sharp, in part due to the low temperature of 20mK, but also due to the details of construction of this particular interferometer.  Larger devices have larger separation in the spectral features (since $f_0\\propto A$) and, consequently, better resolution, but the amplitudes of the higher frequency features at $4f_0$ and $6f_0$  are reduced in the largest devices tested. A repetition of this measurement on the same device but days later is shown in Figure~\\ref{fig:2f0_peak}(b) (presented there as a part of another study to be discussed later), with the dominant peaks persisting at the same frequencies. Further 5\/2 power spectra are displayed in Supplementary Materials, Figure~\\ref{fig:S5-3-1} and \\ref{fig:S5-3-2}.\n\n\n\\begin{figure}[thb]\n\\centering\n    \\includegraphics[width=0.95\\columnwidth]{model_spectra.pdf}\n    \\caption{Modelling FFT spectra at $\\nu=5\/2$ and $7\/2$. In a magnetic field sweep at $\\nu=5\/2$  and $\\nu=7\/2$  rapid oscillations due to the even--odd effect will be modulated by slower oscillations due to $e\/4-e\/2$ interference. At $\\nu=5\/2$ this implies modulating oscillations at $5f_0$ by oscillations at $f_0$, resulting in peaks at $4f_0$ and $6f_0$. At $\\nu=7\/2$, $1.5 f_0$ modulation of $7f_0$ even--odd oscillations should produce peaks at $5.5f_0$ and $8.5 f_0$. Panels (a) and (b) show measured power spectra at these filling fractions: (a) $\\nu=5\/2$, sample~2, preparation~2 (same as in Figure~\\ref{fig:power_spectra}(b)); (b) $\\nu=7\/2$, sample~6, preparation~15, device type~b; $T \\sim 20$mK in both cases.\\\\\n    The measured power spectra are compared with simple models. The black trace in panel~(c) is the FFT of the expression $\\Delta R_L =\\cos(2\\pi Bf_0)\\times\\cos(2\\pi B(5f_0 ))$ taken using the magnetic field window similar to the one used to obtain power spectra in the experiment.  The red trace is the FFT of $\\Delta R_L = \\cos(2\\pi Bf_0)$ demonstrating the $f_0$ oscillation peak. The spectra are scaled to match the locations and amplitudes of their peaks at frequency $f_0$.\\\\\n    Panel (d) shows a similar modulation of the $7f_0$ even--odd oscillations by $1.5f_0$ oscillations expected for the $e\/4-e\/2$ interference at $\\nu=7\/2$.  In addition, the minima of $R_L$ are reproducibly formed away from the re-entrant quantum Hall phases in the high quality material used in this study, resulting in an additional low-frequency modulation at about $0.4 f_0$. To capture these effects we model the resistance as $\\Delta R_L = \\left[0.25 \\cos(2\\pi B(0.4f_0)) + 0.75\\cos(2\\pi B (1.5f_0))\\right] \\times \\cos(2\\pi B(7f_0))$. The FFT of this expression results in the spectrum shown in panel~(d), compared directly to measured data at $\\nu=7\/2$ in panel~(b).  Again, an FFT of $\\Delta R_L = \\cos(2\\pi Bf_0)$ is the red trace panel~(d).  (See also Supplementary Materials, Section~\\ref{sec:S5-5} and Figure~\\ref{fig:S5-5-2}.)\n    }\n    \\label{fig:interf_model}\n\\end{figure}\n\nBefore attempting a similar analysis of the finer spectral features of interference oscillations at $\\nu=7\/2$ accessible through a wider Fourier transform window, let us reiterate that\nthe common observed feature in multiple sample preparations, interference measurements, and their respective spectra such as those shown in Figures~\\ref{fig:Hall_trace}(c) and \\ref{fig:7_halves_interf} (with another example shown in Figure~\\ref{fig:interf_model}(b) and more spectra presented in Section~\\ref{sec:S5-4} of Supplementary Materials) is the large spectral weight concentrated around $7f_0$, the frequency of the predicted even-odd effect. Yet upon closer inspection of the spectrum in Figure~\\ref{fig:7_halves_interf}(d), another common feature appears to be a sharp minimum right at that frequency $7f_0$. (A similar feature is also observed in samples at $\\nu=5\/2$:  See e.g. Fig.~\\ref{fig:S6-2}.) The most likely origin for this dip is some low-frequency modulation of the even-odd oscillations. As has been mentioned in the introduction, the $e\/4-e\/2$ interference with fixed fermion parity would result in a split peak,  $(7\\pm 1.5)f_0$. However, there are also other potential sources of modulation of the dominant $7f_0$ frequency; with the overall shape of the $R_L$ resistance minimum a likely culprit (see Figure~\\ref{fig:S5-5-2} in Section~\\ref{sec:S5-5} of Supplementary Materials). This shape accounts for a modulating frequency of about  $0.4f_0$, with the corresponding spectral feature encircled in Figure~\\ref{fig:7_halves_interf}(e). This modulation by  $0.4f_0$,  is then combined with the modulation by $1.5f_0$ due to $e\/4-e\/2$ interference oscillations. The comparison between the actual observed spectrum and the spectrum modeled by including both of these modulations is shown in Figures~\\ref{fig:interf_model}(b) and (d). Note that the model uses a finite Fourier transform window similar to that used in processing the experimental data. For comparison, Figures~\\ref{fig:interf_model}(a) and (c) show the observed and modeled spectra at $\\nu=5\/2$ where only the modulation due to  $e\/4-e\/2$ interference is taken into account.\n\n\nWhile the high-frequency oscillations are of the foremost importance for demonstrating the non-Abelian nature of the  $\\nu=5\/2$ and $7\/2$ states, the low-frequency features that we have been able to investigate in this study also reveal some important information.  We should mention that the ability to discern lower-frequency spectral features is another key new element of the present study; in the past they were hard to extract due to the narrow magnetic field range where these oscillations had been seen. Utilizing higher purity heterostructures and somewhat different interferometer design we have been able to better isolate the quantum Hall state at $\\nu=5\/2$ from the surrounding compressible states, which resulted in more pronounced, broader minima in  $R_L$ supporting longer runs of higher-amplitude interference oscillations than before. This allowed us to observe and analyze those lower-frequency spectral features, thus addressing one of the shortcomings of our earlier study~\\cite{Willett2013b}.\n\nIt is instructive to compare the low-frequency features between the  $\\nu=5\/2$  and $7\/2$ states. The two states are theoretically expected to correspond to the same topological order, with the same charge and statistics of its excitations. However, due to its different filling factor, the periodicity of AB interference at $\\nu=7\/2$  is expected to be different from 5\/2, as can be seen from Eqs.~(\\ref{eq:AB_phase2})--(\\ref{eq:interferenece_general}). Specifically, the interference of $e\/4$ edge excitations around $e\/2$ bulk quasiparticles should now occur with the periodicity of $2\\Phi_0\/3$ (instead of $\\Phi_0$), and thus the expected spectral peak at $1.5f_0$, nicely distinguishing it from potential electron contribution still occurring at $f_0$. Lastly, $e\/2-e\/2$ interference should manifest itself through a peak at $3f_0$. The oscillation spectrum shown in Figure~\\ref{fig:7_halves_interf}(e) is consistent with these expectations.  See also Section~\\ref{sec:S5-4} of Supplementary Materials.\n\n\\begin{figure}[h!tb]\n\\centering\n    \\includegraphics[width=0.72\\columnwidth]{figure07bcd.pdf}\n    \\caption{\n(a)-(b) $\\nu=5\/2$ oscillation spectra at two different gate voltage settings (sample~2, preparations~1 and 2, device type~b, $T \\sim 20$mK).  The spectrum in panel~(b) shows four peaks at   $f_0$, $2f_0$, $4f_0$ and $6f_0$, consistent with $e\/4$ and $e\/2$ interference.  The $2f_0$ peak is absent when the backscattering gate voltage $V_b$ is changed from $-9$V to $-3.5$V, increasing the tunneling distance $d$ for interfering quasiparticles.  The side gate voltage $V_s$ was adjusted to preserve the area $A$. (Supplementary Materials, Section~\\ref{sec:S5-7b}, Figure~\\ref{fig:S5-7-2} demonstrate the overall similar transport between $\\nu=2$ and $3$ for these gate settings.)\\\\\n(c) Ratio of the peak value at $2f_0$ to that at $f_0$ for a series of backscattering gate voltages $V_b$ in the same device (sample 2, preparations~1 to 5, $T \\sim 20$mK).  The increase of the peak amplitude ratio in response to the constriction narrowing (more negative $V_b$) may be reflecting relative tunneling amplitudes for $e\/2$ and $e\/4$ quasiparticles.\n}\n    \\label{fig:2f0_peak}\n\\end{figure}\n\nThe $f_0$ spectral peak at $\\nu=5\/2$ decays rapidly with increasing temperature up to $T\\approx 80$mK, which is roughly the onset temperature of the deep minimum in $R_L$. (See Supplementary Materials, Section~\\ref{sec:S5-7c}, Figures~\\ref{fig:S5-7-3} and  \\ref{fig:S5-7-4} for more details.) This strongly implies that oscillations at this frequency are, indeed, due to the physics of this fractional quantum Hall state. In particular, this hints at the $e\/4-e\/2$ interference contribution to these oscillations, in addition to the omnipresent electron contribution.\n\nThe last process whereby $e\/2$ quasiparticles braid around bulk $e\/2$ quasiparticles should result in spectral features at $2f_0$ for $\\nu=5\/2$ and at $3f_0$ for $7\/2$. While the observed oscillation spectra at $\\nu=7\/2$ contain hints of the $3f_0$ feature -- see e.g. Figure~\\ref{fig:S5-4-2} of Supplemental Materials for the best example -- Figure~\\ref{fig:7_halves_interf} illustrates the problematic nature of reliably discerning this feature from the noisy  background. Meantime its $2f_0$ counterpart in the $\\nu=5\/2$ spectra obtained in the samples without additional Al purification appears far more prominent, as can been seen in Figure~\\ref{fig:power_spectra}. We should note that this feature is systematically seen at frequencies somewhat \\emph{below} $2f_0$ -- see e.g. Figure~\\ref{fig:sup_peak_fit} showing its observed position for four different samples\/preparations. While we are not certain about the reasons for this discrepancy (it is a subject of an ongoing study), the prominence of this spectral feature enables us to implement additional tests.\nAn important property of the observed $\\nu=5\/2$ oscillation spectra presented here is changing of the magnitude of the $2f_0$ peak as a function of the separation between the inner edge currents in the backscattering constrictions (distance $d$ in schematic of Figure~\\ref{fig:interferometer}).  Figures~\\ref{fig:2f0_peak}(a) and (b) show two complete power spectra in the same interferometer at $\\nu=5\/2$ but with different applied voltages $V_b$ and $V_s$: the more negative is the $V_b$ value, the narrower the constrictions acting as ``beam splitters'' are, which should change the backscattering (tunneling) amplitudes. $V_s$ can be adjusted so that for two different $V_b$ values, the interferometer areas $A$ are kept essentially the same.  Standard longitudinal resistance ($R_L$) measurements (see  Supplementary Materials, Section~\\ref{sec:S5-7b}, Figure~\\ref{fig:S5-7-2}) show little difference for the two different gate configurations.  Note that in the device with voltage $V_b=-3.5$V, the $2f_0$ peak is essentially absent, whereas in the device with $V_b=-9.0$V a large $2f_0$ peak is present.  We therefore conclude that the width $d$ of the constriction (the separation between the QH edges \u2013 see Figure~\\ref{fig:interferometer}) is an important factor controlling the presence of the peak at $2f_0$.  The $V_b$ voltages were adjusted to a range of values between the two shown in the full spectra of Figures~\\ref{fig:2f0_peak}(a) and (b); the resulting plot of the measured ratio between the peak amplitudes at $2f_0$ and $f_0$ as a function of $V_b$ is shown in Figure~\\ref{fig:2f0_peak}(c). According to our findings, as $V_b$ becomes more negative (the backscattering distance d becomes smaller) the amplitude ratio increases.  This finding is consistently observed in multiple different heterostructure wafers and multiple different devices.\n\nThis result can be understood as a consequence of the difference in tunneling probability of $e\/4$ and $e\/2$ edge excitations as a function of the tunneling distance d between the edges in the constrictions that form the interferometer. Theoretical analysis of the tunneling process of the excitations at 5\/2 predicts that the amplitude of the  $e\/2$  tunneling process is suppressed by comparison to the $e\/4$  tunneling due to the larger momentum transfer required for backscattering at a gated narrowing, with the effect becoming more pronounced as the tunneling distance increases~\\cite{Bishara2009a}.  This effect is illustrated in~\\cite{Chen2009} where the dependence of the ratio between the $e\/4$ and $e\/2$ tunneling amplitudes as a function of the tunneling distance has been investigated numerically for the Moore-Read state. Our experimental findings are consistent with this picture, showing the amplitude of the $2f_0$ peak attributed to the $e\/2$ interference decreasing much faster with the distance than that of the peak at $f_0$. Note that the physics behind this effect should be insensitive to the precise nature of the $\\nu=5\/2$ state; it is a consequence of the difference between different quasiparticles' charges. This should be contrasted with the temperature and bias voltage dependence of these tunneling processes, which is expected to be governed by the quasiparticles' scaling exponents. Those considerations would lead to different predictions for the anti-Pfaffian vs. both the Moore-Read and the PH-Pfaffian states, potentially allowing for discriminating between these states~\\cite{Bishara2009a}. Our study is not sensitive to this distinction.\n\nWe should also comment here that contrary to the analysis of~\\cite{Keyserlingk2015}, we do not expect to be able to distinguish between the Moore-Read, anti-Pfaffian and PH-Pfaffian states based on the location of the high-frequency spectral peaks. For the reasons mentioned earlier (and further elaborated in the Supplementary Information), the Abelian phase associated with the $e\/4-e\/4$ braiding (which is different in these states) does not directly contribute to the AB phase observed in the experiment; the interference itself is destroyed if an $e\/4$ quasiparticle braids another unpaired one whereas only the overall fusion channel matters for paired ones. Therefore the observed high-frequency peaks are merely indicative of the non-Abelian nature of the state but their positions cannot be used to discriminate between the candidate states. That said, they contain the information about temporal stability of the non-Abelian fusion channels. For the case of stable overall fermion parity at $\\nu=5\/2$  we expect to see peaks at $4f_0$ and $6f_0$. This scenario appears to be in agreement with the observed enhanced spectral features at these frequencies reported here.  However, in both past~\\cite{Willett2013b} and present studies (see Section~\\ref{sec:S6} of Supplementary Materials) some sweeps result in only a $5f_0$ peak, consistent with random (but not rapidly fluctuating) fusion channel of additional pairs of $e\/4$ quasiparticles introduced into the interferometer. The conditions that affect the temporal stability of the fermion parity are the subject of future investigation.\n\n\n\n\\subsubsection{Summary and Conclusions:}\n\tUsing magnetic field sweeps in quantum Hall Fabry--P\\'{e}rot interferometers at $\\nu=5\/2$ and $7\/2$  filling we have observed large amplitude, reproducible interference oscillations at frequencies inconsistent with Abelian Aharonov--Bohm interference but specific for the non-Abelian even--odd effect for the respective quantum Hall states.  With new ultra-high purity heterostructures we have observed such oscillations for the first time in the  $\\nu=7\/2$ QH state, providing the first evidence for the non-Abelian statistics of its charge\u2013$e\/4$ excitations. Furthermore, the oscillations attributed to the non-Abelian $e\/4$ quasiparticles have been shown to be remarkably stable, indicating stability of their fusion channel \u2013 the fermion parity \u2013 which in turn may have profound consequences for their applications for topological quantum computation. This interpretation is further strengthened by the observation of occasional $\\pi$ phase jumps, the predominant type of instability observed in our traces. These are consistent with the change of the fermion parity inside the interferometer which should manifest itself in a $\\pi$ phase shift of the non-Abelian interference oscillations but have no effect on the Abelian interference. Moreover, the fact that we observe them infrequently and can actually associate a timescale (hours!) to the parity flips is perhaps the most promising outcome of our study, demonstrating the potential of such quantum Hall systems for quantum computing.\n\nThe Fourier transforms of the magnetic field dependence of the resistance around those filling factors show not only peaks corresponding to the non-Abelian even-odd effect associated with  $e\/4$ quasiparticles but also a set of spectral peaks consistent with those anticipated for all possible combinations of interfering quasiparticle types expected to be present in these systems, thus also accounting for the Abelian interference effects.\n\nIn our study we also demonstrate means of controlling the interference processes.  The Abelian  $e\/2-e\/2$ interference is shown to be suppressed by reducing the edge quasiparticle backscattering, thus allowing us to distill different contributions to the interference oscillations.  This gate control is presently the subject of intense investigation, and these are essential steps in an effort to develop devices for topological quantum computation.\n\n\\subsubsection{Acknowledgments}\n\nWe are grateful for some illuminating discussions with Parsa Bonderson, who has also been our co-author on related earlier projects. We acknowledge the hospitality of KITP supported in part by the NSF under Grant No. NSF PHY-1748958. We also acknowledge the Aspen Center for Physics, which is supported by National Science Foundation grant PHY-1607611.\nThe work at Princeton University (L.N.P., K.W.W., K.W.B.)  was funded by the Gordon and Betty Moore Foundation through the EPiQS initiative Grant GBMF4420, and by the National Science Foundation MRSEC Grant DMR 1420541 (Y.J.C.).\n\n\n\n\n\n\n\n\n\n\napsrev4-2.bst 2019-01-14 (MD) hand-edited version of apsrev4-1.bst\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction: Magnetism Matters!}\n\nMost astronomical research does not explicitly incorporate the\neffects of magnetic fields. Nevertheless, there are fundamental\nreasons why we need to try to understand the various roles played\nby magnetism in astrophysical processes. \n\nZweibel \\cite{zwe06}\nprovides two specific, clear motivations for further studies of\ncosmic magnetic fields.\nFirst and foremost, the origin of magnetic fields in the Universe is\na fundamental and unsolved cosmological problem \\cite{gr01,wid02}.\nWe do not know whether the first magnetic fields emerged through\nexotic processes such as phase transitions or string cosmology, or\nthrough standard plasma physics such as turbulence, instabilities or the\nbattery effect. It is unclear whether the diffusion of these seed fields\nthroughout the Universe was then a top-down or a bottom-up process. Were\nmagnetic fields in the early Universe strong enough to moderate the\nformation of large-scale structure?  What role could these fields have\nplayed in the formation of the first stars and galaxies?\n\nSecond, magnetic fields are the key to answering many long-standing\nproblems in plasma physics and astrophysics \\cite{mel80b,gll05}.  On the\nlargest scales, the coherent magnetic fields that stretch over enormous\nphysical scales  in galaxies and clusters allow us to test the extremes\nof dynamo theory and turbulence. Magnetic fields are at the core of\nany viable theory for the acceleration and propagation of cosmic rays.\nMagnetism drives the physics, geometry and evolution of active galactic\nnuclei (AGN). And finally, a whole variety of important physical processes\nthat drive galactic ecology, including star formation, thermal conduction,\ndiffusion and accretion, all rely heavily on the strength and geometry\nof the ambient magnetic field.\n\nIf our goal as astronomers is to better understand the Universe,\nthen we need to include magnetic fields in our observations and in\nour models. In this paper, I outline some of the recent progress\nand future prospects in this area. In \\S\\ref{sec_map}, \nI briefly review the ways in which radio data can map magnetic\nfields out to large distances. In \\S\\ref{sec_mw} \\& \\S\\ref{sec_lmc},\nI show some recent applications to the large-scale magnetic fields\nof the Milky Way and of the Large Magellanic Cloud (LMC), respectively.\nIn \\S\\ref{sec_ska}, I discuss some of the experiments on magnetic\nfields at intermediate and high redshifts that can be conducted\nwith the Square Kilometre Array (SKA).\n\n\\section{Mapping Magnetic Fields}\n\\label{sec_map}\n\nThere are various indirect approaches to studying magnetic fields in\nastrophysical sources \\cite{zh97}. Measurements of optical starlight\npolarisation (e.g., \\cite{hei96}), polarised radio synchrotron emission\n(e.g., \\cite{bh96}) and infrared\/sub-mm dust polarisation (e.g.,\n\\cite{wkc+00}) all characterise the structure of the magnetic field in the\nplane of the sky, $B_\\perp$.  While this provides information on two of\nthe three spatial components of the field vector, a crucial limitation\nof these approaches is that they all only provide the orientation of $B$, but\nnot its direction. For example, simple compression of a tangled magnetic\nfield can produce a set of polarisation vectors which will appear quite\nordered, but which have no spatial coherence.\n\nZeeman splitting is distinct from these other approaches, in\nthat it measures the line-of-sight component of the magnetic field\n($B_\\parallel$), and can determine the direction of this field (e.g.,\n\\cite{th86,rs90}).  However, most Zeeman experiments require long\nintegrations, and often probe localised regions of relatively high\ngas density.\n\nThe remaining approach to studying magnetic fields is {\\em Faraday rotation},\nwhich has proven to be a very powerful probe of $B_\\parallel$. Faraday\nrotation is a change in the linear polarisation angle of a radio signal as it\npropagates through a magnetised plasma. The orientation of the \nobserved electric field vector is:\n\\begin{equation}\n\\Theta = \\Theta_0 + {\\rm RM}~\\lambda^2,\n\\label{eqn_rm1}\n\\end{equation}\nwhere $\\lambda$ is the observing wavelength, $\\Theta_0$ is the intrinsic\npolarisation angle emitted by the source, and $\\Theta$ is the observed\npolarisation angle. The rotation measure (RM) is a path integral through the\nmagneto-ionised foreground:\n\\begin{equation}\n{\\rm RM}~= K \\int n_e B_\\parallel dl\n\\label{eqn_rm2}\n\\end{equation}\nwhere the free electron density, $n_e$, and line-of-sight field strength,\n$B_\\parallel$, are integrated from the observer to the source along a line\nelement $dl$, and $K$ is a constant. An example of Faraday rotation seen in\nthe polarised emission from a radio pulsar is shown in Figure~\\ref{fig_psr}.\n\n\\begin{figure}[b!]\n\\centerline{\\psfig{file=fig_psr.eps,width=0.7\\textwidth,clip=,angle=270}}\n\\caption{Faraday rotation in the Galactic interstellar medium, seen\ntoward the radio pulsar B1154--62 \\protect\\cite{gmg98}. The data\npoints indicate multi-channel spectropolarimetric measurements in\nthe 20~cm band made using the Australia Telescope Compact Array\n(ATCA), while the dashed line shows the best fit of\nEquation~(\\protect\\ref{eqn_rm1}) to these data. The slope of this line\nis RM~$=+495\\pm6$~rad~m$^{-2}$, indicating that the mean\nmagnetic field along this sightline is directed toward us.}\n\\label{fig_psr}\n\\end{figure}\n\nFaraday rotation has three important advantages. First, as for Zeeman\nsplitting, it provides a direction for $B_\\parallel$ along a given\nsightline, allowing us to test for field coherence if we have \nRM data for several adjacent positions on the sky. Second, since Faraday\neffects are strongest in the radio part of the spectrum, interstellar\nextinction can be disregarded, and magnetic fields can thus be probed out\nto cosmological distances. Finally, Faraday rotation is an ``absorption''\nexperiment, in the sense that the signal-to-noise ratio of the measurement\nis determined by the flux of the polarised background source, rather than\nany properties of the region being studied. Measurements of magnetic\nfields in regions that are not easily observable directly, such as the\nGalactic halo and the intergalactic medium (IGM), then become possible,\nprovided that appropriate background sources can be identified.\n\n\\section{The Galactic Magnetic Field}\n\\label{sec_mw}\n\nIt has long been established that there are magnetic fields of significant\nstrength throughout the interstellar medium of the Milky Way\n\\cite{fer49,dg49}. Points of general consensus are that the field\nhas both large-scale (ordered) and small-scale (random) components,\nthat the magnetic field is concentrated in and is generally oriented\nparallel to the disk of the Galaxy, and that the ordered component\nof the field probably broadly follows the spiral arms. Studies of pulsar\nRMs, extragalactic RMs and optical starlight polarisation show that\nthe large-scale field within a kpc of the sun is directed clockwise\n(as viewed from the North Galactic Pole) \\cite{hei96,man72}, but that\nthe field in the Sagittarius-Carina spiral arm, a few kpc closer to\nthe Galactic Centre, is oriented counterclockwise \\cite{sk80,tn80}.\nThis clearly indicates that a large-scale {\\em reversal}\\ of the\nlarge-scale magnetic field occurs somewhere between these two arms.\nThe presence and properties of such reversals provide crucial constraints\non dynamo theories and on the origin of galactic magnetic fields\n(see \\cite{shu05}).\n\nFor more distant parts of the Galaxy, especially beyond the Galactic\nCentre, the geometry of the field is still unclear, mainly due to\nthe lack of known pulsars in these regions. Recent studies have\ncome to a variety of conclusions as to the overall Galactic field\ngeometry. Just to provide a few examples: Weisberg et al.\\ \\cite{wck+04}\nhave argued that the field follows the spiral arms, with a reversal\nbetween each arm; Han et al.\\ \\cite{hml+06} have proposed that the\nfield is a spiral, but with reversals on both sides of every spiral\narm, so that fields in arm and inter-arm regions are directed in\nopposite directions; Vall\\'ee \\cite{val05} has suggested that the\nfield is purely azimuthal rather than a spiral, with reversals in\nconcentric rings. As noted by Vall\\'ee \\cite{val02}, the situation\nis not unlike the early exploration of Australia, during which half\nthe coastline had been mapped in detail, with the more distant parts\ntrailing off into ``terra incognita'', as shown in\nFigure~\\ref{fig_thevenot}.\n\n\n\n\\begin{figure}[t!]\n\\centerline{\\psfig{file=fig_thevenot_xfig.eps,width=0.7\\textwidth}}\n\\caption{An early map of Australia, drawn by the French cartographer\nMelchis\\'edech Th\\'evenot\n(``\\protect\\href{http:\/\/nla.gov.au\/nla.map-rm689a}{Hollandia Nova\ndetecta 1644; Terre Australe decouuerte l'an 1644}'', Map RM 689A,\n\\cite{the63}).  Reproduced with permission of the National Library\nof Australia.}\n\\label{fig_thevenot}\n\\end{figure}\n\n\n\\subsection{Polarised Extragalactic Sources}\n\\label{sec_xgal}\n\nMost of the work mentioned above has relied heavily on pulsar RMs.\nPulsars have several advantages for such studies: they are at known\ndistances (albeit with significant uncertainty for individual sources),\nthey are within the Galaxy (so that pulsars at a range of distances\ncan be used to make differential magnetic field measurements), and both\ndispersion and Faraday rotation of their signals can be measured (so that\nthe $n_e dl$ term in Eqn.~[\\ref{eqn_rm2}]) can be independently\nestimated).\n\nHowever, pulsars also have their limitations. Most pulsars are\nrelatively nearby. This means that individual H\\,{\\sc ii}\\ regions\nand other dense gas clumps along the line of sight can make a large\ncontribution to the overall RM, making it more difficult to probe\nthe properties of large-scale magnetic fields (e.g.,  \\cite{mwkj03}).\nThe distance estimates to individual pulsars from standard models\n(e.g., \\cite{cl02}) can sometimes be significantly in error, making\nit difficult to match the magnetic field direction inferred from a\npulsar's RM to Galactic structure such as spiral arms. Pulsars are\na small population of faint sources (there are less than 2000 known\npulsars, with a median 1.4~GHz flux of $\\sim0.5$~mJy), so that the\ndensity and total number of RM sightlines that they can provide are\nboth relatively low.  Finally, even in regions such as along the\nGalactic plane, where the sky density of pulsar RMs is higher than\naverage, the RM sample cannot be easily smoothed to bring out\nlarge-scale structure, because each pulsar is at a different distance.\n\nTo make substantial further progress in studying the Milky Way's\nmagnetism, an additional approach is therefore needed.  We have\nconsequently undertaken a coordinated effort to greatly expand the\nnumber of polarisation and RM measurements for extragalactic sources\n(i.e., radio galaxies and AGN). These sources are at large distances,\nand so provide a measurement of the Faraday rotation through the\nentire Milky Way along a particular direction. In fact, the observed\nRM is the sum of multiple contributions: from the source itself,\nfrom the IGM, from the Earth's ionosphere, and from the Milky Way.\nHowever, in most cases, and certainly when averaged over a large\nsample of such data, the first three terms are small and can be\ndisregarded, and the total RM can be used as a useful probe of the\nGalaxy's magnetism.\n\nIt is important to emphasise that RM data for extragalactic\nsources only provides an integral of $n_e B_\\parallel dl$, as per\nEquation~(\\ref{eqn_rm2}). Thus if the actual value of $B_\\parallel$ is\nof interest, a model is needed of $n_e$ as a function of $l$ (e.g.,\n\\cite{cl02}).  However,\nmany important problems require knowledge only of the geometry of $B$,\nwithout accurate estimates of its strength.  Since the sign of the\nRM is determined by the sign of $B_\\parallel$, the overall magnetic\nconfiguration can be studied from the RMs alone, without needing to\ninvert the integral.\n\nBrown et al.\\ \\cite{btj03,bhg+07} have recently completed two large\nsurveys of extragalactic RMs behind the Galactic plane: 380 RMs in\nthe outer Galaxy, derived from the Canadian Galactic Plane Survey\n(CGPS)\\footnote{\\href{http:\/\/www.ras.ucalgary.ca\/CGPS\/}{http:\/\/www.ras.ucalgary.ca\/CGPS\/}}\nand 148 RMs in the inner Galaxy, measured as part of the Southern\nGalactic Plane Survey\n(SGPS).\\footnote{\\href{http:\/\/www.atnf.csiro.au\/research\/HI\/sgps\/}{http:\/\/www.atnf.csiro.au\/research\/HI\/sgps\/}}\nThe results are shown in Figure~\\ref{fig_rms}; the vast improvement\nin the sample size compared to previous data-sets reveals a striking\ncoherence in the large-scale magnetic field configuration. For the\nCGPS, the RMs are almost all negative, showing that the Perseus\nspiral arm (through which these sources are viewed) has a clockwise\nmagnetic field \\cite{bt01}. For the SGPS, the situation is considerably\nmore complicated, with alternating regions of positive and negative\nRMs, with regions of low $|{\\rm RM}|$ between them.  The implications\nof the RM structure seen in the SGPS are considered in the following\nsection.\n\n\\begin{figure}\n\\centerline{\\psfig{file=fig_rms-2.eps,width=1.0\\textwidth}}\n\\medskip\n\\centerline{\\psfig{file=fig_rms-1.eps,width=1.0\\textwidth}}\n\\caption{Distribution of RMs in two strips along the\nGalactic plane, corresponding to the survey regions\nof the CGPS (upper panel) and SGPS (lower panel) \\cite{btj03,bhg+07}.\nThe circles mark the positions of RM measurements: blue for previously\nmeasured extragalactic RMs, red for pulsar RMs, and green for new\nCGPS\/SGPS data. The radius of each circle corresponds to the magnitude\nof the RM for that source; filled circles indicate positive RMs and\nopen circles show negative RMs.  Figures are courtesy of Jo-Anne Brown.}\n\\label{fig_rms}\n\\end{figure}\n\n\\subsection{The Magnetic Geometry of the Milky Way}\n\nFigure~\\ref{fig_sgps} shows a smoothed version of the extragalactic\nRM data from the lower panel of Figure~\\ref{fig_rms}. The spatial\ncoherence of Faraday rotation as a function of Galactic longitude\nis very clear.  There are local maxima of $|{\\rm RM}|$ in three directions:\n$\\ell \\approx 292^\\circ$, $\\ell \\approx 312^\\circ$ and $\\ell \\approx\n338^\\circ$, as shown by the dashed vertical lines in Figure~\\ref{fig_sgps}.\nThese sightlines are tangent to the Carina, Crux and Norma spiral\narms, respectively. On either side of these peaks are directions\nwhere $|{\\rm RM}| \\approx 0$~rad~m$^{-2}$, shown by dotted lines\nin Figure~\\ref{fig_sgps}. Superimposed on this oscillatory behaviour,\nis an overall large-scale signature: for $\\ell \\la 304^\\circ$, RMs\nare almost all positive, and for $\\ell \\ga 304^\\circ$, RMs are\npredominantly negative (apart from some high RMs close to the\nGalactic Centre). This dividing line is marked in red in\nFigure~\\ref{fig_sgps}.\n\n\\begin{figure}\n\\centerline{\\psfig{file=fig_brown_egs_xfig.eps,width=\\textwidth,clip=}}\n\\caption{RM vs.\\ Galactic longitude for extragalactic RMs in the\nSGPS region. The purple diamonds represent smoothed RM data,\nwith error bars indicating the standard error of the mean in each bin.\nThe dotted (dashed) lines indicate approximate longitudes of minimum\n(maximum) $|{\\rm RM}|$ in the smoothed distribution. The red line marks the\napproximate transition from predominantly positive RMs ($\\ell \\la 304^\\circ$)\nto predominantly negative RMs ($\\ell \\ga 304^\\circ$).\nAdapted from \\cite{bhg+07}.}\n\\label{fig_sgps}\n\\end{figure}\n\n\nQualitatively, these patterns in RM immediately reveals some\nproperties of the global magnetic field structure of the Galaxy.\nThe peaks in $|{\\rm RM}|$ along each spiral arm demonstrate that\nthe field strength and\/or gas density are high in these regions,\nwhile the low values of $|{\\rm RM}|$ indicate low values of these quantities\nin the inter-arms (see \\cite{bhg+07} for a detailed discussion).\nThe change in overall sign of RM at $\\ell \\approx 304^\\circ$ suggests\nthe presence of a large scale reversal in the field along this sightline.\nof this direction.\n\n\\begin{figure}[b!]\n\\vspace{5cm}\n\\caption{A simple model of the magnetic field in the southern Galaxy,\nfrom a joint fit to the RMs of 149 extragalactic sources and 120\npulsars. Coloured regions show the total strength of the model\nmagnetic field in separate regions bounded by the green lines.  The\nlimiting longitudes of the SGPS are shown as black lines, while the\ngrey scale represents the NE2001 electron density model of Cordes\n\\& Lazio \\cite{cl02}.  Adapted from \\cite{bhg+07}.}\n\\label{fig_model}\n\\end{figure}\n\nThis is quantified in Figure~\\ref{fig_model}, where we show a joint\nfit to extragalactic and pulsar RMs in the southern Galaxy, allowing\na series of concentric, spiral, annuli to each have a differing\nfield strength and field direction.\nThe best fit shows that the Galactic magnetic field is\nprimarily clockwise, except for a strong counterclockwise field in\nthe Scutum-Crux spiral arm (and possibly also in the molecular ring\nin the inner Galaxy).  Sightlines at $\\ell > 304^\\circ$ pass through\nthe Scutum-Crux arm, and are dominated by negative RMs; at $\\ell < 304^\\circ$,\nfield lines are directed toward the observer, and RMs are consequently\npositive.  The quality of the fit is indicated in Figure~\\ref{fig_fit},\nwhere we compare RM data to the predictions of the model in\nFigure~\\ref{fig_model}.  For extragalactic data, the model RMs and\nthe data match very well.  For pulsars, the scatter is larger (mainly\nbecause pulsar RM data cannot be meaningfully smoothed), but the\nmajor features are reproduced.\n\n\\begin{figure}[b!]\n\\centerline{\\psfig{file=fig_brown_sgps_rms.eps,width=\\textwidth}}\n\\caption{RM vs.\\ Galactic longitude for extragalactic sources (upper;\npurple points) and pulsars (lower; black points) in the SGPS region;\nthe extragalactic data have been smoothed as in\nFig.~\\protect\\ref{fig_sgps}.  The green symbols show the RMs predicted\nby the best-fit model of \\protect\\cite{bhg+07} at the positions of\neach of the observed sources. The model data for extragalactic\nsources have been smoothed in the same way as for the observations.\nAdapted from \\protect\\cite{bhg+07}.}\n\\label{fig_fit}\n\\end{figure}\n\nThis global fit is at odds with earlier studies utilising smaller\ndata-sets, in that it suggests that the Galaxy can be modelled with\na predominantly clockwise field, plus a single reversed region.\nThis structure is in line with what is seen also for other spiral\ngalaxies, but needs to be verified by a better mapping of extragalactic\nRMs in the first Galactic quadrant.\n\n\n\n\\section{The Magnetic Field of the Large Magellanic Cloud}\n\\label{sec_lmc}\n\nThe LMC is also particularly amenable to extragalactic RMs as a\nprobe of its magnetic field, because of its large angular extent\n($\\sim6^\\circ$) on the sky. Gaensler et al.\\ \\cite{ghs+05} re-analysed\narchival LMC data taken with the ATCA, and extracted polarisation\nand RMs for 292 background sources. The results, shown in\nFigure~\\ref{fig_lmc}, show that RMs are generally positive on the\neastern half of the galaxy, and negative on the western half.\nAnalysed in more detail, these RMs reveal a sinusoidal pattern as\na function of azimuth, implying a coherent, spiral, pattern in the\nLMC's magnetic field, with a strength of about 1~$\\mu$G \\cite{ghs+05}.\n\n\\begin{figure}\n\\centerline{\\psfig{file=fig_lmc_compress.eps,width=\\textwidth}}\n\\caption{RMs of extragalactic sources behind the Large Magellanic\nCloud \\cite{ghs+05}.  The image shows the distribution of emission measure toward\nthe LMC in units of pc~cm$^{-6}$. The symbols show the position,\nsign and magnitude of extragalactic RMs: filled (open) circles\ncorrespond to positive (negative) RMs, while asterisks indicate RMs\nwhich are consistent with zero within their errors.  The diameter\nof each circle is proportional to the magnitude of the RM.}\n\\label{fig_lmc}\n\\end{figure}\n\nThe presence of this relatively strong, ordered, field is somewhat\nsurprising in the LMC. Standard turbulent dynamo theory requires\n5--10~Gyr to amplify a weak primordial seed field to microgauss\nlevels, but the repeated tidal interactions between the LMC, Milky\nWay and Small Magellanic Cloud should disrupt any field that might be\nslowly built up through this process. The coherent field revealed\nin Figure~\\ref{fig_lmc} must have been amplified and organised rapidly,\nin only a few hundred million years. One possibility is a cosmic\nray dynamo (e.g., \\cite{hkol04}), which should thrive in the vigorous\nstarburst environment supplied by the LMC.\n\n\n\n\\section{Magnetism with the Square Kilometre Array}\n\\label{sec_ska}\n\n\\subsection{The Rotation Measure Grid}\n\nThe results presented in \\S\\ref{sec_mw} \\& \\S\\ref{sec_lmc} can be\ngreatly expanded upon with a larger sample of RMs (see also Kronberg,\nthese proceedings). With the SKA, we envisage an all-sky ``rotation\nmeasure grid'' \\cite{bg04,gbf04}, which would be derived from a\n1.4~GHz full-Stokes continuum survey. For an SKA field of view of\n5~deg$^2$, six months of observing would result in an RMS sensitivity\nof $\\approx$$0.1$~$\\mu$Jy~beam$^{-1}$. To estimate the yield of the\nresulting RM grid, one needs to consider ``$\\log N - \\log P$'',\ni.e., the differential source counts in linear polarisation, analogous\nthe usual $\\log N - \\log S$ function in total intensity \\cite{bg04,tsg+07}.\nThis results in a distribution like that shown in Figure~\\ref{fig_grid}.\nWhile there are uncertainties in extrapolating to the low flux\nlevels expected for the SKA (see discussion by \\cite{tsg+07}), we\ncan roughly predict that the RM grid should yield about $\\sim10^8$\nsightlines, with a typical spacing between measurements of $\\sim1'$.\nA simulation of the polarised sky as might be seen with the SKA\nis shown in Figure~\\ref{fig_sim}.\nSuch a data-base will provide a fantastic probe of all manner of\nextended foreground sources, either individually (like the case of\nthe LMC) or as a statistical ensemble (see \\cite{sab+08} and Arshakian\net al., these proceedings).\n\n\\begin{figure}\n\\centerline{\\psfig{file=fig_logNlogP_xfig.eps,width=\\textwidth}}\n\\caption{The predicted flux distribution of extragalactic radio\nsources in both total intensity (solid lines) and in linear\npolarisation (dashed lines), adapted from \\cite{bg04}.  The upper\npanel shows the differential source count distribution, while the\nlower panel shows the integral distribution.  Approximate flux\nlimits for wide-field surveys with the EVLA and with the SKA are indicated.}\n\\label{fig_grid}\n\\end{figure}\n\n\\begin{figure}\n\\centerline{\\psfig{file=fig_ska_sim_compress.eps,width=\\textwidth}}\n\\caption{A simulation of a one-hour SKA observation in linear\npolarisation, for a field of view of 1~deg$^2$. The angular resolution\nis a few arcsec, and the gray scale is in units of $\\mu$Jy. The\nimage was created by squaring the intensity values of an NVSS field\nto generate a Ricean noise distribution, and then adjusting the\nflux and spatial scales to simulate the SKA source density predicted\nby Fig.~\\protect\\ref{fig_grid}.  The insets show some hypothesised\ndistributions of position angle vs.\\ $\\lambda^2$ for three sources\nin the field, from which RMs can then be calculated.}\n\\label{fig_sim}\n\\end{figure}\n\n\\subsection{The Magnetic Universe}\n\nOne of the main applications of the SKA's RM grid will be to study\nthe growth of galactic-scale magnetic fields as a function of cosmic\ntime.  The expectation is that the sightlines to distant extragalactic\nsources should generally intersect one or more foreground galaxies,\nas is seen in the Ly-$\\alpha$ and Mg\\,{\\sc ii} absorption lines in\nthe optical spectra of quasars. Such intervenors should generate\nan RM signature in the background source, and this signature should\npotentially evolve with redshift. In particular, Equation~(\\ref{eqn_rm2}),\nif rewritten to take into account cosmological effects, contains a\n$(1+z)^{-2}$ dilution term because of redshift of the emitted\nradiation, but in some models can also contain co-moving terms with\ndependencies $n_e \\propto (1+z)^3$ and $B_\\parallel \\propto (1+z)^2$ \\cite{wid02}.\nThe overall RM may then potentially evolve as rapidly as RM $\\propto\n(1+z)^3$, in which case we expect that filaments\nand absorbers should begin to show an increasingly large RM at\nhigher $z$. If we can obtain a large sample of both RMs (from the\nSKA) and accompanying redshifts (from the next generation of\nphotometric and spectroscopic optical surveys), we can apply a\nvariety of statistical tests  to the distribution of RM vs.\\ $z$\nto map the magnetic evolution of the Universe to $z \\sim 3$\n\\cite{kol98,bbo99,kbm+08}.\n\n\n\\subsection{Magnetic Fields at $z > 5$}\n\nThere is already good evidence that microgauss-strength magnetic\nfields exist out to redshifts $z \\sim 1-2$ \\cite{kbm+08,kpz92}.\nIf we can extend such data out to $z \\ga 5$, we can potentially\nobtain strong constraints on how large-scale magnetic fields\nwere created and then amplified. Such measurements\ncan be made by obtaining RMs for polarised sources at very high redshifts.\n\nIndeed, radio emission has already been detected from two classes\nof sources at $z > 6$: gamma-ray burst afterglows \\cite{fck+06} and\nquasars \\cite{mbhw06}.  We currently lack the sensitivity to detect\nlinear polarisation and RMs from these objects, but such measurements\nshould be possible with the SKA. Furthermore, since the cosmic\nmicrowave background is linearly polarised, deep observations at\nthe upper end of the SKA frequency range may be able to measure RMs\nagainst it \\cite{kl96b,shm04}.  Such an experiment, while challenging,\nwould probe the integrated Faraday rotation over almost all of the\nUniverse's history. In considering such measurements, it is important\nto note that an RM measurement to a high-$z$ source does not provide\nany direct constraints on high-$z$ magnetic fields on its own, since\nthe observed RM will also contain contributions from low-$z$\ncomponents of the sightline, and from the Milky Way foreground.\nOnce a high-$z$ RM data point has been obtained, deep radio and\noptical observations of that field can yield a large number of  RMs\nand redshifts for adjacent foreground objects (see Fig.~\\ref{fig_sim}).\nWhen the corresponding foreground contribution is then removed, the\nhigh-$z$ magnetic field can be isolated studied.\n\n\\section{Conclusions}\n\nCosmic magnetism is a vigorous and rapidly developing field. What\nmakes this area particularly relevant for the SKA is that magnetic\nfields at cosmological distances are uniquely probed at radio\nwavelengths.  By studying the evolution of magnetic fields over the\nUniverse's history, we can simultaneously address a variety of major\ntopics in fundamental physics and astrophysics. This work also has\nstrong synergies with other astronomy and astroparticle experiments,\nsuch as {\\em Planck}, LSST, {\\em JWST}, HESS and Auger \\cite{ewvs06,apr07}.\n\nIn the coming years, a host of SKA pathfinders will begin to finally\nreveal the depth and detail of the polarised sky \\cite{rbb+06,jbb+07},\nculminating in an exploration of the full Magnetic Universe with\nthe Square Kilometre Array.\n\n\n\n\\acknowledgments\n\nI thank my various collaborators for their contributions to the\nwork reported here, in particular Jo-Anne Brown for providing the\nmaterial for several figures.  This work has been supported by the\nAustralian Research Council (grant FF0561298) and by the National\nScience Foundation (grant AST-0307358).\n\n\\providecommand{\\href}[2]{#2}\\begingroup\\raggedright","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nThe existence of higher structures (such as Gerstenhaber algebras or Batalin-Vilkovisky algebras) on cohomology or homology of a certain algebraic structure was initiated by M. Gerstenhaber in the study of Hochschild cohomology of associative algebras \\cite{gers}. Later, a more sophisticated approach of his result was given by Gerstenhaber and Voronov using operad with multiplication in connections with Deligne's conjecture \\cite{gers-voro}. \nHowever, it can only ensure the existence of a Gerstenhaber structure. In differential geometry and noncommutative geometry, one wants a more concrete structure of differential calculus to understand the full account of the picture \\cite{nest-tsy,tamar-tsy}. A pair $(\\mathcal{A}, \\Omega )$ consisting of a Gerstenhaber algebra $\\mathcal{A}$ and a graded space $\\Omega$ is called a calculus if $\\Omega$ carries a module structure over the algebra $\\mathcal{A}$ (given by a map $i$) and a module structure over the Lie algebra $\\mathcal{A}^{\\bullet + 1}$ (given by a map $\\mathcal{L}$) and a differential $B : \\Omega_\\bullet \\rightarrow \\Omega_{\\bullet +1}$ that mixes $\\mathcal{L}, i$ and $B$ by the Cartan-Rinehart homotopy formula. For a smooth manifold $M$, the pair $(\\mathcal{X}^\\bullet (M), \\Omega^\\bullet (M))$ of multivector fields and differential forms; for an associative algebra $A$, the pair $(H^\\bullet (A), H_\\bullet (A))$ of Hochschild cohomology and homology yields differential calculus structure.  In \\cite{kowal} Kowalzig extends the result of Gerstenhaber and Voronov by introducing a cyclic comp module over an operad (with multiplication). Such a structure induces a simplicial homology on the underlying graded space of the comp module and certain action maps. When passing onto the cohomology and homology, one gets a differential calculus structure.\n\nIn this paper, we first construct a new example of differential calculus in the context of hom-associative algebras. A hom-associative algebra is an algebra whose associativity is twisted by a linear homomorphism \\cite{makh-sil-0}. Such twisted structures were first appeared in the context of Lie algebras to study $q$-deformations of Witt and Virasoro algebras \\cite{hart}. Hom-associative algebras are widely studied in the last 10 years from various points of view. In \\cite{amm-ej-makh,makh-sil} Hochschild cohomology and deformations of hom-associative algebras are studied, whereas, homological perspectives are studied in \\cite{hasan}. The homotopy theoretic study of hom-associative algebras are considered in \\cite{das3}. In \\cite{das1} the present author showed that the space of Hochschild cochains of a hom-associative algebra $A$ carries a structure of an operad with a multiplication yields the cohomology a Gerstenhaber algebra (see also \\cite{das2}). Here we show that the space of Hochschild chains of $A$ forms a cyclic comp module over the above-mentioned operad. The induced simplicial homology coincides with the Hochschild homology of $A$. Hence, following the result of Kowalzig, we obtain a differential calculus structure on the pair of Hochschild cohomology and homology of $A$. See Section \\ref{section-3} for details clarification.\n\nFinally, as an application, we obtain a Batalin-Vilkovisky algebra structure on the Hochschild cohomology of a regular unital symmetric hom-associative algebra. This generalizes the corresponding result for associative algebras obtained by Tradler \\cite{tradler}.\n\nThroughout the paper, $k$ is a commutative ring of characteristic $0$. All linear maps and tensor products are over $k$.\n\n\\section{Noncommutative differential calculus and cyclic comp modules}\nIn this section, we recall noncommutative differential calculus and cyclic comp modules over non-symmetric operads. We mention how a cyclic comp module induces a noncommutative differential calculus. Our main references are \\cite{gers-voro,kowal}. We mainly follow the sign conventions of \\cite{kowal}.\n\n\\begin{defn}\n(i) A Gerstenhaber algebra over $k$ is a graded $k$-module $\\mathcal{A} = \\oplus_{i \\in \\mathbb{Z}} \\mathcal{A}^i$ together with a graded commutative, associative product $\\cup : \\mathcal{A}^p \\otimes \\mathcal{A}^q \\rightarrow \\mathcal{A}^{p+q}$ and a degree $-1$ graded Lie bracket $[~, ~]: \\mathcal{A}^p \\otimes \\mathcal{A}^q \\rightarrow \\mathcal{A}^{p+q-1}$ satisfying the following Leibniz rule\n\\begin{align*}\n[f, g \\cup h ] = [f , g ] \\cup h + (-1)^{(p-1) q} g \\cup [f, h], ~~ \\text{ for } f \\in \\mathcal{A}^p, g \\in \\mathcal{A}^q, h \\in \\mathcal{A}.\n\\end{align*}\n\n(ii) A pair $(\\mathcal{A}, \\Omega)$ consisting of a Gerstenhaber algebra $\\mathcal{A} = ( \\mathcal{A}, \\cup, [~, ~])$ and a graded $k$-module $\\Omega$ is called a precalculus if there is a graded $(\\mathcal{A}, \\cup)$-module structure on $\\Omega$ given by $i : \\mathcal{A}^p \\otimes \\Omega_n \\rightarrow \\Omega_{n-p}$ and a graded Lie algebra module by $\\mathcal{L} : \\mathcal{A}^{p+1} \\otimes  \\Omega_n \\rightarrow \\Omega_{n-p}$ satisfying\n\\begin{align}\\label{eqn-t}\ni_{[f, g]} = i_f \\circ \\mathcal{L}_g - (-)^{p (q+1)} \\mathcal{L}_g \\circ i_f, ~ \\text{ for } f \\in \\mathcal{A}^p, g \\in \\mathcal{A}^q.\n\\end{align}\n\n(iii) A precalculus $(\\mathcal{A}, \\Omega)$ is said to be a calculus if there is a degree $+1$ map $B : \\Omega_\\bullet \\rightarrow \\Omega_{\\bullet + 1}$ satisfying $B^2 =0$ and the following Cartan-Rinehart homotopy formula holds\n\\begin{align*}\n\\mathcal{L}_f = B \\circ i_f - (-1)^p~ i_f \\circ B, ~ \\text{ for } f \\in \\mathcal{A}^p.\n\\end{align*}\n\\end{defn}\n\n\\begin{defn}\nA non-symmetric operad $\\mathcal{O}$ in the category of $k$-modules consists of a collection $\\{ \\mathcal{O}(p) \\}_{p \\geq 1}$ of $k$-modules together with $k$-bilinear maps (called partial compositions) $\\circ_i : \\mathcal{O}(p) \\otimes \\mathcal{O}(q) \\rightarrow \\mathcal{O}(p+q-1)$, for $1 \\leq i \\leq p$, satisfying the following identities\n\\begin{align*}\n(f \\circ_i g) \\circ_j h = \\begin{cases} (f \\circ_j h) \\circ_{i+p-1} g ~~~&\\mbox{if } j <i\\\\\nf \\circ_i (g \\circ_{j-i+1} h) ~~~&\\mbox{if } i \\leq j < q+i\\\\ \n(f \\circ_{j-q+1} h) \\circ_i g ~~~&\\mbox{if } j \\geq q+i \\end{cases}\n\\end{align*}\nfor $f \\in \\mathcal{O}(p),~ g \\in \\mathcal{O}(q),~ h \\in \\mathcal{O}(r)$;\nand there is a distinguished element $\\mathds{1} \\in \\mathcal{O}(1)$ satisfying $\\mathds{1} \\circ_1 f = f = f \\circ_i \\mathds{1}$, for $f \\in \\mathcal{O}(p)$ and $1 \\leq i \\leq p$.\n\\end{defn}\n\nIn an operad $\\mathcal{O}$, there is a degree $-1$ bracket $[~, ~]: \\mathcal{O}(p) \\otimes \\mathcal{O}(q) \\rightarrow \\mathcal{O}(p+q-1)$ given by\n\\begin{align*}\n[f, g] := \\sum_{i=1}^p (-1)^{(q-1)(i-1)} f \\circ_i g - (-1)^{(p-1)(q-1)} \\sum_{i=1}^q (-1)^{(p-1)(i-1)} g \\circ_i f.\n\\end{align*}\n\nLet $\\mathcal{O} = (\\mathcal{O}, \\circ_i, \\mathds{1})$ be a non-symmetric operad. An element $\\pi \\in \\mathcal{O}(2) $ is called a multiplication on $\\mathcal{O}$ if it satisfies $\\pi \\circ_1 \\pi = \\pi \\circ_2 \\pi$. Note that a multiplication induces a differential $\\delta_\\pi : \\mathcal{O}(n) \\rightarrow \\mathcal{O}(n+1),~ \\delta_\\pi (f) = [\\pi, f].$\nWe denote the corresponding cohomology groups by $H^n_\\pi (\\mathcal{O})$, for $n \\geq 1$. \nFurther, a multiplication $\\pi$ induces a degree $0$ product $f \\cup g := (\\pi \\circ_2 f) \\circ_1 g$. In \\cite{gers-voro} the authors showed that these two operations passes onto the cohomology $H^\\bullet_\\pi (\\mathcal{O})$ and make it a Gerstenhaber algebra.\n\n\\begin{defn}\n(i) A comp module over an operad $\\mathcal{O} = (\\mathcal{O}, \\circ_i, \\mathds{1}) $ consists of a sequence of $k$-modules $\\{ \\mathcal{M} (n) \\}_{n \\geq 0}$ together with $k$-bilinear operations (called comp module maps) $\\bullet_i : \\mathcal{O} (p) \\otimes \\mathcal{M}(n) \\rightarrow \\mathcal{M}(n-p+1)$, for $p \\leq n$ and $1 \\leq i \\leq n-p+1$, satisfying the following identities\n\\begin{align}\\label{eqn-p}\nf \\bullet_i ( g \\bullet_j x ) = \\begin{cases} g \\bullet_j ( f \\bullet_{i+q-1} x ) ~~~ & \\mbox{ if } j < i \\\\\n( f \\circ_{j-i+1} g) \\bullet_i x ~~~ & \\mbox{ if } j-p \\leq i \\leq j \\\\\ng \\bullet_{j-p+1} ( f \\bullet_i x) ~~~ & \\mbox{ if } 1 \\leq i \\leq j-p, \\end{cases}\n\\end{align}\nfor $f \\in \\mathcal{O}(p), g \\in \\mathcal{O}(q)$ and $x \\in \\mathcal{M}(n)$. It is called unital comp module if\n\\begin{align}\\label{eqn-q}\n\\mathds{1} \\bullet_i x = x, ~ \\text{ for } i =1, \\ldots, n.\n\\end{align}\n\n\n(ii) A cyclic (unital) comp module over $\\mathcal{O}$ is a unital comp module $\\mathcal{M}$ equipped with an additional comp module map $\\bullet_0 : \\mathcal{O}(p) \\otimes \\mathcal{M}(n) \\rightarrow \\mathcal{M} (n-p+1)$, for $p \\leq n+1$ such that the relations (\\ref{eqn-p}) and (\\ref{eqn-q}) hold $i=0$ as well; and a $k$-linear map $t : \\mathcal{M}(n) \\rightarrow \\mathcal{M}(n)$, for $n \\geq 1$, satisfying $t^{n+1} = \\mathrm{id}$ and\n\\begin{align}\\label{eqn-r}\nt ( f \\bullet_i x) = f \\bullet_{i+1} t(x), ~ \\text{ for } i= 0, 1, \\ldots, n-p.\n\\end{align}\n\\end{defn}\n\nLet $(\\mathcal{O}, \\pi)$ be a non-symmetric operad with a multiplication and $\\mathcal{M}$ be a cyclic unital comp module over $\\mathcal{O}$. Then there is a simplicial boundary map $b : \\mathcal{M}(n) \\rightarrow \\mathcal{M}(n-1)$, for $n \\geq 1$, given by\n\\begin{align}\\label{simplicial-boun}\nb (x) = \\sum_{i=0}^{n-1} (-1)^i~ \\pi \\bullet_i x + (-1)^n ~\\pi \\bullet_0 t (x).\n\\end{align}\nWe denote the corresponding homology groups by $H_\\bullet (\\mathcal{M}).$ Moreover, there is a cap product $i_f := f \\cap \\_ : \\mathcal{M}(n) \\rightarrow \\mathcal{M}(n-p)$, for $f \\in \\mathcal{O}(p)$ given by\n\\begin{align}\\label{cap-pro}\ni_f x = ( \\pi \\circ_2 f) \\bullet_0 x, ~\\text{ for } x \\in \\mathcal{M}(n)\n\\end{align}\nand a Lie derivative $\\mathcal{L}_f : \\mathcal{M}(n) \\rightarrow \\mathcal{M}(n-p+1)$, for $f \\in \\mathcal{O}(p)$ given by\n\\begin{align}\\label{lie-pro}\n\\mathcal{L}_f x := \\begin{cases}  \\sum_{i=1}^{n-p+1} (-1)^{(p-1)(i-1)} f \\bullet_i x + \\sum_{i=1}^p (-1)^{n (i-1) + p-1} f \\bullet_0 t^{i-1} (x) ~~ & \\mbox{ if } p < n+1\\\\\n(-1)^{p-1} \\sum_{i=0}^n (-1)^{in} f \\bullet_0 t^i (x)  ~~ & \\mbox{ if } p = n+1. \\end{cases}\n\\end{align}\nIt has been shown in \\cite[Proposition 4.3, Theorem 4.4]{kowal} that these two operators satisfy the following identities\n\\begin{align}\n&i_{f \\cup g} = i_f \\circ i_g, ~~~~ i_{\\delta_\\pi f} = b \\circ i_f - (-1)^p~ i_f \\circ b, \\label{eqn-ab}\\\\\n&\\mathcal{L}_{[f, g]} = \\mathcal{L}_f \\circ \\mathcal{L}_g - (-1)^{(p-1)(q-1)} \\mathcal{L}_g \\circ \\mathcal{L}_f , ~~~~ \\mathcal{L}_{\\delta_\\pi f} = - b \\circ \\mathcal{L}_f + (-1)^{(p-1)} \\mathcal{L}_f \\circ b. \\label{eqn-cd}\n\\end{align}\nIt follows from the above identities that the cap product and the Lie derivative descend to the simplicial homology $H_\\bullet (\\mathcal{M})$ and the graded homology $H_\\bullet (\\mathcal{M})$ is a module over both the algebra $(H^\\bullet_\\pi (\\mathcal{O}), \\cup)$ and the graded Lie algebra $(H^{\\bullet + 1}_\\pi (\\mathcal{O}), [~, ~])$. It has been further proved in \\cite[Theorem 4.5]{kowal} that the induced operations on (co)homology satisfies the identity (\\ref{eqn-t}) to make the pair $(H^\\bullet_\\pi (\\mathcal{O}), H_\\bullet (\\mathcal{M))}$ is a precalculus.\n\n\nSome additional structure on the operad $\\mathcal{O}$ makes the above precalculus into a calculus. We start with the following.\n\\begin{defn}\\label{defn-unit-opera} An operad with multiplication $(\\mathcal{O}, \\pi)$ is called unital if there is a $k$-module $\\mathcal{O}(0)$ so that the partial compositions $\\circ_i$ extend to $\\mathcal{O}(0)$, and there is an element $e \\in \\mathcal{O}(0)$ satisfying $\\pi \\circ_1 e = \\pi \\circ_2 e = \\mathds{1}$.\n\\end{defn}\n\nIf $(\\mathcal{O}, \\pi)$ is unital and $\\{ \\mathcal{M}(n) \\}_{n \\geq 0}$ is a cyclic comp module over $\\mathcal{O}$ (in the sense that (\\ref{eqn-p}) and (\\ref{eqn-r}) holds for $p=0$ as well), then one may define a differential $B : \\mathcal{M}(n) \\rightarrow \\mathcal{M}(n+1)$, called the Connes boundary operator, by\n\\begin{align*}\nB := \\sum_{i=0}^n (-1)^{in}  (\\mathrm{id} -t ) t ( e \\bullet_{n+1} t^i (x)).\n\\end{align*}\nLike classical Hochschild case, one may consider normalized cochain complex $\\{ \\overline{\\mathcal{O}}, \\delta_\\pi \\}$ that induces the same cohomology with $H^\\bullet_\\pi (\\mathcal{O})$. Similarly, a normalized chain complex $\\{ \\overline{\\mathcal {M}}, b \\}$ can be considered which induces the same homology with $H_\\bullet (\\mathcal{M})$ \\cite{kowal}.\nIt has been further shown in \\cite{kowal} that on the normalized (co)chain complexes\n\\begin{align}\\label{cartan-pre-id}\n\\mathcal{L}_f = [B, i_f ] + [b, S_f] - S_{\\delta_\\pi f}, \\text{ for } f \\in \\overline{\\mathcal{O}}(p),\n\\end{align}\nwhere the map $S_f : \\mathcal{M} (n) \\rightarrow \\mathcal{M} (n-p+2)$ is given by\n\\begin{align*}\nS_f = \\sum_{j=1}^{n-p+1} \\sum_{i=j}^{n-p+1} (-1)^{n(j-1) + (p-1)(i-1) } ~e \\bullet_0 ( f \\bullet_i t^{j-1} (x)), \\text{ for } 0 \\leq p \\leq n.\n\\end{align*}\nand $S_f = 0$, for $p > n$. Thus, it follows from (\\ref{cartan-pre-id}) that the Cartan-Rinehart homotopy formula holds on the induced (co)homology. Hence the pair $(H^\\bullet_\\pi (\\mathcal{O}), H_\\bullet (\\mathcal{M}))$ is a differential calculus.\n\n\n\n\\section{Hom-associative algebras and calculus structure}\\label{section-3}\nIn this section, we first recall hom-associative algebras and their Hochschild (co)homologies \\cite{makh-sil-0}, \\cite{amm-ej-makh}, \\cite{hasan}. In the next, we show that the pair of cohomology and homology forms a precalculus. Under some additional conditions on the hom-associative algebra, the precalculus turns out to be a  noncommutative differential calculus.\n\n\\begin{defn}\nA hom-associative algebra is a $k$-module $A$ together with a $k$-bilinear map $\\mu : A \\otimes A \\rightarrow A, (a, b) \\mapsto a \\cdot b $ and a $k$-linear map $\\alpha : A \\rightarrow A$ satisfying the following hom-associativity:\n\\begin{align*}\n(a \\cdot b ) \\cdot \\alpha (c) = \\alpha (a) \\cdot ( b \\cdot c ), ~ \\text{ for } a, b, c \\in A.\n\\end{align*} \n\\end{defn}\nA hom-associative algebra as above is denoted by the triple $(A, \\mu, \\alpha )$. It is called multiplicative if $\\alpha ( a \\cdot b ) = \\alpha (a) \\cdot \\alpha (b)$, for $a, b \\in A$. In the rest of the paper, by a hom-associative algebra, we shall always mean a multiplicative hom-associative algebra.\nIt follows from the above definition that any associative algebra is a (multiplicative) hom-associative algebra with $\\alpha = \\mathrm{id}_A$.\n\nA hom-associative algebra $(A, \\mu, \\alpha )$ is said to be unital if there is an element $1 \\in A$ such that $\\alpha (1) = 1$ and $a \\cdot 1 = 1 \\cdot a = \\alpha (a),$ for all $a \\in A$.\n\n\\begin{exam}\\label{hom-alg-exam}\nLet $(A, \\mu)$ be an associative algebra over $k$ and $\\alpha : A \\rightarrow A$ be an algebra homomorphism. Then $(A, \\mu_\\alpha =\\alpha \\circ \\mu , \\alpha )$ is a hom-associative algebra, called obtained by composition. If the associative algebra $(A, \\mu)$ is unital and $\\alpha$ is an unital associative algebra morphism then the hom-associative algebra $(A, \\mu_\\alpha =\\alpha \\circ \\mu , \\alpha )$ is unital with the same unit.\n\\end{exam}\n\nLet $(A, \\mu, \\alpha)$ be a hom-associative algebra. A bimodule over it consists of a $k$-module $M$ and a linear map $\\beta : M \\rightarrow M$ with actions $l : A \\otimes M \\rightarrow M,~ (a,m) \\mapsto am$, and $r: M \\otimes A \\rightarrow M, ~ (m,a) \\mapsto ma$ satisfying $\\beta (am) = \\alpha (a) \\beta(m),~ \\beta (ma) = \\beta(m) \\alpha (a)$ and\nthe following bimodule conditions are hold\n\\begin{align*}\n(a \\cdot b)\\beta( m) = \\alpha(a) (bm) ~~~~ (am)\\alpha(b) = \\alpha(a)(mb) ~~~~ (ma) \\alpha(b) = \\beta(m) (a \\cdot b), ~ \\text{ for } a, b \\in A, m \\in M.\n\\end{align*} \nA bimodule can be simply denoted by $(M, \\beta)$ when the actions are understood. It is easy to see that $(A, \\alpha)$ is a bimodule with left and right actions are given by $\\mu$. \n\nLet $(A, \\mu, \\alpha)$ be a hom-associative algebra and $(M, \\beta)$ be a bimodule over it. The group of $n$-cochains of $A$ with coefficients in $(M, \\beta)$ is given by $C^n_\\alpha ( A, M) := \\{ f : A^{\\otimes n} \\rightarrow M |~ \\beta \\circ f = f \\circ \\alpha^{\\otimes n} \\},$ for $n \\geq 1$. The coboundary map $\\delta_\\alpha : C^n_\\alpha (A, M) \\rightarrow C^{n+1}_\\alpha (A, M)$ given by\n\\begin{align}\n(\\delta_\\alpha f)(a_1, \\ldots, a_{n+1}) :=~& \\alpha^{n-1}(a_1)  f ( a_2, \\ldots, a_{n+1})  + (-1)^{n+1} f (a_1, \\ldots, a_n) \\alpha^{n-1} (a_{n+1}) \\\\\n~&+ \\sum_{i=1}^n (-1)^i~ f ( \\alpha (a_1), \\ldots, \\alpha (a_{i-1}), a_i \\cdot a_{i+1}, \\alpha (a_{i+2}), \\ldots, \\alpha (a_{n+1})). \\nonumber\n\\end{align}\nThe corresponding cohomology groups are denoted by $H^n_\\alpha (A, M)$, for $n \\geq 1$. When the bimodule is given by $(A, \\alpha)$, the corresponding cochain groups are denoted by $C^n_\\alpha (A)$ and the cohomology groups are denoted by $H^n_\\alpha (A)$, for $n \\geq 1$.\n\nIn this paper, we only require the Hochschild homology of $A$ with coefficients in itself. For homology with coefficients, see \\cite{hasan}. The $n$-th Hochschild chain group of $A$ with coefficients in itself is given by $C_n^\\alpha (A) := A \\otimes A^{\\otimes n}$, for $n \\geq 0$ and the boundary operator $d^\\alpha : C_n^\\alpha (A) \\rightarrow C_{n-1}^\\alpha (A)$ given by\n\\begin{align}\\label{hoch-hom-boundary}\nd^\\alpha ( a_0 \\otimes a_1 \\cdots a_n) :=~& a_0 \\cdot a_1 \\otimes \\alpha (a_2) \\cdots \\alpha (a_n) + (-1)^n a_n \\cdot a_0 \\otimes \\alpha (a_1) \\cdots \\alpha (a_{n-1}) \\\\\n~&+ \\sum_{i=1}^{n-1} (-1)^i~ \\alpha (a_0) \\otimes \\alpha (a_1) \\cdots (a_i \\cdot a_{i+1}) \\cdots \\alpha (a_n). \\nonumber\n\\end{align}\nThe corresponding homology groups are denoted by $H_n^\\alpha (A)$, for $n \\geq 0$.\n\n\nLet $(A, \\mu, \\alpha )$ be a hom-associative algebra. It has been shown in \\cite{das1} that the collection of Hochschild cochains $\\mathcal{O} = \\{ \\mathcal{O}(p) = C^p_\\alpha (A) \\}_{p \\geq 1}$ forms a non-symmetric operad with partial compositions\n\\begin{align}\\label{hom-ope}\n(f \\circ_i g ) ( a_1, \\ldots, a_{p+q-1}) = f ( \\alpha^{q-1} (a_1), \\ldots, \\alpha^{q-1} ( a_{i-1}), g (a_i, \\ldots, a_{i+q-1}), \\ldots, \\alpha^{q-1} ( a_{p+q-1})),\n\\end{align}\nfor $f \\in \\mathcal{O}(p), g \\in \\mathcal{O}(q)$ and the identity element $\\mathds{1} = \\mathrm{id}_A.$ Moreover, the element $\\mu \\in \\mathcal{O}(2) = C^2_\\alpha (A)$ is a multiplication in the above operad. The differential induced by $\\mu$ is same as $\\delta_\\alpha$ up to a sign. Hence the graded space of Hochschild cohomology $H_\\alpha^\\bullet (A)$ carries a Gerstenhaber structure.\n\n\\subsection{Precalculus structure}\n\nLet $\\mathcal{M}(n) := A \\otimes A^{\\otimes n}$, for $n \\geq 0$ be the $n$-th Hochschild chain group of $A$. For convenience, we denote the basic elements of $\\mathcal{M}(n)$ by $a_0 \\otimes a_1 \\cdots a_n$ when there is no confusion causes. For $p \\leq n$ and $1 \\leq i \\leq n-p+1$, we define maps $\\bullet_i : \\mathcal{O}(p) \\otimes \\mathcal{M}(n) \\rightarrow \\mathcal{M}(n-p+1)$ by\n\\begin{align*}\nf \\bullet_i ( a_0 \\otimes a_1 \\cdots a_n ) := \\alpha^{p-1}( a_0 ) \\otimes \\alpha^{p-1}( a_1) \\cdots \\alpha^{p-1}(a_{i-1}) f (a_i, \\ldots, a_{i+p-1}) \\alpha^{p-1}(a_{i+p}) \\cdots \\alpha^{p-1}( a_n). \n\\end{align*} \n\\begin{prop}\nWith these notations, $\\mathcal{M} = \\{ \\mathcal{M}(n) \\}_{n \\geq 0}$ is a unital comp module over the operad $\\mathcal{O}$.\n\\end{prop}\n\n\\begin{proof}\nWe have to verify the identities (\\ref{eqn-p}) and (\\ref{eqn-q}). First, for $j < i$, we have\n\\begin{align*}\n&f \\bullet_i ( g \\bullet_j (a_0 \\otimes a_1 \\cdots a_n )) \\\\&= f \\bullet_i \\big( \\alpha^{q-1}(a_0) \\otimes \\alpha^{q-1}(a_1) \\cdots g (a_j, \\ldots, a_{j+q-1}) \\cdots \\alpha^{q-1} (a_n) \\big) \\\\\n&= \\alpha^{p+q-2} (a_0) \\otimes \\alpha^{p+q-2} (a_1) \\cdots \\alpha^{p+q-2} (a_{j-1}) \\alpha^{p-1} ( g (a_j, \\ldots, a_{j+q-1} )) \\cdots \\\\& \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad f (\\alpha^{q-1} (a_{i+q-1}), \\ldots, \\alpha^{q-1}(a_{i+p+q-2})) \\cdots \\alpha^{p+q-2} (a_n).\n\\end{align*}\nOn the other hand,\n\\begin{align*}\n&g \\bullet_j ( f \\bullet_{i+q-1} (a_0 \\otimes a_1 \\cdots a_n )) \\\\\n&= g \\bullet_j \\big( \\alpha^{p-1} (a_0) \\otimes \\alpha^{p-1} (a_1) \\cdots f (a_{i+q-1} , \\ldots, a_{i+p+q-2}) \\cdots \\alpha^{p-1} (a_n)    \\big)\\\\\n&= \\alpha^{p+q-2} (a_0) \\otimes \\alpha^{p+q-2} (a_1) \\cdots \\alpha^{p+q-2} (a_{j-1})  g ( \\alpha^{p-1}(a_j), \\ldots, \\alpha^{p-1}(a_{j+q-1}) ) \\cdots \\\\& \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\alpha^{q-1}(f ( a_{i+q-1}, \\ldots, a_{i+p+q-2})) \\cdots \\alpha^{p+q-2} (a_n).\n\\end{align*}\nHence $f \\bullet_i ( g \\bullet_j (a_0 \\otimes a_1 \\cdots a_n )) = g \\bullet_j ( f \\bullet_{i+q-1} (a_0 \\otimes a_1 \\cdots a_n ))$. Similarly, for  $j-p <i \\leq j$, we can verify that\n\\begin{align*}\n&f \\bullet_i ( g \\bullet_j (a_0 \\otimes a_1 \\cdots a_n )) \\\\\n&= \\alpha^{p+q-2} (a_0) \\otimes \\alpha^{p+q-2} (a_1) \\cdots f \\big(\\alpha^{q-1} (a_i), \\ldots, g (a_j, \\ldots, a_{j+q-1}), \\ldots, \\alpha^{q-1} (a_{i+p+q-2})    \\big) \\cdots \\alpha^{p+q-2} (a_n)\\\\\n&= (f \\circ_{j-i+1} g) \\bullet_i (a_0 \\otimes a_1 \\cdots a_n ).\n\\end{align*}\nFinally, for $1 \\leq i \\leq j-p$, we have\n\\begin{align*}\n&f \\bullet_i ( g \\bullet_j (a_0 \\otimes a_1 \\cdots a_n )) \\\\ \n&= \\alpha^{p+q-2} (a_0) \\otimes \\alpha^{p+q-2} (a_1) \\cdots f (\\alpha^{q-1}(a_i), \\ldots, \\alpha^{q-1}(a_{i+p-1})) \\cdots \\alpha^{p-1} (g (a_j , \\ldots, a_{j+q-1})) \\cdots \\alpha^{p+q-2} (a_n)\\\\\n&= g \\bullet_{j-p+1} ( f \\bullet_i (a_0 \\otimes a_1 \\cdots a_n )).\n\\end{align*}\nIt is also easy to see that $\\mathds{1} \\bullet_i (a_0 \\otimes a_1 \\cdots a_n ) = (a_0 \\otimes a_1 \\cdots a_n )$, for all $a_0 \\otimes a_1 \\cdots a_n \\in \\mathcal{M}(n)$. Hence the proof.\n\\end{proof}\n\nWe define additional maps $\\bullet_0 : \\mathcal{O}(p) \\otimes \\mathcal{M}(n) \\rightarrow \\mathcal{M}(n-p+1)$, for $p \\leq n+1$ and $t : \\mathcal{M} (n) \\rightarrow \\mathcal{M}(n)$ by\n\\begin{align*}\nf \\bullet_0 ( a_0 \\otimes a_1 \\cdots a_n ) :=~& f(a_0, \\ldots, a_{p-1}) \\otimes \\alpha^{p-1}( a_p) \\cdots \\alpha^{p-1}( a_n),\\\\\nt (  a_0 \\otimes a_1 \\cdots a_n ) :=~& a_n \\otimes a_0 a_1 \\cdots a_{n-1}.\n\\end{align*}\n\n\\begin{prop}\nWith these additional structures, $\\mathcal{M}$ is a cyclic comp module over $\\mathcal{O}$.\n\\end{prop}\n\n\\begin{proof}\nSimilar to the proof of the previous proposition, one shows that $\\bullet_0$ satisfy the identities (\\ref{eqn-p}) and (\\ref{eqn-q}). The operator $t$ obviously satisfies $t^{n+1} = \\mathrm{id}$. Finally, for any $1 \\leq i \\leq n-p$,\n\\begin{align*}\nt ( f \\bullet_i ( a_0 \\otimes a_1 \\cdots a_n ) ) =~& t ( a_0 \\otimes a_1 \\cdots f (a_i, \\ldots, a_{i+p-1}) \\cdots a_n )\\\\\n=~& a_n \\otimes a_0 \\cdots f (a_i, \\ldots, a_{i+p-1})  \\cdots a_{n-1} \\\\\n=~& f \\bullet_{i+1} (a_n \\otimes a_0 a_1 \\cdots a_{n-1}) = f \\bullet_{i+1} t (a_0 \\otimes a_1 \\cdots a_n ).\n\\end{align*}\nThe same holds for $i=0$. Hence the proof.\n\\end{proof}\n\n\nConsider the operad $\\mathcal{O}$ with the multiplication $\\pi \\in \\mathcal{O}(2)$.  As $\\mathcal{M}$ is a cyclic comp module over $\\mathcal{O}$, the corresponding simplicial boundary map $b $ as of (\\ref{simplicial-boun}) is precisely given by the Hochschild boundary operator $d^\\alpha$ given in (\\ref{hoch-hom-boundary}). Hence $H_\\bullet (\\mathcal{M})$ is given by the Hochschild homology $H_\\bullet^\\alpha (A)$ of the hom-associative algebra $A$. \n\nNote that, in this example, the corresponding cap product (\\ref{cap-pro}) and Lie derivative (\\ref{lie-pro}) are given by\n\\begin{center}\n $i_f ( a_0 \\otimes a_1 \\cdots a_n ) =  \\alpha^{p-1} (a_0) \\cdot f (a_1, \\ldots, a_p) \\otimes \\alpha^p (a_{p+1}) \\cdots \\alpha^p (a_n),$\n \\end{center}\n \\begin{align*}\n \\mathcal{L}_f ( a_0 \\otimes& a_1 \\cdots a_n ) = \\sum_{i=1}^{n-p+1} (-1)^{(p-1)(i-1)} ~\\alpha^{p-1} (a_0) \\otimes \\alpha^{p-1} (a_1) \\cdots f (a_i, \\ldots, a_{i+p-1}) \\cdots \\alpha^{p-1} (a_n)\\\\\n &+ (-1)^{p-1} f (a_0, \\ldots, a_{p-1}) \\otimes \\alpha^{p-1} (a_p) \\cdots \\alpha^{p-1} (a_n) \\\\\n& + \\sum_{i=2}^{p-1} (-1)^{n(i-1) + p-1} ~f(a_{n-i+2}, \\ldots, a_{n-i+p+1}) \\otimes \\cdots \\alpha^{p-1} (a_n) \\alpha^{p-1}(a_0) \\cdots \\alpha^{p-1} (a_{n-i+1})\\\\\n& + (-1)^{(n+1)(p-1)}~ f(a_{n-p+2}, \\ldots, a_n, a_0) \\otimes \\alpha^{p-1}(a_1) \\cdots \\alpha^{p-1} (a_{n-p+1}), ~ \\text{ for } p < n+1.\n\\end{align*}\n\nHence we get the following.\n\n\\begin{thm}\nLet $(A, \\mu, \\alpha )$ be a hom-associative algebra. Then the pair $(H^\\bullet_\\alpha (A), H_\\bullet^\\alpha (A))$ of Hochschild cohomology and Hochschild homology of $A$ forms a precalculus.\n\\end{thm}\n\n\\subsection{Calculus structure}\n\nA hom-associative algebra $(A, \\mu, \\alpha)$ is said to be regular if $\\alpha$ is invertible. In such a case, it can be easily checked that $(A, \\mu_{\\alpha^{-1}} = \\alpha^{-1} \\circ \\mu)$ is an associative algebra and $\\alpha$ is an algebra morphism. Moreover, the hom-associative algebra $(A, \\mu, \\alpha)$ is then obtained by composition (Example \\ref{hom-alg-exam}).\n\nLet $(A, \\mu, \\alpha)$ be a regular unital hom-associative algebra with unit $1 \\in A$. Then the corresponding operad with multiplication $(\\mathcal{O}, \\mu)$ considered in (\\ref{hom-ope}) is unital in the sense of Definition \\ref{defn-unit-opera} with $\\mathcal{O}(0) = \\{ a \\in A | \\alpha (a) = a \\}$ and the partial compositions $\\circ_i : \\mathcal{O}(p) \\otimes \\mathcal{O}(q) \\rightarrow \\mathcal{O}(p+q-1)$, for $p, q \\geq 0$ given by the same formula (\\ref{hom-ope}). The element $1 \\in \\mathcal{O}(0)$ is unit as\n\\begin{align*}\n(\\mu \\circ_1 1 )(a) = \\mu (1, \\alpha^{-1}(a)) = 1 \\cdot \\alpha^{-1} (a) = a ~~~ \\text{~ and ~} ~~~\n(\\mu \\circ_2 1 )(a) = \\mu (\\alpha^{-1}(a), 1) = \\alpha^{-1} (a) \\cdot 1 = a.\n\\end{align*}\n\n\\begin{remark}\nIn this case, the Hochschild cochain complex is defined from degree $0$ with $C^0_\\alpha (A) = \\mathcal{O}(A)$ and $\\delta_\\alpha : C^0_\\alpha (A) \\rightarrow C^1_\\alpha (A)$ by $\\delta_\\alpha (a) (b) =  \\alpha^{-1}(b) \\cdot a - a \\cdot \\alpha^{-1}(b)$, for $a \\in C^0_\\alpha (A)$ and $b \\in A$. More generally, if a hom-associative algebra $A$ is regular and a bimodule $(M, \\beta)$ is also regular (i.e. $\\beta$ is invertible) then the Hochschild cohomology of $A$ with coefficients in $(M, \\beta)$ can be defined from degree $0$.\n\\end{remark}\n\nIt is also easy to see that the Hochschild chains $\\{ \\mathcal{M} (n) \\}_{n \\geq 0}$ is infact a cyclic comp module over the unital operad $\\mathcal{O}$ in the sense that the identities (\\ref{eqn-p}) and (\\ref{eqn-r}) holds for $p= 0$. \nHence by the result of the previous section, we deduce the following.\n\\begin{thm}\nLet $(A, \\mu, \\alpha)$ be a regular unital hom-associative algebra. Then the pair  $(H^\\bullet_\\alpha (A), H_\\bullet^\\alpha (A))$ is a noncommutative differential calculus.\n\\end{thm}\n\n\\section{Application}\n\nUsing the calculus of the previous section and some additional hypothesis, we construct a Batalin-Vilkovisky algebra structure on the Hochschild cohomology of a hom-associative algebra. In particular, the Hochschild cohomology of a regular unital symmetric hom-associative algebra carries a Batalin-Vilkovisky algebra structure. We start with the following definition.\n\n\\begin{defn}\nA Gerstenhaber algebra $(\\mathcal{A}, \\cup, [~, ~])$ is said to be a Batalin-Vilkovisky algebra (BV algebra in short) if there is a BV-generator, i.e. a degree $-1$ map $\\triangle : \\mathcal{A}^\\bullet \\rightarrow \\mathcal{A}^{\\bullet -1}$ satisfying $\\triangle^2 = 0 $ and\n\\begin{align*}\n[f, g ] = - (-1)^p  \\big( \\triangle ( f \\cup g) -  \\triangle (f) \\cup g - (-1)^p f \\cup \\triangle (g)   \\big), ~ \\text{ for } f \\in \\mathcal{A}^p, ~ g \\in \\mathcal{A}.\n\\end{align*}\n\\end{defn}\n\n\\begin{lemma}\\label{fin-lem}\nIn a calculus $(\\mathcal{A}, \\Omega )$, we have for $f \\in \\mathcal{A}^p$ and $g \\in \\mathcal{A}^q$,\n\\begin{align*}\ni_{[f, g]} x = (-1)^{q+1}~ i_{f \\cup g} B (x) + (-1)^{p+1} B (i_{f \\cup g} x) + i_f B ( i_g x ) + (-1)^{pq+p+q}~ i_g  B ( i_f x). \n\\end{align*}\n\\end{lemma}\n\n\\begin{proof}\nWe have\n\\begin{align*}\ni_{[f, g]} =~& [ i_f, \\mathcal{L}_g ] = i_f \\circ \\mathcal{L}_g - (-1)^{ p (q+1)} \\mathcal{L}_g \\circ i_f \\\\\n=~& i_f \\circ ( B \\circ i_g - (-1)^q ~i_g \\circ B) -(-1)^{p (q+1)} ( B \\circ i_g - (-1)^q ~i_g \\circ B) \\circ i_f  \\\\\n=~& i_f \\circ B \\circ i_g + (-1)^{q+1}~ i_{f \\cup g} \\circ B  - (-1)^{p(q+1)} (-1)^{pq} B \\circ i_{f \\cup g}  + (-1)^{p (q+1) + q} i_g \\circ B \\circ i_f.\n\\end{align*}\nHence the proof.\n\\end{proof}\n\n\\begin{prop}\\label{final-prop}\nLet $(A, \\mu, \\alpha)$ be a regular unital hom-associative algebra. If there is an element $c \\in H^\\alpha_d (A)$ such that $B (c) = 0$ and the maps $H^n_\\alpha (A) \\rightarrow H^\\alpha_{d-n} (A),~ f \\mapsto i_f c$ are $H^\\bullet_\\alpha (A)$-module isomorphisms, then the map \n\\begin{align*}\n\\triangle : H^\\bullet_\\alpha (A) \\rightarrow H^{\\bullet -1}_\\alpha (A) ~~\\text{ defined by } ~~ i_{\\triangle (f)} c = B (i_f c)\n\\end{align*}\nis a BV-generator on the Gerstenhaber algebra $H^\\bullet_\\alpha (A)$. In other words, $H^\\bullet_\\alpha (A)$ is a BV algebra.\n\\end{prop}\n\n\\begin{proof}\nWe have from Lemma \\ref{fin-lem} that\n\\begin{align*}\ni_{[f, g]} c =~& - (-1)^ p ~ B \\circ i_{f \\cup g} (c) + i_f \\circ B \\circ i_g (c) + (-1)^{pq+p+q} ~ i_g \\circ B \\circ  i_f (c) \\\\\n=~& -(-1)^p i_{\\triangle ( f \\cup g)} c + i_f \\circ i_{\\triangle (g)} c + (-1)^{pq + p + q} i_g i_{\\triangle (f)} c \\\\\n=~& - (-1)^p \\big(   i_{\\triangle ( f \\cup g)} c - (-1)^p ~ i_{f \\cup \\triangle (g)} c - i_{\\triangle (f) \\cup g} c \\big).\n\\end{align*}\nHence the result follows from the given hypothesis.\n\\end{proof}\n\nIn the next, we give a dual analogue of Lemma \\ref{fin-lem}, hence a dual version of Proposition \\ref{final-prop}. Before that, we need the followings.\n\nLet $(A, \\mu, \\alpha)$ be a regular unital hom-associative algebra. Then it can be easily checked that the dual space $A^*$ with the linear map $(\\alpha^{-1})^* : A^* \\rightarrow A^*$ is a bimodule over the hom-associative algebra $A$ with left and right actions given by $(a \\theta )(b)  =  \\theta ( \\alpha^{-1} (b \\cdot a))$ and $(\\theta a ) (b) = \\theta ( \\alpha^{-1} (a \\cdot b))$, for $a, b \\in A$ and $\\theta \\in A^*$.\n\nLet $B: C^\\alpha_n (A, A) \\rightarrow C^\\alpha_{n+1}(A,A)$ be the Connes boundary map. Then the dual map $B^* : \\mathrm{Hom}(A^{\\otimes n+1} , A^*) \\rightarrow  \\mathrm{Hom}(A^{\\otimes n} , A^*) $ restricts to a map (denoted by the same notation)  $B^* : C^{n+1}_\\alpha (A, A^*) \\rightarrow C^n_\\alpha (A, A^*)$, for $n \\geq 0$ that also passes onto the cohomology $H^\\bullet_\\alpha (A, A^*)$. The induced map on cohomology is also denoted by $B^*$.\n\nFor $f \\in C^p_\\alpha (A)$ and $m \\in C^{|m|}_\\alpha (A, A^*)$, we define a product $f \\cdot m \\in C^{p+|m|}_\\alpha (A, A^*)$ by\n\\begin{align*}\n(f \\cdot m)(a_1 , \\ldots, a_{p+|m|}) = f(a_1, \\ldots, a_p ) m (a_{p+1}, \\ldots, a_{p+|m|})\n\\end{align*}\nthat induces a product on the cohomology level.\n\n\\begin{lemma}\nLet $(A, \\mu, \\alpha)$ be a regular unital hom-associative algebra. For $f \\in H^p_\\alpha (A), ~ g \\in H^q_\\alpha (A)$ and $m \\in H^\\bullet_\\alpha (A, A^*)$, we have\n\\begin{align*}\n[f, g] \\cdot m = (-1)^{p+1} B^* ( ( f \\cup g) \\cdot m) + f \\cdot B^* ( g \\cdot m) + (-1)^{pq+p+q}~ g \\cdot B^* ( f \\cdot m) + (-1)^{q+1} ( f \\cup g) \\cdot B^* (m).\n\\end{align*}\n\\end{lemma}\n\n\\begin{proof}\nWe have from the definition of $B^*$ that $B^*(m) = (-1)^{|m|} m \\circ B$ and from the Koszul sign convension\n\\begin{align*}\nm ( i_f  x) = (-1)^{|m| p } (f \\cdot m )(x).\n\\end{align*}\nOn the other hand, from Lemma \\ref{fin-lem}, we have\n\\begin{align*}\nm (i_{[f,g]} x) = (-1)^{q+1}~ m (i_{f \\cup g} B(x)) + (-1)^{p+1}~ m \\circ B (i_{f \\cup g} x) + m ( i_f B (i_g x)) + (-1)^{pq+p+q}~ m (i_g B (i_f x)).\n\\end{align*}\nHence we get\n\\begin{align*}\n(-1)^{|m|(p+q-1)} ([f,g] \\cdot m)(x) =~&  (-1)^{q+1 + |m| (p+q) + p+q + |m|}~ B^* ((f \\cup g) \\cdot m)(x) \\\\~&+ (-1)^{p+1+|m| + (|m|+1) (p+q)}~ ((f \\cup g) \\cdot B^*(m)) (x) \\\\\n~&+ (-1)^{|m| p + (p+|m| +1) q + p + |m|}~ (g \\cdot B^*(f \\cdot m))(x) \\\\\n~&+ (-1)^{pq+p+q + |m|q + (q + |m| +1)p + q + |m|}~ (f \\cdot B^*(g \\cdot m ))(x).\n\\end{align*}\nThus the result follows by cancelling the sign $(-1)^{|m|(p+q-1)}$ from both sides.\n\\end{proof}\n\n\\begin{prop}\\label{prop-last-prop}\nLet $(A, \\mu, \\alpha)$ be a regular unital hom-associative algebra. If there is an element $m \\in H^d_\\alpha (A, A^*)$ such that $B^* (m) = 0$ and the maps $H^\\bullet_\\alpha (A) \\rightarrow H^{\\bullet + d}_\\alpha (A, A^*), ~ f \\mapsto f \\cdot m$ are $H^\\bullet_\\alpha (A)$-module isomorphisms, then\n\\begin{align*}\n\\triangle : H^\\bullet_\\alpha (A) \\rightarrow H^{\\bullet -1}_\\alpha (A, A) ~~\\text{ defined by }~~ (\\triangle f) \\cdot m = B^* ( f \\cdot m)\n\\end{align*}\nmakes the Gerstenhaber algebra $H^\\bullet_\\alpha (A)$ into a BV algebra.\n\\end{prop}\n\n\\begin{proof}\nWe have from the previous lemma that\n\\begin{align*}\n[f, g] \\cdot m = (-1)^{p+1} \\triangle ( f \\cup g) \\cdot m  + ( f \\cup \\triangle (g)) \\cdot m + (-1)^{pq + p + q} (g \\cup \\triangle (f)) \\cdot m.\n\\end{align*}\nThe result follows by applying the graded commutativity on the last term of the right-hand side and from the given hypothesis.\n\\end{proof}\n\nA regular unital hom-associative algebra $(A, \\mu, \\alpha)$ is said to be symmetric if there is a hom-bimodule isomorphism $\\Theta : A \\xrightarrow{\\sim} A^*.$ Thus, in a symmetric algebra, there is an isomorphism $H^\\bullet_\\alpha (A) \\xrightarrow{\\sim} H^\\bullet_\\alpha (A, A^*)$ via the map $\\Theta$. Hence the dual map $B^* : H^\\bullet_\\alpha (A, A^*) \\rightarrow H^{\\bullet -1}_\\alpha (A, A^*)$ induces a map (denoted by the same notation) $B^* : H^*_\\alpha (A) \\rightarrow H^{\\bullet -1}_\\alpha (A).$\n\n\\begin{thm}\nLet $(A, \\mu, \\alpha)$ be a regular unital symmetric hom-associative algebra. Then the map $B^* : H^*_\\alpha (A) \\rightarrow H^{\\bullet -1}_\\alpha (A)$ defines a BV algebra structure on the Gerstenhaber algebra $H^\\bullet_\\alpha (A).$\n\\end{thm}\n\n\\begin{proof}\nConsider the element $\\Theta (1) \\in A^* = C^0_\\alpha (A, A^*)$ which is a $0$-cocycle. Hence $m = [ \\Theta (1) ] \\in H^0_\\alpha (A, A^*)$. Since $B^*$ decreases degree, we have $B^* (m) = 0$. Moreover, the maps $H^\\bullet_\\alpha (A) \\rightarrow H^\\bullet_\\alpha (A, A^*),~ f \\mapsto f \\cdot m \\simeq f$ are isomorphisms. With these isomorphisms, the map $\\triangle$ given in Proposition \\ref{prop-last-prop} coincides with $B^*$, as\n\\begin{align*}\n\\triangle (f) \\cong ( \\triangle f) \\cdot m = B^* ( f \\cdot m ) \\cong B^* (f).\n\\end{align*}\nHence the result follows from Proposition \\ref{prop-last-prop}.\n\\end{proof}\n\\begin{remark}\nThis generalizes the corresponding result for associative algebras \\cite{tradler}. However, our proof relies on the noncommutative differential calculus associated with (hom-)associative algebras.\n\\end{remark}\n\\medskip\n\n\\noindent {\\bf Acknowledgements.} The research is supported by the fellowship of Indian Institute of Technology (IIT) Kanpur. The author thanks the Institute for support.\n\n\\medskip\n\n\\noindent {\\bf Data Availability Statement.} Data sharing is not applicable to this article as no new data were created or analyzed in this study.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nIn \\cite{KS} we proved that for every coloring \n\\(F:[\\oo_n]^{<\\oo}\\to\\oo\\) there exists a set \n\\(A\\in [\\oo_n]^{<\\oo}\\) which can be written at least \n\\(2^n-1\\) ways as \\(A=H_0\\cup H_1\\) for some \n\\(H_0\\neq H_1\\), \\(F(H_0)=F(H_1)\\) and that for \n\\(n=1\\) there is in fact a function \\(F\\) for which this is \nsharp. \nHere we show that for every \\(n<\\oo\\) it is consistent that \n\\(2^\\oo=\\oo_n\\) and for some function $F$ as above \nfor every finite set \\(A\\) there are at most \\(2^n-1\\) \nsolutions of the above equation. \nWe use historic forcing which was first used in \\cite{BS} and \\cite{SS} \nthen in \\cite{K} and \\cite{GS}.\nUnder GCH, we improve the positive result of \\cite{KS} by showing \nthat for every  $F$ as above some finite set can be written \nat least \n\\(T_n=\\frac{1}{2}\\sum^n_{i=1}{n+i\\choose n}{n\\choose i}\\) \nways as the union of two sets with the same $F$ value. \n\n\nWith the methods of \\cite{KS} it is easy to show the \nfollowing corollary of our\nindependence result. \nIt is consistent that \\(2^\\oo=\\oo_n\\) and \nthere is a function \\(f:{\\bf R}\\to\\oo\\) \nsuch that if $x$ is a real number then $x$ \ncannot be written more than $2^n-1$ ways as \nthe arithmetic mean of some $y\\neq z$ with \n\\(f(y)=f(z)\\). (\\((y,z)\\) and \\((z,y)\\) are not regarded \ndistinct.) \nAnother idea of \\cite{KS} can be used to modify \nour second result to the following.\nIf GCH holds and  $V$ is a vector space over the rationals \nwith $|V|=\\oo_n$, $f:V\\to\\oo$ \nthen some vector can be written at least $T_n$ ways\nas the arithmetic mean of two vectors with the same \n$f$-value. \n\\bigskip\n\n\\noindent{\\bf Notation} \nWe use the standard  set theory notation. \nIf \\(S\\) is a set, \\(\\kappa\\) a cardinal, then \n$[S]^{\\kappa}=\\{A\\subseteq S:|A|={\\kappa}\\}$, \n$[S]^{<{\\kappa}}=\\{A\\subseteq S:|A|<{\\kappa}\\}$, \n$[S]^{\\leq {\\kappa}}=\\{A\\subseteq S:|A|\\leq{\\kappa}\\}$. \n\\(P(S)\\) is the power set of \\(S\\). \nIf $f$ is a function, $A$ a set, then \n$f[A]=\\{f(x):x\\in A\\}$. \n\n\n\\section{The independence result}\n\n\\begin{thm}\n For\\ $1\\leq n <\\oo$ it is \n consistent that $2^\\oo=\\oo_n$ and there is a \n function $F:[\\oo_n]^{<\\oo}\\to \\oo$ such that \n for every $A\\in [\\oo_n]^{<\\oo}$ there are at most \n $2^n-1$ solutions of $A=H_0\\cup H_1$ with $H_0\\neq H_1$, \n $F(H_0)=F(H_1)$.\n\\end{thm}\n\\bigskip\n\n\n\n\nFor $\\aa<\\omega_n$ fix a bijection ${\\varphi}_\\aa:\\aa\\to |\\aa|$. \nFor $x\\in [\\omega_n]^{<\\omega}$ define \n$\\gg_i(x)$ for $i<k=\\min(n,|x|)$ as follows. \n$\\gg_0(x)=\\max(x)$. \n$$\n  \\gg_{i+1}(x)= \n               {\\varphi}^{-1}_{\\gg_0(x)} \\Bigl(\\gg_i\\bigl({\\varphi}_{\\gg_0(x)}\n                                        [x\\cap \\gg_0(x)]\\bigr)\\Bigr).\n$$\n$\\gg(x)=\\{\\gg_0(x),\\dots,\\gg_{k-1}(x)\\}$. \n\nSo, for example, if $n=0$ then $\\gg(x)=\\emptyset$, if $n=1$, \n\\(x\\neq\\emptyset\\), then \n$\\gg(x)=\\{\\gg_0(x)\\}=\\{\\max(x)\\}$.\n\n\\bigskip\n\\begin{lem}\n Given $s\\in [\\omega_n]^{\\leq n}$ \n there are at most countably many $x\\in [\\omega_n]^{<\\omega}$ \n such that $\\gg(x)=s$.\n\\end{lem}\n\\bigskip\n\\noindent{\\bf Proof} \nBy induction on $n$. \n\\qed\n\\bigskip\n\nLet $\\Phi(s)=\\bigcup\\{x:\\gg(x)\\subseteq s\\}$, \na countable set for \\(s\\in [\\oo_n]^{<\\oo}\\).\n\n\\bigskip\n\\noindent{\\bf Definition} The two sets \n$x$, $y\\in [\\omega_n]^{<\\omega}$ are \n{\\sl isomorphic} if the structures \n$(x;<,\\gg_0(x), \\dots, \\gg_{k-1}(x))$, \n$(y;<,\\gg_0(y), \\dots, \\gg_{k-1}(y))$, are isomorphic, i.e., \n$|x|=|y|$ and the positions of the elements $\\gg_i(x)$, $\\gg_i(y)$ \nare the same. \n\n\\medskip\nNotice that for every finite $j$ there are just finitely many \nisomorphism types of $j$-element sets. \n\\medskip\n\n\nThe elements of $P$, the applied notion of forcing will be \n{\\sl some} structures of the form \n$p=(s,f)$ where $s\\in[\\omega_n]^{<\\omega}$ and \n$f:P(s)\\to \\omega$.\n\nThe only element of $P_0$ is ${\\bf 1}_P=(\\emptyset,\\langle \\emptyset,0\n\\rangle)$,  it will be the largest element of $P$. \nThe elements of $P_1$ are of the form \n$p=(\\{\\xi\\},f)$ where $f(\\emptyset)=0\\neq f(\\{\\xi\\})$ \nfor $\\xi<\\omega_n$. \n\nGiven $P_t$, \n$p=(s,f)$ is in  $P_{t+1}$ if the following is true. \n$s=\\Delta\\cup a\\cup b$ is a disjoint decomposition. \n$p'=(\\Delta\\cup a,f')$ and $p''=(\\Delta\\cup b,f'')$ are in $P_t$ \nwhere $f'=f|P(\\Delta\\cup a)$, $f''=f|P(\\Delta\\cup b)$. \nThere  is $\\pi:\\Delta\\cup a\\to \\Delta\\cup b$, an isomorphism \nbetween $(\\Delta\\cup a,<,P(\\Delta\\cup a),f')$ and \n$(\\Delta\\cup b,<,P(\\Delta\\cup b),f')$. \n$\\pi|\\Delta$ is the identity. \nFor $H\\subseteq \\Delta\\cup a$ the sets $H$ and \n$\\pi[H]$ are isomorphic. \n$a\\cap \\Phi(\\Delta)=b\\cap \\Phi(\\Delta)=\\emptyset$. \n$f-f'-f''$ is one-to-one and takes only values outside \n${\\rm Ran}(f')$ (which is the same as ${\\rm Ran}(f'')$). \n$P=\\bigcup\\{P_t:t<\\omega\\}$. \nWe make $p\\leq p',p''$ and the ordering on $P$ is the one \ngenerated by this. \n\\bigskip\n\\begin{lem}\n $(P,\\leq)$ is ccc.\n\\end{lem}\n\\bigskip\n\\noindent{\\bf Proof} \nAssume that $p_\\aa\\in P$ ($\\aa<\\omega_1$). \nWe can assume by thinning and using the $\\Delta$-system lemma \nand the pigeon hole principle that the following hold. \n$p_\\aa\\in P_t$ for the same $t<\\omega$. \n$p_\\aa=(\\Delta\\cup a_\\aa,<,P(\\Delta\\cup a_\\aa),f_\\aa)$ where the structures \n$(\\Delta\\cup a_\\aa,<,f_\\aa)$ and \n$(\\Delta\\cup a_\\bb,<,f_\\bb)$ are isomorphic for $\\aa,\\bb<\\oo_1$, \n$\\{\\Delta,a_\\aa:\\aa<\\oo_1\\}$ pairwise disjoint. \nWe can also assume that if $\\pi$ is the \nisomorphism between \n$(\\Delta\\cup a_\\aa,<,f_\\aa)$ and \n$(\\Delta\\cup a_\\bb,<,f_\\bb)$ then $H$ and $\\pi[H]$ \nare isomorphic for $H\\subseteq \\Delta\\cup a_\\aa$. \nMoreover, if we assume that $\\Delta$ occupies the \nsame positions in the ordered sets $\\Delta\\cup a_\\aa$ \n($\\aa<\\oo_1$) then $\\pi$ will be the identity on $\\Delta$. \nAs $\\Phi(\\Delta)$ is countable, by removing \ncountably many indices we can also assume that \n$\\Phi(\\Delta)\\cap a_\\aa=\\emptyset$ for $\\aa<\\oo_1$. \nNow any $p_\\aa$ and $p_\\bb$ are compatible as we can \ntake \n$p=(\\Delta\\cup a_\\aa\\cup a_\\bb,<,P(\\Delta\\cup a_\\aa \\cup a_\\bb),f)\n\\leq p_\\aa$, $p_\\bb$ \nwhere $f\\supseteq f_\\aa$, $f_\\bb$ \nis an appropriate extension, i.e., \n$f-f_\\aa -f_\\bb$ is one-to-one and takes values \noutside ${\\rm Ran}(f_\\aa)$. \n\\qed\n\n\\bigskip\n\\begin{lem}\n If $(s,f)\\in P$, $H_0,H_1\\subseteq s$ \n have $f(H_0)=f(H_1)$ then $H_0$, $H_1$ are isomorphic.\n\\end{lem}\n\\bigskip\n\\noindent{\\bf Proof} \nSet  $(s,f)\\in P_t$. \nWe prove the statement by induction on $t$. \nThere is nothing to prove for $t<2$. \nAssume now that $(s,f)\\in P_{t+1}$, \n$s=\\Delta\\cup a\\cup b$, \n$\\pi:\\Delta\\cup a\\to \\Delta\\cup b$ \nas in the definition of $(P,\\leq)$. \nAs $f(H_0)$ is a value taken twice by $f$, both \n$H_0$ and $H_1$ must be subsets of either \n$\\Delta\\cup a$ or $\\Delta\\cup b$. \nWe are done by induction unless $H_0\\subseteq \\Delta\\cup a$ \nand $H_1\\subseteq \\Delta\\cup b$ (or vice versa). \nNow $H_0$ and $\\pi[H_0]$ are isomorphic and \n$f(H_0)=f(\\pi[H_0])=f(H_1)$ so by the inductive hypothesis \n$\\pi[H_0]$ and $H_1$ are ismorphic and then so are $H_0$, $H_1$. \n\\qed\n\n\\bigskip\n\\begin{lem}\n If $(s,f)\\in P$, \n $H_0,H_1\\subseteq s$, \n $f(H_0)=f(H_1)$, $x\\in H_0\\cap H_1$ then $x$ occupies the \n same position in the ordered sets $H_0$, $H_1$.\n\\end{lem}\n\\bigskip\n\\noindent{\\bf Proof} \nSimilarly to the proof of the previous Lemma, \nby induction on $t$, for $(s,f)\\in P_t$. \nWith similar steps, we can assume that \n$(s,f)=(\\Delta\\cup a\\cup b,f)\\leq (\\Delta\\cup a,f'), (\\Delta\\cup b,f'')$, \n$H_0\\subseteq \\Delta\\cup a$, \n$H_1\\subseteq \\Delta\\cup b$. \nNotice that $x\\in \\Delta$. \nNow, as $\\pi(x)=x$, \n$x$ is a common element of $\\pi[H_0]$ and $H_1$ and also \n$f''(\\pi[H_0])=f''(H_1)$. \nBy induction we get that $x$ occupies the same position in \n$\\pi[H_0]$ and $H_1$ so by pulling back we get that this is \ntrue for $H_0$ and $H_1$. \n\\qed\n\n\\bigskip\n\\begin{lem}\n If $(s,f)\\in P$, \n $A\\subseteq s$, $0\\leq j\\leq n$ then \n $A$ can be written at most $2^j-1$ ways as \n $A=H_0\\cup H_1$ with $H_0$, $H_1$ distinct, \n $f(H_0)=f(H_1)$, and $|\\gg(H_0)\\cap \\gg(H_1)|\\geq n-j$.\n\\end{lem} \n\\bigskip\n\\noindent{\\bf Proof} \nBy induction on $j$ and inside that induction, by induction on $t$, \nfor $(s,f)\\in P_t$. \nThe case $t<2$ will always be trivial. \n\nAssume first that $j=0$. \nIn this case our Lemma reduces to the following statement. \nThere are no $H_0\\neq H_1$ such that $\\gg(H_0)= \\gg(H_1)$. \nIn the inductive argument we assume as usual that \n$s=\\Delta\\cup a\\cup b$ and so $(s,f)\\in P_{t+1}$ was created from \n$(\\Delta\\cup a,f')$ and $(\\Delta\\cup b,f'')$, \n$H_0\\subseteq \\Delta\\cup a$, \n$H_1\\subseteq \\Delta\\cup b$. \nAs \n$\\gg(H_0)= \\gg(H_1)$, \n$\\gg(H_0)\\subseteq \\Delta$, but then, \nas $\\Phi(\\Delta)\\cap a=\\emptyset$, $H_0$ can have no points outside \n$\\Delta$ and similarly for $H_1$, so we can go back, say to \n$(\\Delta\\cup a,f')\\in P_t$ which concludes the argument. \n\nAssume now that the statement is proved for \n$j$ and we have $p=(s,f)\\in P_{t+1}$, \n$s=\\Delta\\cup a\\cup b$ and  $p$ was created from \n$p'=(\\Delta\\cup a,f')$ and $p''=(\\Delta\\cup b,f'')$. \nIn $A\\subseteq \\Delta\\cup a\\cup b$ we can assume that \n$y=A\\cap a \\neq\\emptyset$, $z=A\\cap b\\neq \\emptyset$ as \notherwise we can pull back to $p'$ or $p''$. \nBut then, if $A=H_0\\cup H_1$, then,if,  say, \n$H_0 \\subseteq \\Delta\\cup a$, $H_1\\subseteq \\Delta\\cup b$ hold, \nthen necessarily $H_0\\cap a=y$, $H_1\\cap b=z$, \nso $H_0=x_0\\cup y$, $H_1=x_1\\cup z$ where \n$x_0\\cup x_1=x=A\\cap \\Delta$. \nWe can create decompositions of \n$B=x\\cup\\pi[y]\\cup z$ by taking $B=\\pi[H_0]\\cup H_1$. \nBut some of these decompositions will not be different and \nit may happen that we get non-proper (i.e., one-piece) decomposition. \nThis can only happen if $\\pi[y]=z$, and then \nthe two decompositions \n$A=(x_0\\cup y)\\cup(x_1\\cup z)$ and $A=(x_1\\cup y)\\cup(x_0\\cup y)$ \nproduce the same decomposition of $B$, namely, \n$B=(x_0\\cup z)\\cup(x_1\\cup z)$ and there is but one decomposition, \n$A=(x\\cup y)\\cup(x\\cup z)$ which cannot be mapped to a decomposition \nof $B$. If this (i.e., \\(\\pi[y]=z\\)) does not happen, \nwe are done by induction. \nIf this does happen, we know that $\\gg(H_0)=\\gg(x_0\\cup y)$ has \nan element in $y$ (by the argument at the beginning of the proof). \nAs $f(x_0\\cup y)=f(x_1\\cup z)$, by Lemmas 3 and 4, both \n$H_0=x_0\\cup y$ and $H_1=x_1\\cup z$ have an element in the $\\gg$-subset, \nat the same positions  which are mapped onto each other by $\\pi$. \nWe get that $\\gg(x_0\\cup z)\\cap\\gg(x_1\\cup z)$ has at least $n-j$ \nelement, so by our inductive assumption we have at most \n$2^j-1$ decompositions, which gives at most \n$2\\cdot(2^j-1)+1=2^{j+1}-1$ decompositions of $A$. \n\n\\qed\n\n\n\n\\bigskip\nLet $G\\subseteq P$ be a generic subset. \nSet $S=\\bigcup\\{s:(s,f)\\in G\\}$,  \n$F=\\bigcup\\{f:(s,f)\\in G\\}$. \n\n\\bigskip\n\\begin{lem}\n There is a $p\\in P$ such that \n $p\\forc |S|=\\aleph_n$.\n\\end{lem}\n\\bigskip\n\\noindent{\\bf Proof} \nOtherwise ${\\bf 1}\\forc \\sup(S)<\\oo_n$. \nBy ccc, there is an ordinal $\\xi<\\oo_n$ for which  \n${\\bf 1}\\forc \\sup(S)<\\xi$, but this is impossible  as there are \nconditions in $P_1$ forcing that $\\xi\\in S$.  \n\\qed\n\n\\bigskip\nNow we can conclude the proof of the Theorem. \nIf $G$ is generic, and $p\\in G$ with the condition $p$ of Lemma 6, \nthen in $V[G]$   $F$ witnesses the theorem \nby Lemma 5 (for \\(j=n\\)) on the ground set \n\\(S\\). \nAs \\(|S|=\\oo_n\\) we can replace it by \\(\\oo_n\\). \\qed\n\n\\section{The GCH result}\n\nSet \n$$\nT_n= \\frac{1}{2}\\sum^n_{i=1}{n+i\\choose n}{n\\choose i}.\n$$\nSo \\(T_1=1\\), \\(T_2=6\\), \n\\(T_3=31\\). \nIn general, \\(T_n\\) is asymptotically\n\\(c(3+2\\sqrt{2})^n\/\\sqrt{n}\\)  for some $c$.\n\\begin{thm}\n {\\em (GCH)} If $F:[\\oo_n]^{<\\oo}\\to \\oo$ \n then some $A\\in [\\oo_n]^{<\\oo}$ has \n at least $T_n$ decompositions as \n $A=H_0\\cup H_1$, $H_0\\neq H_1$, \n $F(H_0)=F(H_1)$.\n\\end{thm}\n\\bigskip\n\\noindent{\\bf Proof} \nBy the Erd\\H{o}s-Rado theorem (see \\cite{EHMR,ER}) \nthere is a set $\\{x_\\aa:\\aa<\\oo_1\\}$ which is \n$(n-1)$-end-homogeneous, i.e., for some \n$g:[\\oo_1]^{<\\oo}\\to\\oo$, \nif \n$\\aa_1<\\cdots<\\aa_k<\\bb_1<\\cdots<\\bb_{n-1}<\\oo_1$ \nthen \n\n$$\nf(\\{ x_{\\aa_1},\\dots,x_{\\aa_{k}},x_{\\bb_1},\\dots,x_{\\bb_{n-1}}\\}) \n= g(\\aa_1,\\dots,\\aa_k).\n$$\n\nSelect \n$S_1\\in [\\oo_1]^{\\oo_1}$ \nin such a way that $g(\\aa)=c_0$ for $\\aa\\in S_1$. \nSet $\\gg_1=\\min(S_1)$. \nIn general, if $\\gg_i$, $S_i$ are given ($1\\leq i<n$) \npick \n$S_{i+1}\\in [S_i-(\\gg_i+1)]^{\\oo_1}$ \nso that \n$g(\\gg_1,\\dots,\\gg_i,\\aa)=c_i$ \nfor $\\aa\\in S_{i+1}$ and set \n$\\gg_{i+1}=\\min(S_{i+1})$. \nGiven $\\gg_1,\\dots,\\gg_n$ and \n$S_n$ let \n$\\gg_{n+1},\\dots,\\gg_{2n}$ be the $n$ \nleast elements \nof $S_n-(\\gg_n+1)$. \n\nOur set will be \n$A=\\{x_{\\gg_1},\\dots,x_{\\gg_{2n}}\\}$. \nFor $0\\leq i<n$ the color of any $(n+i)$-element \nsubset of $A$ containing $x_{\\gg_1},\\dots,x_{\\gg_i}$ will be $c_i$. \nWe can select \n$\\frac{1}{2}{2n-i\\choose n}{n\\choose i}$ \ndifferent pairs of those sets which cover $A$. \nIn toto, we get $T_n$ decompositions of $A$. \\qed\n\n \n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\chapter{Introduction}\n\nIn their classic work, Gross and Witten[\\GW], and independently Wadia[\\Wadia],\nshowed\nthat lattice QCD in two dimensions contains a third order phase transition\nin the large $N$ limit.  Basically, what happens is that the partition function\nfor the plaquettes reduces to a product of partition functions for the\nindividual plaquettes.  The\ntheory on each one is equivalent to a $d=0$ unitary matrix model,\nwith the potential given by\n$$V={N\\over g^2a^2}\\tr(U+U^\\dagger).\\eqn\\matpot$$\nThis theory is equivalent to zero-dimensional field theory of $N$ eigenvalues\nwith values that lie on a circle and a\npotential\n$$V=-(N\/2g^2a^2)\\sum_i(e^{i\\theta_i}+e^{-i\\theta_i})\n-\\sum_{i<j}\\log\\left(\\sin{\\theta_i-\\theta_j\\over2}\\right)^2.\\eqn\\Vpot$$\nThe theory becomes critical by tuning the lattice size such that the highest\nfermion of the Dyson gas lies at the top of the potential.  For fixed\n$g$, as the lattice size is decreased, the potential becomes very deep\nand the fermions move further away from the critical configuration.\nThus in the limit of vanishingly small $a$, the fermions should only\nsee a quadratic potential.\n\nVery recently there has been a revived interest in large $N$ ${\\rm QCD}_2$.  This\nprogram was started by Gross[\\Gross], who was looking for a string formulation\nof the problem.  Further work in this direction was done by one of\nthe authors[\\JM] and finally, the complete theory was worked out by Gross\nand Taylor, who showed that ${\\rm QCD}_2$\\ is essentially a string theory for\na target space with arbitrary genus.  The basic insight in handling\nthis problem is to\nequate $U(N)$ or $SU(N)$ representations with sums of maps of\nworld-sheets into the target space.  It turns out that the number\nof boxes for a Young tableau of a representation is equal to the number\nof coverings of the surface.  For surfaces with genus $g>1$, organizing\nthe partition function by the number of covers also nicely organizes\nthe sum into powers of $1\/N$.  Hence it is very easy to compute the\nleading order contribution in the large $N$ limit for these surfaces.\nFor the torus, there can be leading order contributions from any number\nof coverings, but it is easy to sum their contribution.\nHowever, in the\ncase of the sphere finding the leading order contribution in $1\/N$\nis not so easy, namely because all $SU(N)$ or $U(N)$ representations\ncontribute to the leading order behavior, and computing these contributions\ninvolves computing $1\/N$ corrections to the dimensions of these\nrepresentations.\n\nRecently, Douglas and Kazakov (DK) discovered a clever way to solve\nthe problem of the sphere by treating the rows of the young tableau\nand the number of boxes in each row as continuous variables[\\DK].  Doing\nthis they were able to compute the leading behavior which led them\nto a surprising result:  the theory contains\na third order phase transition, similar to  the case of the\nlattice version of Gross, Witten\nand Wadia, but for the continuum limit of the theory.\n\nHowever, the $U(N)$ case has an interesting feature--- the sum\nover representations is not asymptotic.  In order to correct for this,\none must sum over an infinite number of charge sectors.  While this\nwill eventually lead to overcounting for finite $N$, the answer will\nat least be asymptotic in the large $N$ limit.  These different charge\nsectors can be thought of as being extra solutions to the large $N$\nequations of motion.  These different sectors are conjugate to the $U(1)$\ninstantons.  Because of this, one should be able to find\nthe such solutions using the DK analysis.\nIn this paper we do precisely this by generalizing an ansatz described\nby DK.\n\nThere has also been some recent work by Witten in ${\\rm QCD}_2$, although from\na different perspective[\\Witten].  He has shown that the partition function on\nany Riemann surface is given by a sum over the saddle points.  An interesting\nquestion is how the DK phase transition appears from this point of view.\n\nIn section two we review recent work on 2d QCD and construct a free fermion\npicture for this theory.  In section three we present an alternative derivation\nof Witten's result that the partition function is a sum over saddle points.\nWe then show how the continuum phase transition occurs in this dual picture\nof ${\\rm QCD}_2$.  In section 4 we consider the new solutions\nfor $U(N)$ ${\\rm QCD}_2$ and generalize the analysis of DK for\nthese solutions.  We consider the two cases where the area is near its\ncritical value, and when the area is very large.  In section 5 we compare\nthese solutions to corresponding solutions for lattice $SU(N)$ ${\\rm QCD}_2$.\nIn section 6 we present our conclusions.  We include an appendix with some\nuseful equations for elliptic integrals.\n\n\n\\chapter{Review of ${\\rm QCD}_2$}\n\nLet us first review how ${\\rm QCD}_2$\\ on a cylinder is the same as a\ntheory of free fermions by showing that it can be reduced to a one-dimensional\nunitary matrix model[\\MP-\\ROI].\nIn the gauge $A_0=0$, the Hamiltonian is given as\n$$H=\\half\\int_0^L dx\\tr F_{01}^2=\\half\\int_0^L dx\\tr \\dot A_1^2\\eqn\\Hamgauge$$\nwith the overdot denoting a time derivative.\nThe $A_0$ equation of motion is now the constraint\n$$D_1F_{10}=\\partial_1\\dot A_1+ig[A_1,\\dot A_1]=0.\\eqn\\gconstraint$$\nDefine a new variable $V(x)$,\n$$V(x)=W_0^x\\dot A_1(x)W_x^L,\\eqn\\Vdef$$\nwhere\n$$W_a^b={\\rm P}e^{ig\\int_a^bdxA_1}.\\eqn\\Wdef$$\nThen \\Hamgauge\\ can be written as\n$$\\partial_1V(x)=0,\\eqn\\Veq$$\nso $V(x)$ is a constant.  Thus $V(0)=V(L)$, which implies that\n$$[W,\\dot A_1(0)]=0,\\eqn\\WAcomm$$\nwhere $W\\equiv W_0^L$ and we have used the periodicity of $A_1$ in $x$.\n\n{}From the definitions \\Vdef\\ and \\Wdef, we find the relation\n$$\\dot W=ig \\int_0^L dx W_0^x\\dot A_1(x)W_x^L=ig \\int_0^L dx V(x),\\eqn\\Wdeq$$\nand therefore using \\Veq\\ and \\WAcomm, we derive\n$$\\dot W=igLW\\dot A_1(0)=igL\\dot A_1(0)W.\\eqn\\WdeqII$$\n\\WdeqII\\ then implies that\n$$[W,\\dot W]=0.\\eqn\\WWdeq$$\n\nBecause $V(x)=V(0)$, $\\dot A_1(x)$ satisfies\n$$\\dot A_1(x)=W_0^x\\dot A_1(0)W_x^0.\\eqn\\Adeq$$\nThus, using this relation along with \\WdeqII, we can rewrite the Hamiltonian\nin \\Hamgauge\\ as\n$$H=-{1\\over 2g^2L}\\tr(W^{-1}\\dot W)^2.\\eqn\\Hammm$$\nIf the gauge group is $U(N)$, with the $U(1)$ coupling given by $g\/N$, then\n\\Hammm\\ is the Hamiltonian for the one-dimensional unitary matrix model.\nThe constraint in \\WWdeq\\ reduces the space of states to singlets[\\APA].\nHence, the problem is reducible to the eigenvalues of $W$.\n\nUpon quantization, this problem is equivalent to a system of $N$\nnonrelativistic fermions living on a circle, with the Hamiltonian given by\n$$H=-\\left({g^2L\\over2}\\right)\\sum_{i=1}^N{\\partial^2\\over\\partial\\theta_i^2},\n\\qquad\\qquad 0\\le\\theta_i<2\\pi.\\eqn\\Hamferm$$\nThe fermionization is due to the Jacobian of the change of variables from\n$W$ to its eigenvalues, introducing the Vandermonde-type\ndeterminant in the wavefunction of the states, which in the unitary\nmatrix case reads\n$$ {\\widetilde \\Delta} = \\prod_{i<j} \\sin{\\theta_i - \\theta_j \\over 2}\n= {\\Delta (e^{i\\theta_i}) \\over \\prod_i e^{i(N-1) \\theta_i}},\\eqn\\VM$$\nwhere $\\Delta(\\lambda_i ) = \\prod_{i<j} (\\lambda_i - \\lambda_j )$ is the\nstandard Vandermonde determinant.\nNotice that each factor in \\VM\\ is antiperiodic on the circle.\nThus, if $N$ is even the fermions have antiperiodic boundary conditions.\nLikewise, if $N$ is odd they have periodic boundary conditions.\nThis can be understood in terms of transporting a fermion once around the\ncircle, passing by $N-1$ other fermions along the way\nand therefore picking up $N-1$ minus signs.\nHence, in either case, the ground state is built by filling all states\nwith wave numbers between $-N\/2+1\/2$ and $N\/2-1\/2$, inclusive.\nSubtracting off the ground state energy, one easily sees that this spectrum\nreproduces that found for the different representations of $U(N)$.\n\nIf the gauge group is $SU(N)$, because $A_1$ is now traceless\n$W$ will also obey the condition $\\det W = 1$. Therefore the center of\nmass coordinate for the fermions is absent and we must mod it out\nof the theory. This means that we need to identify states in which all\nfermions have their momentum shifted by the same amount.\nMoreover, we must subtract the energy of the center of mass from the\nenergy of each state in the theory.\n\nNow consider the partition function for the sphere.\nIn terms of the fermions, we want a matrix element that\ncorresponds to the sphere topology.  The obvious thing to do is\nto map the end points of a cylinder to two points and therefore $W=1$ there.\nIn terms of the\nfermions, this corresponds to computing the inner product of $N$ fermions\nwhose position is at the point $x=0$ on the circle at time $t=0$ with\nthe fermions at the same point at time $t=T$.  This inner product\n$Z$ is given by\n$$Z=\\langle x_i ;t=T|y_i ;t=0\\rangle,\\eqn\\matrxelm$$\nin the limit that all $x_i\\to 0$ and $y_i\\to 0$.\n\nTo calculate $Z$, one can insert a complete set of momentum states for\neach particle.  The wave function at $t=0$ and $t=T$ must be antisymmetric\nunder the exchange $x_i\\to x_j$ or $y_i\\to y_j$.  Therefore, the matrix element\nin \\matrxelm\\ has the factor\n$$\\det|e^{ip_ix_j}|\\det|e^{-ip_iy_j}|.\\eqn\\wffact$$\nAs $x_i\\to 0$ and $y_i\\to 0$, the factor in \\wffact\\ approaches\n$$\\prod_{i<j}(x_i-x_j)(y_i-y_j)(p_i-p_j)^2,\n\\eqn\\wffactII$$\nThe matrix element in \\matrxelm\\ is therefore given by\n$$Z=C \\sum_{p_i}\n\\prod_{i>j}(p_i-p_j)^2(x_i-x_j)(y_i-y_j)\n\\exp(-\\half g^2LT\\sum_i p_i^2),\\eqn\\Zreduce$$\nwhere $C$ is an unimportant constant.\nNot surprisingly, this term approaches zero in this limit.\nBut in order to find the sphere contribution, one should notice that\nthe fermion wavefunction at the end points are {\\it more}\nsingular than a $\\delta$-function, namely\n$$\\psi (x_i ) = \\widetilde\\Delta (x_i ) \\delta (W-1) = {1 \\over \\Delta (x_i )}\n\\delta (x_i ) \\eqn\\Sing$$\nwhere a factor of $1\/\\widetilde\\Delta^2 (x_i )$ was produced by the change of\nvariables from $W$ to $x_i$.\nTherefore, it is necessary to divide the expression in \\Zreduce\\ by these\nextra Vandermonde determinants, that is,\n$$\\prod(x_i-x_j)(y_i-y_j),$$\nleaving a finite expression.  Hence the sphere partition function is\n$$Z_{\\rm sphere}=C\\sum_{p_i}\n\\prod_{i>j}(p_i-p_j)^2 \\exp(-\\half g^2LT\\sum_i p_i^2).\\eqn\\Zsphere$$\n\nWe can compare this to the sphere partition function of Migdal and\nRusakov[\\Migdal,\\Rus], which is given by\n$$Z_{MR}=\\sum_R (d_R)^2 \\exp(-g^2AC_{2R})\\eqn\\MRsph$$\nwhere the sum is over all representations of $U(N)$ or $SU(N)$ and\n$d_R$ is the dimension of the representation and $C_{2R}$ is the\nquadratic casimir.\nThe correspondence of the fermion states with the $U(N)$ represenatations\nis as follows[\\Mike]:\nIf we describe a representation by a Young tableau, then the number\nof boxes in row $i$, $n_i$ is the momentum shift\nof the fermion with the $i^{\\rm th}$ highest momentum\nabove its ground state value.\nIn terms of boxes, the casimir is given by\n$$C_{2R}=\\half\\left(N\\sum_i n_i+\\sum_i n_i(n_i-2i+1)\\right)\\eqn\\casimir$$\nwhich one can easily checked is reproduced by the fermions after\nsubtracting off the ground state energy.  The dimension of the\nrepresentation is given by\n$$\\eqalign{d_R&=\\prod_{i>j}\\left(1-{n_i-n_j\\over i-j}\\right)\\cr\n&=\\prod_{i>j}(i-j)^{-1}\\prod_{i>j}\\Bigl((n_j-j)-(n_i-i)\\Bigr).}\\eqn\\dimrep$$\nThe first product in the second line of \\dimrep\\ is a representation\nindependent term and is thus an unimportant\nconstant.  The second term is just $p_j-p_i$ for the fermions.\nThe total momentum is the $U(1)$ charge for a representation.\nHence we find full agreement with the result of Migdal and Rusakov.\n\n\\chapter{Classical Solutions}\n\nAfter a cursory inspection of \\Zsphere\\ it would appear that $Z_{\\rm sphere}$\nis simply the partition function for a $d=0$ matrix model in a\nquadratic potential. Unlike the lattice case, the potential never turns over,\nso one might not expect a phase transition.  As was shown by Douglas and\nKazakov, this is not correct.  The point is that the variables $p_i$ that\nappear in \\Zsphere\\ are discrete, hence the density of eigenvalues will\nbe bounded.  When this bound is reached, a phase transition occurs. Thus,\nin the strong coupling phase we simply have condensation of the fermions\nin their momentum lattice, which gives a very simple physical understanding\nof the phase transition mechanism. The critical value of the area is reached\nwhen the density of $p_i$, as given by the Wigner semicircle law which is\nvalid in the continuoum, reaches somewhere the lattice bound, namely one,\nthus reproducing the DK result.\n\nThere should be a classical field configuration, termed master field, which\ndominates the path-integral in each phase. In fact, as was shown by\nWitten[\\Witten]\nusing the localization theorem of Duistermaat and Heckman (DH),\nthe full ${\\rm QCD}_2$\\ path integral can be\nwritten as a suitable sum over classical saddle points. In what follows we\nwill give a very simple demonstration of Witten's result using the fermion\npicture and identify the classical configuration which dominates, that is,\nthe master field.\n\nThe key observation is that the exact propagator of a free particle is\nproportional to the exponential of the action corresponding to the classical\n(straight) path connecting the initial and final points. Since ${\\rm QCD}_2$\\ on the\ntorus is equivalent to $N$ free fermions, its partition function will also\nbe given by an appropriate classical path of the free fermions. The things to\nbe taken into account, however, are\n\ni) The particles live on a circle; therefore there are several possible\nclassical paths for each of them, differing by their winding around the\ncircle with fixed initial and final positions.\n\nii) The particles are fermions; thus one should also consider paths where\nthe final positions of the particles have been permuted, weighted by a\nfermionic factor $(-)^C$, where $C$ is the number of times the paths of\nthe particles cross.\n\nThe total partition function will then be the (weighted) sum of the actions\nof all these classical configurations. Since to each\npath corresponds a (diagonal) matrix $W(t)$, and to that (up to gauge\ntransformations) a classical field configuration satisfying the field\nequations of motion, this is the sum over saddle points of the action\nof Witten.\n\nFor the sphere the same picture holds, with the difference that all paths\nstart and end at the point $x=0$, and that each path is further weighted\nby an extra factor, due to the division by the Vandermonde determinants as\nexplained in the previous section. This is, again, the sum over saddle\npoints of Witten, and the extra weighting factors are the determinants\nwhich appear in the DH theorem. These paths are characterized by their\nwinding numbers $\\{ n_i \\}$ (up to permutation) and thus this is a sum of\nthe form\n$$ Z_{\\rm sphere} = \\sum_{n_i} w( n_i ) \\exp \\Big(-{2\\pi^2 \\over g^2 LT}\n\\sum_i n_i^2 \\Big)\\eqn\\Zcl$$\nwhere $w( n_i )$ are the (as yet undetermined) weighting factors.\n\nThe easiest way to obtain the full expression in \\Zcl\\ is to Poisson resum\n\\Zsphere. Using the formula\n$$ \\sum_n f(n) = \\sum_n {\\tilde f} (2\\pi n)\\eqn\\resum$$\nwhere $f(x)$ is any function and $\\tilde f$ is its Fourier\ntransform, we obtain\n$$Z_{\\rm sphere} = C \\sum_{n_i} F_2 (2\\pi n_i ),\\eqn\\Zres$$\nwhere\n$$ F_2 ( x_i ) = \\int \\prod_i dp_i e^{-i \\sum_i x_i p_i } \\Delta^2 ( p_i )\n\\exp(-\\half g^2LT\\sum_i p_i^2).\\eqn\\FF$$\nTo find the Fourier transform appearing in \\FF, we first note that\n$$ F_1 \\equiv \\int \\prod_i dp_i e^{-i \\sum_i x_i p_i } \\Delta ( p_i )\n\\exp(-\\half \\alpha \\sum_i p_i^2) = C \\Delta ( x_i )\n\\exp(-{1 \\over 2\\alpha} \\sum_i x_i^2) .\\eqn\\F$$\nTo prove this, notice that\n$$ F_1 = \\Delta ( -\\partial_{p_i}) \\int \\prod_i dp_i e^{-i \\sum_i x_i p_i }\n\\exp(-\\half \\alpha \\sum_i p_i^2) = P ( x_i )\n\\exp(-{1 \\over 2\\alpha} \\sum_i x_i^2).\\eqn\\FP$$\n$P( x_i )$ is a polynomial of degree $N(N-1)\/2$; moreover it is completely\nantisymmetric in $x_i$. Therefore, up to a normalization, it is the\nVandermonde. The constant $C$ in \\F\\ can be found explicitly, it is however\nirrelevant for this discussion since it will amount to an overall coefficient\nin the final result. Using the convolution property of the Fourier transform\nof a product, in combination with \\F, we find\n$$\\eqalign{ F_2 ( x_i ) &= (F_1 \\otimes F_1 ) ( x_i )\\cr\n& = C \\int \\prod_i dy_i\n\\Delta ({x_i - y_i \\over 2}) \\Delta ({x_i + y_i \\over 2})\n\\exp\\Big(-{1 \\over 4g^2LT}\\sum_i [(x_i+y_i)^2 + (x_i-y_i)^2 ]\\Big)\\cr\n&= C \\exp({1 \\over 2g^2LT} \\sum_i x_i^2 ) \\int \\prod_i dy_i \\prod_{i<j}\n(y_{ij}^2 - x_{ij}^2 ) \\exp \\Big(-{1 \\over 2g^2LT}\\sum_i y_i^2 \\Big)}\n\\eqn\\FFF$$\nwhere $x_{ij}=x_i - x_j$ and $y_{ij}=y_i-y_j$. Substituting \\FFF\\ in\n\\Zres\\ we recover expression \\Zcl, with the weights $w( n_i )$ given by\n$$ w( n_i ) = C \\int \\prod_i dy_i \\prod_{i<j} (y_{ij}^2 - n_{ij}^2 ) \\exp\n\\Big(-{1 \\over 2g^2LT}\\sum_i y_i^2 \\Big) .\\eqn\\weights$$\nThe above determines the expression of the sphere partition function in\nterms of classical saddle point configurations.\n\nIn the large $N$ limit one particular classical configuration in \\Zcl\\\nshould dominate. To determine this, let us ignore for the moment the fact\nthat the $n_i$ are discrete and replace the sum in \\Zcl\\ with an\nintegral. Then we can bring the expression for $Z_{\\rm sphere}$ into the form\n$$ Z_{\\rm sphere} = C \\int \\prod_i ds_i dt_i \\Delta (s_i) \\Delta (t_i)\n\\exp \\Big(-\\half N \\sum_i ( s_i^2 + t_i^2 )\\Big),\\eqn\\Zlarge$$\nwhere we rescaled $g^2$ to $g^2 \/N$, in order to have a nontrivial large $N$\nlimit, and changed variables to $s=(y+2\\pi n)\/\\sqrt{2g^2 LT}$,\n$t=(y-2\\pi n)\/\\sqrt{2 g^2 LT}$.\nWe see that in \\Zlarge\\ all explicit dependence on the coupling constant and\narea have disappeared. The area enters this picture indirectly, through the\ndiscreteness of $n_i$. (In fact, if it were not for this discreteness the\nabove integral would vanish.) Due to this, the plane $(s,t)$ is discrete in\nthe $s-t$ direction and consists of parallel diagonal lines with distance\n$D=2\\pi \/\\sqrt{g^2 LT}$. If this spacing is such that the distribution of\n$s$ and $t$ contributing to the above integral in the large $N$ limit is\nentirely within the lines $n=1$ and $n=-1$, then only the sector $n=0$\ncontributes. This signals a phase transition. Unlike the quantum\ncase, there is no exclusion principle for the windings $n_i$ and thus no\nmaximum density to be saturated. The phase transition in the classical\nexpansion is more like a Bose condensation to the ground state $n =0$.\n\nTo estimate the critical area, substitute the Vandermondes in \\Zlarge\\\nwith their absolute value, which corresponds to putting the product of\ndifferences appearing in \\weights\\ into absolute values. This has no effect\nfor the $n_i=0$ term, while it overestimates the contribution of the other\nsectors. \\Zlarge\\ then becomes the product of two independent\nintegrals involving a gaussian factor and one power of the Vandermonde.\nThe large $N$ saddle point of these integrals is found in a standard way\nand the distribution of $s_i$ and $t_i$ is, in fact, again a Wigner\nsemicircle with radius $R=\\sqrt2$. This on the $(s,t)$ plane creates a\n``square\" distribution with side $2R$ in each direction. When this square\nlies entirely within the lines $n=\\pm 1$, the sectors $n\\neq 0$ do not\ncontribute and we are in the Boson condensate phase. For an estimate of the\ncritical area, assume that the transition occurs when the corners of the\nsquare start touching the lines $n=\\pm 1$, that is, when the half-diagonal\nbecomes $D$. Putting $2R\\sqrt2 \/2 = D$ we obtain $g^2 A_{\\rm crit}=\\pi^2$.\nSomewhat surprisingly, our estimation has given the precise result of\nDouglas and Kazakov. It is not entirely clear from this argument why\nthe transition should occur precisely at this point.\n\nIn any event, we see that the classical configuration in the weak-coupling\n(small area)\nphase is $n_i=0$. This corresponds to the master field $A=0$, up to\ngauge transformations. As a check, notice that in this phase the discreteness\nof $p_i$ in the quantum expression is irrelevant (no saturation) and thus\nwe can substitute the sums in \\Zsphere\\ by integrals. This, upon resummation,\nbecomes the $n_i=0$ term in \\Zcl, as can be seen by the expression for $w(0)$\nobtained from \\weights. Only in the strong coupling phase will we have a\nnontrivial master field. This is implicitly determined by \\Zcl\\ and \\weights,\nalthough its explicit expression is not known.\n\n\n\\chapter{$U(1)$ Sectors}\n\nThere is one difficulty with the string picture as it now stands for\nthe $U(N)$ gauge group, namely, the sum is not asymptotic with the exact\nanswer in the limit $1\/N\\to0$.\nThis is because there exist states with finite energy, but which will never\nappear in the perturbative sum in \\MRsph.\nThese are the states that correspond to $N$ fermions with momenta\nshifted by a constant finite amount.  Such states have finite energy in the\nlarge $N$ limit, but do not show up in a perturbative\nsum over surfaces since the corresponding Young tableau has at least\n$N$ boxes.  Moreover, these states are local minima, in the sense that in order\nto find a state with lower energy, it is necessary to shift the total momentum\nby a large amount.\n\nWe can rectify this problem by including other sectors in the sum.  For finite\n$N$, this will eventually lead to overcounting, but the answer will be\nasymptotic.  To this end, let us define $Z_m$\n$$Z_m=\\sum_{\\rm Reps}d_R^2\\exp\\left(-{Ag^2\\over2}\\sum_i(n_i+m)\\right)\n\\exp\\left(-{Ag^2\\over2N}\\sum_i(n_i-2i+1+m)(n_i+m)\\right),\\eqn\\Zmeq$$\nwhere we have basically added $m$ boxes to each row.  After some manipulation,\n$Z_m$ reduces to\n$$Z_m=\\sum_{\\rm Reps}d_R^2e^{-{Ag^2\\over2}(1+2m\/N)n_R}\ne^{-{Ag^2\\over2N}\\tilde n_R}e^{-Ag^2m^2\\over2},\n\\eqn\\ZmeqII$$\nHence we see that $Z_m$ has the same form as $Z_0$, except that\nthe area term that appears in front of $n$ has been shifted and there is\nan extra factor of $m^2$ in the energy.\nBut the $\\tilde n$ term is the same, meaning that\nthe Gross-Taylor rules are exactly the same as in the $Z_0$ case,\nexcept that the area that will be used for the Nambu-Goto term has\nbeen renormalized, with a different shift for the chiral and antichiral\nsectors.\nThe sum\n$${\\cal Z}=\\sum_mZ_m\\eqn\\Zstring$$\nis now an asymptotic sum for the ${\\rm QCD}_2$\\ string.\n\nWe can interpret the sum in \\Zstring\\ as a sum over different\nclassical string solutions, since around each sector $m$ there is a\nsum over $1\/N$, the string coupling.   These different sectors are\nbasically the conjugates of the $U(1)$ instanton sectors.\nWe can Poisson resum \\Zstring, giving the expression\n$${\\cal Z}=\\sqrt{\\pi\\over Ag^2}\\sum_m\\sum_{\\rm Reps}d_R^2\ne^{-({Ag^2\\over2}+{2\\pi im\\over N})n_r} e^{-{Ag^2\\over2N}\\tilde n_R}\ne^{Ag^2n_R^2\\over2N^2}\ne^{-{m^2\\over2g^2A}}.\\eqn\\Zprsm$$\nThe $m^2$ term now has a factor of $1\/g^2$ in front of it, which is the\ncontribution one would expect from $U(1)$ instantons.  Furthermore, the\narea that now appears in the Nambu-Goto action is now complex\\foot{\nThis might be useful for formulating ${\\rm QCD}_2$\\ as a topological field\ntheory.  We thank S. Cordes and C. Vafa for informing us of their work\nin this area.}, and there\nis also an additional $n^2$ term.  This last term has been discussed before\nin the $SU(N)$ case and was attributed to contributions from tubes and handles\non the world-sheet.\n\n\n\nSince there are other string solutions corresponding to different values\nof $m$, we might expect to be able to generalize the work of Douglas\nand Kazakov to find the leading order contributions around these solutions.\nTo this end, consider the solutions to the equations of motion for\nthe path integral in \\Zsphere,\n$${Ap_i\\over2N}=\\sum_{j\\ne i}{1\\over p_i-p_j},\\eqn\\eqmdis$$\nwhere we have absorbed the QCD coupling into the area and rescaled it\nby a factor of $N$.  In the large $N$ limit we can define the new\nvariables\n$x=i\/N$, and $h(x)=p_i\/N$, giving the new equation,\n$$Ah\/2={\\rm P}\\int d\\lambda {u(\\lambda)\\over h-\\lambda},\\eqn\\eqmcont$$\nwhere $u(\\lambda)$ is the density of eigenvalues, $u(\\lambda)=dx\/d\\lambda$.\nBut unlike the eigenvalues of a matrix model in a quadratic potential,\nthe $p_i$ are discrete, therefore the density $u(\\lambda)$ is bounded,\nsatisfying $u(\\lambda)\\le1$.\n\nDK treated this problem\nby dividing up the possible real values of $\\lambda$ into three regions,\nwhere $u(\\lambda)=0$, $0<u(\\lambda)<1$ and $u(\\lambda)=1$.  They choose\nthe ansatz that the first region\noccurs for $|\\lambda|>a$, the second for $b<|\\lambda|<a$, and the third\nfor $|\\lambda|<b$.  If we define $\\widetilde u(\\lambda)$ to be the density of\neigenvalues minus the contribution from the region where $u(\\lambda)=1$,\nthen \\eqmcont\\ can be rewritten as\n$$Ah\/2+\\log{h-b\\over h+b}={\\rm P}\\int d\\lambda{\\widetilde u(\\lambda)\\over h-\\lambda},\n\\eqn\\eqmtcont$$\nThe problem has been reduced to a two cut eigenvalue problem.  To solve\nthis, define the function\n$$f(h)=\\int d\\lambda{\\widetilde u(\\lambda)\\over h-\\lambda}.\\eqn\\feq$$\n$f(h)$ then must satisfy\n$$f(h)={1\\over 2\\pi i}\\sqrt{(h^2-a^2)(h^2-b^2)}\n\\oint ds {As\/2-\\log{b-s\\over b+s}\\over (h-s)\\sqrt{(a^2-s^2)(b^2-s^2)}},\n\\eqn\\fcontour$$\nwhere the contour surrounds the cuts from the square roots, but not\nthe cut from the log nor the singularity at $h$.\nPulling the contour back, \\fcontour\\ leads to the equation\n$$f(h)=h{A\\over2}+\\log{h-b\\over h+b}-\\sqrt{(a^2-h^2)(b^2-h^2)}\n\\int_{-b}^b ds {1\\over(h-s)\\sqrt{(a^2-s^2)(b^2-s^2)}}.\\eqn\\feqcon$$\n\nExpanding about large $h$, one can then find relations between $a$,\n$b$ and $A$.  Before doing this, let us go back and reexamine\nthe DK ansatz.  They of course make the reasonable\nassumption that the solution to the equations of motion is symmetric\nabout $h=0$.  This is quite sensible for the ground state, but should\nnot be true for solutions that correspond to sectors with nonzero\nvalues of $m$.  Therefore, let us relax their ansatz somewhat and\nassume that $0<u(\\lambda)<1$ for the regions $b<\\lambda<a$ and $d<\\lambda<c$,\nwhere $a$, $b$, $c$ and $d$ are to be determined.  Then we can then\nproceed as before, reaching the equation\n$$f(h)=h{A\\over2}+\\log{h-b\\over h-c}-\n\\int_{c}^b ds {\\sqrt{(a-h)(b-h)(c-h)(d-h)}\n\\over(h-s)\\sqrt{(a-s)(b-s)(s-c)(s-d)}}.\\eqn\\feqnew$$\n{}From \\feq\\ and (A.6) one finds that the density of eigenvalues is given by\n$$\\eqalign{u(\\lambda)&={\\sqrt{(a-\\lambda)(\\lambda-b)(\\lambda-c)(\\lambda-d)}\\over\\pi}\\int_c^b\nds{1\\over(\\lambda-s)\\sqrt{(a-s)(b-s)(s-c)(s-d)}}\\cr\n&={2\\over\\pi(\\lambda-c)(\\lambda-d)}\n{\\sqrt{(a-\\lambda)(\\lambda-b)(\\lambda-c)(\\lambda-d)}\\over\\sqrt{(a-c)(b-d)}}\\cr\n&\\qquad\\qquad\\qquad\\times\n\\left((c-d)\\Pi({b-c\\over b-d}{\\lambda-d\\over \\lambda-c},q)+(\\lambda-c)K(q)\\right),}\n\\eqn\\density$$\nwhere $K$ and $\\Pi$ are the complete elliptic integrals of the first\nand third kind.\n\nLet us now expand the righthand side \\feqnew\\ in powers of $1\/h$ and compare\nit to the expansion in \\feq.  Matching the terms of order $h$ gives\nthe equation\n$${A\\over2}-\\int_c^b ds {1\\over\\sqrt{(a-s)(b-s)(s-c)(s-d)}}=0.\\eqn\\hexpI$$\nUsing equation (A.1) in the appendix gives\n$$A={4K(q)\\over\\rho},\\eqn\\hIre$$\nwhere\n$$\\rho=\\sqrt{(a-c)(b-d)},\\qquad\\qquad q=\\sqrt{(a-d)(b-c)\\over(a-c)(b-d)}.$$\nNote that for the case $c=-b$, $a=-d$, the modulus $q$ that appears\nin \\hIre\\ is different than the corresponding equation in [\\DK].\n\nMatching the $h^0$ terms in the expansion of \\feq\\ and \\feqnew\\ leads\nto the equation\n$$\\half\\int_c^b ds {a+b+c+d\\over\\sqrt{(a-s)(b-s)(s-c)(s-d)}}=\n\\int_c^b ds {s\\over\\sqrt{(a-s)(b-s)(s-c)(s-d)}}.\\eqn\\hexpII$$\nUsing (A.1) and (A.2), we can rewrite this equation as\n$${c-d\\over\\rho}\\Pi(\\alpha,q)={a+b+c-d\\over2\\rho}K(q),\\eqn\\hIIre$$\nwhere $\\alpha=\\sqrt{b-c\\over b-d}$.  In [\\DK], the $h^0$ equation is\ntrivially satisfied due to symmetry.\n\nMatching the $h^{-1}$ term in the two equations gives\n$$\\eqalign{\\int ds \\widetilde u(s)+b-c&=-\\int_c^b\nds{s^2\\over\\sqrt{(a-s)(b-s)(s-c)(s-d)}}\n\\cr\n&+\\half\\int_c^b ds {s(a+b+c+d)\\over\\sqrt{(a-s)(b-s)(s-c)(s-d)}}\\cr\n&+{1\\over8} \\int_c^b ds {a^2+b^2+c^2+d^2-2ab-2ac-2ad-2bc-2bd-2cd\n\\over\\sqrt{(a-s)(b-s)(s-c)(s-d)}}.}\\eqn\\hexpIII$$\nThe integral on the lefthand side of \\hexpIII\\ is the total number\nof eigenvalues minus those in the region between $c$ and $b$, divided\nby $N$.  But the number between $c$ and $b$ is just $N(b-c)$, thus the\nlefthand side is unity.\nUsing (A.1), (A.2) and (A.3), and then using the equation\nin \\hIIre, we can reduce \\hexpIII\\ to\n$$1={(a-b-c+d)^2\\over4\\rho}K(q)+\\rho E(q),\\eqn\\hIIIre$$\nwhere $E(q)$ is the complete elliptic integral of the second kind.\n\nThe next term in the expansion in \\feq\\ is\n$$h^{-2}\\int ds \\widetilde u(s)s.$$\nBut this is just the sum of the momenta for this particular solution\ncoming from the regions where the density is less than one,\ndivided by $N^2$.  This is its contribution to the $U(1)$ charge divided by\n$N^2$.   Let us call this rescaled charge $\\widetilde Q$.\nUsing equations (A.1)-(A.4), \\hIIre\\ and \\hIIIre, the $h^{-2}$\nexpansion leads to the equation\n$$\\widetilde Q+\\half(b^2-c^2)=Q={W\\over4}+{K(q)\\over16}XYZ,\\eqn\\hIVre$$\nwhere\n$$\\eqalign{W&=a+b+c+d\\qquad\\qquad\\qquad X=a-b-c+d\\cr\n           Y&=a+b-c-d\\qquad\\qquad\\qquad Z=a-b+c-d.}\\eqn\\neweqs$$\n$(b^2-c^2)\/2$ is the contribution to the rescaled charge from the\neigenvalues that sit between $c$ and $b$, hence the lefthand side of\n\\hIVre\\ is the total rescaled charge.  The charge for the lowest\nenergy state in sector $m$ is $mN$, hence $Q=m\/N$.\n\nFinally, we can find the specific heat, which is the next term in the expansion\nof $f(h)$.\nUsing (A.1)-(A.5), \\hIIre\\ and \\hIIIre, we find,\nafter a fair amount of algebraic manipulation\n$$\\eqalign{F'(A,Q)-1\/24&=\\int ds u(s)s^2\\cr\n&={\\rho E(q)\\over48}(3W^2+X^2+Y^2+Z^2)\\cr\n&\\qquad+{K(q)\\over12\\rho}\n(3W^2X^2+6WXYZ+Y^2Z^2+2X^2Y^2+2X^2Z^2+X^4)\\cr\n&={K(q)\\over12\\rho}\n(6WXYZ+Y^2Z^2+X^2Y^2+X^2Z^2)\\cr\n&\\qquad+{1\\over48}(3W^2+X^2+Y^2+Z^2).}\\eqn\\spheat$$\n\nGiven the area $A$ and the charge $Q$,\nin principle, one should be able to\nsolve for $a$, $b$, $c$ and $d$ using the four equations \\hIre, \\hIIre,\n\\hIIIre\\ and \\hIVre.  However, we should not expect solutions for all\npossible values of $A$ and $Q$.  For instance, from DK, we know that\nthere are no solutions for $Q=0$ and $A<\\pi^2$.\nThis of course is the weak coupling regime.  But for nonzero values\nof $Q$, the minimum value of $A$ should be higher.  In fact, in the\ninfinite area limit, the upper value of $Q$ is bounded.  Naively, this\nshould happen for the sector when the fermion with the smallest or largest\nmomentum is zero. We will see that the naive answer is correct.\n\nTo further analyze the problem, we will consider two regions.  The first\ncorresponds to the area near its critical value for small values of $Q$.\nThe second region occurs for very large values of the area.\nLet us first consider what happens at the critical point.  In this case\n$b=c$, thus $\\alpha=q=0$ and $\\Pi=K=E=\\pi\/2$. From equation \\hIIre\\ we\nfind that $a=-d$ at the critical point.\nTo determine $b$, let us go back to equation \\density.  Since $b=c$, we\nfind that $u(\\lambda)$ is given by\n$$\\eqalign{u(\\lambda)&={\\sqrt{a-\\lambda\\over a+\\lambda}}{1\\over\\sqrt{a^2-b^2}}\n[(\\lambda-b)+(b+a)]\\cr\n&=\\sqrt{a^2-\\lambda^2\\over a^2-b^2}.}\\eqn\\dencrit$$\nIf $b\\ne0$, then for $\\lambda^2<b^2$, $u(\\lambda)>1$.  But this violates the\nansatz that the density is less than or equal to unity.  Therefore,\nwe must choose $b=0$.  Using \\hIIIre\\ and \\hIVre\\ we then see that\n$a={2\\over\\pi}$ and that the charge for this solution is zero.\n\nNow consider small, but nonzero values for $Q$.  In this case,\n$b-c$ must be nonzero.  To this end, let\n$$\\epsilon=b-c,\\qquad\\qquad\\delta=b+c\\eqn\\bpc$$\nUsing \\hIIIre\\ and the asymptotic expansions in the appendix,\nwe find the leading correction to $a$ from its critical value is\n$$\\Delta a=-{\\pi\\over32}(\\epsilon^2+2\\delta^2).\\eqn\\acorr$$\nSince this correction is of order $\\epsilon^2$ and not $\\epsilon$,\nit will not contribute to $a+d$ in leading order.\nThis leading order correction can be found from\n\\hIIre\\ and the asymptotic expansions in the appendix.  A little algebra\nshows that\n$$a+d={\\pi^2\\over32}\\epsilon^2\\delta.\\eqn\\apdcorr$$\nWe can now use \\hIVre, \\bpc\\ and \\apdcorr\\ to find $Q$.\nLet us rewrite \\hIVre\\ as\n$$\\eqalign{Q&=(a+d)\\left({1\\over4}+{K\\over16\\rho}(a-d+b-c)(a-d-b+c)\\right)\\cr\n&\\qquad\\qquad\n+(b+c)\\left({1\\over4}-{K\\over16\\rho}(a-d+b-c)(a-d-b+c)\\right).}\\eqn\\chargere$$\nThe factor multiplying $a+d$ is to leading order $1\/2$.  The term multiplying\n$b+c$ is actually much smaller.   In fact,  this term is third order in $\\epsilon$\nand is given by $-\\pi^3\\epsilon^3\/1024$.  Hence, the leading contribution to\nthe charge actually comes from the $a+d$ term and is\n$$Q={\\pi^2\\over64}\\epsilon^2\\delta.\\eqn\\chcorr$$\n\nThe charge that appears in \\chcorr\\ is limited by the maximum value of\n$\\delta$ given $\\epsilon$.  This bound is determined by enforcing the ansatz\nthat $u(\\lambda)\\le1$.  The region where this ansatz might be violated\nis where $\\lambda\\approx b$ or $\\lambda\\approx c$.  Let us\nconsider the case where $\\lambda$ is near $b$.  Examining equation \\density,\nwe see that the first modulus in the elliptic integral of the third kind\napproaches unity as $\\lambda$ approaches $b$ from above.\nTherefore, if we substitute the asymptotic expansion for $\\Pi$ in (A.11)\nin \\density, we find that the density is given by\n$$\\eqalign{u(\\lambda)&= {2\\over\\pi}{1\\over b-d}\\sqrt{(a-b)(\\lambda-b)\\over(a-c)(b-c)}\n\\Biggl\\{(b-c)K(q)+(c-d)K-E(q){(a-c)(b-d)\\over a-b}\\cr\n&\\qquad\\qquad\\qquad+(c-d){\\pi\\over2}\\sqrt{(a-c)(b-d)\\over(a-b)(c-d)}\n\\sqrt{(b-d)(b-c)\\over(\\lambda-b)(c-d)}\n\\Biggr\\}+{\\rm O}(\\lambda-b)\\cr\n&=1+\\sqrt{\\lambda-b}{2\\over\\pi}\\sqrt{a-b\\over(a-c)(b-c)}\\left(K(q)-E(q){a-c\\over\na-b}\n\\right)+{\\rm O}(\\lambda-b).}\\eqn\\densityapp$$\nHence, in order to ensure that the ansatz is satisfied, it is necessary\nthat the relation\n$${a-c\\over a-b}E(q)-K(q)>0,\\eqn\\EKrel$$\nbe upheld.  Using the asymptotic expansions and the values for $b$, $c$, $a$\nand $d$ given by \\bpc\\ and \\acorr, we find that\n$${a-c\\over a-b}E(q)-K(q)\\approx{\\pi^3\\over32}(2\\delta\\epsilon+\\epsilon^2).\\eqn\\epsde$$\nTherefore, in order for \\EKrel\\ to be satisfied, we must have $\\delta>-\\epsilon\/2$.\nWe can also derive the constraint that $\\delta<\\epsilon\/2$ by examining $u(\\lambda)$\nas $\\lambda\\to c$.  Therefore, we must satisfy\n$$|\\delta|<\\epsilon\/2.\\eqn\\delepscon$$\nWe can rewrite this constraint in terms of the charge and the area. From\n\\hIre\\ and the asymptotic expansion, we have the relation\n$$A-A_c={3\\pi^4\\over64}(\\epsilon^2+2\\delta^2),\\eqn\\Aeps$$\nwhere $A_c$ is the critical value for the area.\nHence, using \\chcorr, \\delepscon\\ and \\Aeps, we have that\nthe maximum allowed charge sector for a given area near its critical value is\n$$Q_{max}={32\\sqrt{2}\\over27\\pi^4}(A-A_c)^{3\/2}.\\eqn\\Qmax$$\nThis relation is a little reminiscent of the maximum charge of a\nReissner-Nordstrom black hole.\n\nNow we wish to examine the behavior deep in the strong coupling regime,\nwhich corresponds to large values of the area.\nIn this case, we expect $b$ to approach $a$ and $c$ to approach $d$.  The\nvalues of $a$ and $d$ are determined by the charge of the sector that we\nare considering.  For this behavior $q\\to1$, and thus $q'\\to0$.  At this\npoint it is convenient to rewrite $\\Pi(\\alpha,q)$ in terms of elliptic\nintegrals of the first and second kind.  Using the relation in the\nappendix, equation \\hIIre\\ can be written as\n$$-E(q)F(\\theta,q)+K(q)E(\\theta,q)={a+b-c+d\\over2\\rho}K(q),\\eqn\\hIIinc$$\nwhere $\\sin\\theta={a-c\\over a-d}$\nUsing \\hIIIre\\ and the asymptotic expansions in (A.12) and (A.13),\nwe find that in this region\n$$a-d\\approx1\\eqn\\adrel$$\nwhich is expected since the the integral of $u(\\lambda)$ should be unity\nand for almost all values of $\\lambda$ between $a$ and $d$, $u(\\lambda)=1$.\nPlugging in leading order asymptotic expansions in (A.12)-(A.15)\ninto \\hIIinc\\ and invoking \\adrel, we then find the following approximate\nequation\n$$-\\log{1+\\sin\\theta\\over\\cos\\theta}+\\log\\sqrt{(a-c)(b-d)\\over(a-b)(c-d)}\\to\n\\log\\sqrt{1\\over a-b}\\approx a\\log\\sqrt{1\\over(a-b)(c-d)}.\\eqn\\logrel$$\nIf we let $a-b=\\epsilon$ and $c-d=\\epsilon^\\mu$, then \\logrel\\ gives\n$$a={1\\over1+\\mu}.\\eqn\\aeq$$\nSince $0<\\mu<\\infty$, we see that the possible values of $a$ range from\n0 to 1.  Of course what this means is that no local solutions exist\nif the fermions are shifted such that all of the momenta are greater\nthan 0 or all are less than 0.  This should not come as a big surprise,\nsince once these limits are reached, then there are small deformations\nof the fermion momenta which lower the energy of the state.\n\nThe charge for these solutions is dominated by the $W$ term in \\hIVre,\nsince $X$ $Y$ and $Z$ are all small.  Clearly in the limit $b\\to a$ and\n$c\\to d$, the charge approaches\n$$Q\\to {a+d\\over2}=a-\\half.\\eqn\\Qlarge$$\n\n\\chapter{Correspondence with Lattice Models}\n\nDouglas and Kazakov have remarked that the phase transition for the continuum\nmodel is similar to the phase transition that occurs for the lattice.  In\nboth cases the phase transition is third order and the equations of motion\nfor the eigenvalues are given by a two cut model.  In some sense, these\ntwo situations are dual to each other, with the weak coupling region of the\nlattice model acting like the strong coupling region of the continuum\ncase.\n\nHowever, it would appear that this correspondence breaks down when we\nconsider the solutions with nonzero values of $Q$.  There are no\ncorresponding solutions for weakly coupled $U(N)$ on the lattice.  But\na little more thought shows that correspondence is between continuum $U(N)$\nand the lattice $SU(N)$ and vice versa.  For instance, the continuum\n$SU(N)$ case does not have these extra solutions, since the $U(1)$ charge\nis not a degree of freedom.  In terms of the fermions, the center of mass\ncoordinate is modded out.  Hence shifting all the fermion momenta by\nthe same amount gives back the same state.  On the other hand, the $SU(N)$\ncase has an extra term in the casimir, $n^2\/N^2$, where $n$ is the number\nof boxes in the representation. But this term does not survive the scaling\nlimit, so we can safely drop it.\n\nFor $SU(N)$ on the lattice, the eigenvalues sit in a potential described\nby \\matpot.  However, unlike the $U(N)$ case, the center of mass position\nfor these eigenvalues must satisfy the constraint that\n$$\\sum_i\\theta_i=2\\pi m,\\eqn\\thcon$$\nwhere $m$ is an integer.  Clearly, the $U(N)$ classical solution is the\nsame as the lowest energy state for the $SU(N)$ case, which will have\n$m=0$.  But suppose we consider a case where $0<m<N\/2$.  Since $m$ is\nan integer, we cannot smoothly deform this to the $m=0$ classical\nsolution.  Hence each value of $m$ must have a classical minimum.  We\ncould also have values of $m$ which are less than 0, but shifting the\ntotal position by $2\\pi N$ just shifts all eigenvalues around the\ncircle, hence these correspond to the same solution.\nThe maximum value of $m$ for a given weak coupling is determined by\nthe value which puts an eigenvalue at the top of the potential. Clearly as\nthe coupling becomes stronger, the number of classical solutions will\ndecrease.\n\nLet us discuss this situation in  a little more detail.  We can impose the\nconstraint in \\thcon\\ by inserting a $\\delta$-function into the path integral\nwith the form\n$$\\delta(\\sum_i\\theta_i-2\\pi m)={\\lim\\atop\\epsilon\\to0}{1\\over\\sqrt{2\\pi\\epsilon}}\ne^{-{1\\over2\\epsilon}(\\sum_i\\theta_i-2\\pi m)^2}.\\eqn\\thdf$$\nHence, using \\Vpot\\ we find the equation of motion\n$$N\\beta\\sin\\theta_i-\\sum_{j\\ne i}\\cot{\\theta_i-\\theta_j\\over2}\n-{1\\over\\epsilon}(\\sum_i\\theta_i-2\\pi m)=0.\\eqn\\eqmot$$\nwhere $\\beta=1\/(a^2g^2)$.\nUnder the usual rescaling, one ends up with the equation\n$$\\beta\\sin\\theta-{1\\over\\epsilon}(\\int\\mu(\\phi)\\phi-2\\pi m\/N)={\\rm P}\\int\\mu(\\phi)\n\\cot{\\theta-\\phi\\over2},\\eqn\\eqscaled$$\nwhere $\\mu(\\phi)$ is the density of eigenvalues.  As $\\epsilon\\to0$, the last\nterm on the lhs of \\eqscaled\\ is divergent unless the integral is very close\nto $2\\pi m$.  To this end, let us replace this entire term by a constant\n$\\alpha$.  Next define the function $f(\\theta)$,\n$$f(\\theta)=\\int d\\phi \\mu(\\phi) \\cot {\\theta-\\phi\\over2}.\\eqn\\fth$$\nGiven this definition, the function is analytic everywhere in the strip\n$-\\pi<{\\rm re}\\theta<\\pi$, except for a cut along the real line.  $f(\\theta)$ is\nclearly invariant under $\\theta\\to\\theta+2\\pi$.\n\n$f(\\theta)$ can be solved for\nby analyzing its behavior as ${\\rm Im}\\theta\\to\\infty$.  In particular, using \\fth\\\nthe large ${\\rm Im}\\theta$ behavior for $f(\\theta)$ is\n$$f(\\theta)=-i\\int d\\phi\\mu(\\phi)-ie^{i\\theta}\\int d\\phi e^{-i\\phi}\\mu(\\phi)+\n{\\rm O}(e^{2i\\theta}).\\eqn\\fthas$$\nLet us assume that the eigenvalues sit on the strip $\\Delta-a<\\lambda<\\Delta+a$.  Then\nfrom \\eqscaled\\ and \\fth, $f(\\lambda\\pm i\\epsilon)$ satisfies\n$$f(\\lambda\\pm i\\epsilon)=\\beta\\sin(\\lambda)+\\alpha\\mp2i\\pi\\mu(\\lambda).\\eqn\\fcut$$\nChoosing the periodic function\n$$f(\\theta)=\\beta\\sin\\theta+\\alpha-2\\beta\\cos{\\theta+\\Delta\\over2}\\sqrt{\\sin^2{\\theta-\\Delta\\over2}-\n\\sin^2{a\\over2}},\\eqn\\ffunc$$\none finds that the asymptotic behavior is matched if\n$$\\beta\\sin^2{a\\over2}\\cos\\Delta=1,\\eqn\\asyone$$\nand\n$$\\beta\\cos^2{a\\over2}\\sin\\Delta=-\\alpha.\\eqn\\asytwo$$\n{}From \\fcut\\ and \\ffunc\\ one learns that the density of eigenvalues is given\nby\n$$\\mu(\\lambda)={\\beta\\over\\pi}\\cos{\\lambda+\\Delta\\over2}\n\\sqrt{\\sin^2{a\\over2}-\\sin^2{\\lambda-\\Delta\\over2}}.\\eqn\\eigden$$\nIn order that the density of states is positive, one must satisfy\nthe inequalities $\\Delta+a\\le\\pi$ and $\\Delta-a\\ge-\\pi$.  One can also easily show\nthat the integral of $\\sin\\phi$ weighted by the density of eigenvalues\nsatisfies\n$$\\int\\mu(\\phi)\\sin\\phi=\\cos^2{a\\over2}\\sin\\Delta.\\eqn\\sinwt$$\nHence $\\Delta$ basically measures the anisotropy of the solution about the\nbottom of the well.\n\n\nAs $\\beta\\to\\infty$, it is clear that the allowed solutions for $\\Delta$ lie\nanywhere between $-\\pi\/2$ and $\\pi\/2$.  This is similar to the situation\nin the previous section, where any charge between $-N\/2$ and $N\/2$ is\nallowed as the area approaches infinity.    However, near the critical point,\nonly a small range for $\\Delta$ is allowed.  Clearly, from \\asyone, the critical\nvalue of $\\beta$ is $\\beta_c=1$.  Therefore, the only\nallowed $\\Delta$ at the critical point is $\\Delta=0$.\n\nMoving slightly away from the critical point, one has $a=\\pi-\\epsilon$ and a\nmaximum value of $\\Delta$ given by $\\Delta=\\epsilon$.  At the maximum, the integral\nof the angles weighted by the density of eigenvalues is then given by\n$$\\eqalign{\n\\int d\\phi\\mu(\\phi)\\phi&={\\beta\\over\\pi}\\int_{\\Delta-a}^{\\Delta+a}\nd\\phi \\phi\\cos{\\phi+\\Delta\\over2}\n\\sqrt{\\sin^2{a\\over2}-\\sin^2{\\phi-\\Delta\\over2}}\\cr\n&=\\Delta-{\\beta\\over\\pi}\\sin\\Delta\\int_{-a}^{a}d\\phi \\phi\\sin\\phi\\sqrt{\\sin^2{a\\over2}-\n\\sin^2{\\phi\\over2}},}\\eqn\\intth$$\nwhere we have shifted the integration variables and have used the fact that\nthe integral over the density of states is one.  For small $\\epsilon$ we find\nthat the leading order term in \\intth\\ is\n$$\n\\int d\\phi\\mu(\\phi)\\phi={1\\over8}\\epsilon^2\\Delta\\log{1\\over\\epsilon}+{\\rm O}(\\epsilon^2\\Delta).\n\\eqn\\intlead$$\n$\\beta$ is the inverse area, hence we find that the area for the plaquette\nsatisfies\n$$A_c-A={1\\over4}(\\epsilon^2+2\\Delta^2).\\eqn\\Alat$$\nTherefore, the maximum charge in terms of the area, behaves\nlike\n$$Q_{\\rm max}\\sim (A_c-A)^{3\/2}\\log{1\\over A_c-A}.\\eqn\\Qmlat$$\nHence, unlike the continuum case, the maximum charge in terms of the\narea has a scaling violating piece.\nHowever, the integral over $\\sin\\phi$ has a maximum that behaves like\n$(A_c-A)^{3\/2}$, which is closer to the behavior of the preceeding section.\n\n\n\\chapter{Discussion}\n\nIn this paper we have given an alternative derivation of Witten's\nproof that the QCD partition function is given by a sum over\nsaddle points.\nWe then showed how the sphere phase transition occurs in the\ndual picture of QCD string theory.\n\nWe next demonstrated how to find new solutions for the $U(N)$\n${\\rm QCD}_2$\\ sphere partition function corresponding to the different $U(1)$\nsectors.  We have also shown how these solutions relate to solutions\nfound for the lattice versions of $SU(N)$.\nIn some sense, it is surprising that the nonzero charge solutions can actually\nbe found.  One might have expected that since there is nothing in the\nmachinary of section 4 that explicitly states that the eigenvalues\nlie on the lattice, then nothing should prevent the nonzero charge\nsolutions from sliding down to the absolute minimum.\nBut actually, this information is in there, because of the $\\log$ term\nthat appears in \\fcontour. This term states that there is a region\nwhere the eigenvalues have a constant density, that is, they lie on\nthe lattice.  The boundaries of this region essentially add another\ndegree of freedom.  This then allows us to find solutions with nonzero\ncharges.\n\nIt is hoped that the results presented here might have some use in\ninvestigating random matrix model theory.  Perhaps one can consider\ncases where the entries of the Hamiltonian are restricted to be\nintegers.  One might then find similar behavior to that shown here.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}