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{"text":"\\section{Introduction}\nThis demo file is intended to serve as a ``starter file''\nfor IEEE Computer Society conference papers produced under \\LaTeX\\ using\nIEEEtran.cls version 1.8b and later.\nI wish you the best of success.\n\n\\hfill mds\n \n\\hfill August 26, 2015\n\n\\subsection{Subsection Heading Here}\nSubsection text here.\n\n\n\\subsubsection{Subsubsection Heading Here}\nSubsubsection text here.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Conclusion}\nThe conclusion goes here.\n\n\n\n\n\n\n\n\\ifCLASSOPTIONcompsoc\n \n \\section*{Acknowledgments}\n\\else\n \n \\section*{Acknowledgment}\n\\fi\n\n\nThe authors would like to thank...\n\n\n\n\n\n\n\n\n\\section{Introduction}\n\\label{simu1}\n\nPure CMOS hardware based Spiking Neural Networks (SNN) designs have turned out to be less efficient owing to the large computation\/memory requirement \\cite{jo2010nanoscale}. Several hybrid CMOS-nanodevice designs involving emerging non-volatile memory nanodevices have been proposed as efficient SNN hardware alternatives compared to pure CMOS design. The emerging non-volatile memory devices offer added efficiency as they are ultra compact, low-power and intrinsically mimic properties of biological neurons and synapses \\cite{burr2017neuromorphic}. Design exploration of such dedicated hybrid SNN hardware (CMOS-nanodevice) becomes a very complex and challenging multi-dimensional problem due to following reasons:\n(i) need to capture different variants of spatio-temporal learning rules, neuron and synaptic behavior models \\cite{andrew2003spiking,ghosh2009spiking,gerstner2002spiking}, (ii) wide variety of nanodevice\/nanocircuit options for emulating neuro-synaptic characteristics with different underlying physics of operation, (iii)`lack of maturity at the level of device fabrication, (iv) almost negligible standardization when it comes to choice of synapse, neuron and learning-rule models, (v) wide-variety of application tasks, and (vi) lack of standard benchmarking. \nComprehensive simulation frameworks or CAD tools which allow flexibility at the level of hardware-algorithm co-optimization thus become a necessity for addressing a domain like dedicated SNN hardware. Evaluating the impact of emerging device technology on network performance and associated hardware overheads by using a high-level simulation framework which takes in account details at the levels of device, circuit and algorithm simultaneously is essential to optimize performance and cut down on expensive ASIC manufacturing cycles. In literature, different biological SNN simulator frameworks have been reported such as: SpikeNET \\cite{delorme2003spikenet}, Brian \\cite{goodman2008brian}, Nengo \\cite{bekolay2014nengo}, CARLsim3 \\cite{beyeler2015carlsim}, NEST \\cite{gewaltig2007nest}, NEURON \\cite{carnevale2006neuron} etc. A limited number of simulation frameworks that deal with the implementation details of nanodevices in neural networks have also been reported, for instance: Xnet \\cite{bichler2013design} and N2D2\\cite{N2D2} are C++ based event-driven simulators, N2S3\\cite{boulet2017n2s3} is a Scala-based event driven simulator, NeuroSim+ \\cite{cpy17} is a device-to-algorithm framework for evaluating performances of emerging memory devices as synapses in Multi-Layer Perceptron (MLP) architectures. Most of these simulators can be either classified as Clock-Driven (CD) or Event-Driven (ED) based on when they update variables (at all time-steps or at certain event triggered time-steps). In this work we present a new simulation tool developed in MATLAB called MASTISK. MASTISK is an acronym for \\textit{MAchine-Learning and Synaptic-plasticity Technology Integration Simulation frameworK}. According to etymology of the word MASTISK (pronounced as \\textit{mas-tea-she-q}), it means 'brain' in Sanskrit. Key distinguishing features of the current version of MASTISK, compared to other neuromorphic simulators reported in literature \\cite{bichler2013design,N2D2,boulet2017n2s3}, are: \n\\begin{itemize}\n\\item{It can model a wide variety of synaptic circuits, including those in which non-identical pulses are required for conductance modulation.}\n\\item{It allows users to emulate LIF\/IF neuron circuits using nanodevices.}\n\\item{It allows emulation of any arbitrarily shaped neuron spike thus leading to different flavors of STDP or other spatio-temporal learning rules.}\n\\item{The tool has a parametric file interface, which requires the user to only specify a set of parameters defining the network topology, without the need for writing any additional code or script.}\n\\item{The tool provides an automated parameter tuning framework based on genetic algorithms to assist users in optimizing the network performance.}\n\\end{itemize}\n\n\n\n\n\n\\section{The MASTISK Framework}\n\\label{mastisk}\n\nMASTISK is designed using a CD simulation approach. Following sub-sections describe the simulator- workflow, dataflow and cases. \n\n\n\\begin{figure}[!ht]\n\\centering\n\\includegraphics[scale=0.38]{algo_flow2.png}\n\\caption{Generic workflow of MASTISK}\n\\label{workflow}\n\\end{figure}\n\n\n\\begin{figure}[!t]\n\\centering\n\\includegraphics[scale=0.3]{algo_flow3.png}\n\\caption{(a) Flow of data and spikes along a generic network initiated by MASTISK. (b) Code overview of the basic functioning of the framework.}\n\\label{network1}\n\\end{figure}\n\n\n\n\\subsection{Simulation Workflow}\n\\label{simu2}\nKey sequence for running any simulation in the proposed MASTISK framework is shown in Fig. \\ref{workflow}. First step requires a set of override parameters from the user corresponding to the devices and circuits being used as building blocks as well as the network parameters. The parameters are stored in the properties of different structures as shown in Fig. \\ref{workflow}. In SNNs the input data is encoded into multiple spike trains and the distribution of their frequencies determine the type of encoding used. MASTISK offers features for poisson, fixed frequency and custom AER sensor data based spike encoding. Following this, training and inference takes place which provides \\textit{training\\_accuracy}, \\textit{testing\\_accuracy} and the structure \\textit{net} containing the network topology and weights of synapses, as output. The parameter tuning interface is optional and can be used to determine the set of parameters that leads to optimum network performance.\n\n\\subsection{Simulator Dataflow}\nThe logical flow of information in the network and the simulator backbone is shown in Fig. \\ref{network1}. After spike encoding of input is completed the output is computed for each layer sequentially beginning with layer 1 for the number of time steps N given by Eq. \\ref{tstep},\n\n\\begin{equation}\n\\label{tstep}\nTotal\\ time\\ steps\\ (N) = \\dfrac{Total\\ training\\ time\\ (T)}{Simulation\\ time\\ step\\ (dt)}\n\\end{equation}\n\nwhere dt is the simulator time-step and T is the total time for which the network needs to be trained. The function \\textit{get\\_prevlayerSpikes()} is used to obtain the spike information of the previous layer. The function \\textit{getsynapsestate()} is used to obtain the modes (explained in Section \\ref{simu3}) of the synapses and \\text{get\\_totalCurrent()} is used to obtain the total current entering each neuron. The neuron membrane potential and weights of synapses are updated using \\textit{neuron\\_update()} and \\textit{weight\\_update()} respectively.\n\n\\subsection{Simulation Cases}\nThe framework in its current form offers four different simulation scenarios: (1) Biological SNN, (2) nanodevice based synapses, (3) nanodevice based LIF neurons, and (4) nanodevice based synapse and neurons, (Fig. \\ref{edisonn}). Case (II) is for devices that show controlled conductance modulation properties like RRAM, PCM etc, in response to applied signal in the form of voltage or current. Case (III) is for devices that mimic the Leaky-Integrate-Fire (LIF) properties of membrane potential in a biological neuron. Few proposals incorporate all-nanodevice based implementation of SNN (neuronal and synaptic) \\cite{sengupta2016vision,pantazi2016all}, represented by case (IV). \n\n\n\n\n\\begin{figure}[!t]\n\\centering\n\\includegraphics[scale=0.35]{algo_flow1.png}\n\\caption{The different facets of MASTISK.}\n\\label{edisonn}\n\\end{figure}\n\n\\section{Framework Capabilities}\n\\label{frame}\n\\begin{figure}[!b]\n\\centering\n\\includegraphics[width=\\columnwidth]{2_networks.png}\n\\caption{ Example SNN with (a) Two layers and intra-layer inhibitory connections, (b) Three layers and inter-layer inhibitory connectons.}\n\\label{network2}\n\\end{figure}\n\n\n\\subsection{Network Topology Modeling}\nThe structure \\textit{layers} is used to define the network topology. \nOne of the most commonly used feed-forward SNN topologies in literature is shown in Fig.\\ref{network2} (a). The number of layers and number of neurons in each layer are set by the \\textit{num} and \\textit{neurons} properties. MASTISK offers the flexibility to keep a certain set of weights fixed using \\textit{layers.plastic} (Fig. \\ref{network2} (a)), which in some cases can lead to improved performance and faster convergence in learning \\cite{bengio2007greedy}. The output labeling layer in SNNs can be set using \\textit{layers.label}. Different types of synaptic connections supported are: all-to-all, one-to-one and sparse, specified by \\textit{layers.conn\\_type} (Fig. \\ref{network2} (a)). In case of sparse connectivity the degree of sparseness is specified by the value of \\textit{layers.sparse}. Indices of start and end layers between which inhibitory connections exist are stored in \\textit{layers.inh\\_conn}. \n\n\n\\subsection{Spike Modeling}\nThe capability to model arbitrary spike shapes is highly desirable in a neuromorphic simulator. MASTISK applies piece-wise linear approximation to implement spike modeling. The points of non-differentiability in the spike waveform (both timestamp and amplitude) are stored in an array: \\textit{ckt\\_params.pre\\_volt}. The simulator divides a complex spike into several pieces by using the specified starting (t$_{i}$,V$_{i}$) and end points (t$_{i+1}$,V$_{i+1}$) of each piece. If t$_{i}$ $\\neq$ t$_{i+1}$, then points V$_{i}$ and V$_{i+1}$ are joined by a line as shown in Fig. \\ref{network2} (b). If t$_{i}$ = t$_{i+1}$ and V$_{i}$ $\\neq$ V$_{i+1}$, then there is a discontinuity or abrupt change between V$_{i}$ and V$_{i+1}$. In this case, V$_{i+1}$ becomes the starting point for the pieces following t$_{i+1}$.\n \n\n \n\\begin{figure}[!t]\n\\centering\n\\includegraphics[scale=0.38]{general_synapse.png}\n\\caption{Generic nanodevice based synaptic circuit implemented in MASTISK.}\n\\label{general_synapse}\n\\end{figure}\n\n\n\n\n\\subsection{Synaptic device and architecture modeling}\n\\label{simu3}\nGeneric architecture of a synaptic circuit that can be implemented in MASTISK is shown in Fig. \\ref{general_synapse}. MASTISK considers three different types of neuronal spikes: post-spike1 (propagated back to synaptic circuit), post-spike2 (pre-spike for the next layer of synapses) and Inhib-spike (optional and used only when inhibition is to be implemented). Synaptic circuit and applied spikes depend on the synaptic device being used and this in turn effects the learning rule being implemented. Modeling such device specific constraints would require extensive code modifications in available simulation frameworks \\cite{bichler2013design,N2D2,boulet2017n2s3}. In order to resolve this issue, we have split the operating states of a synapse in four modes: (1) Idle state, (2) Spike Transmission, (3) Potentiation and (4) Depression. In idle state there is neither any current flow through the synaptic device nor any conductance change. In spike-transmission state only current flows through the synapse and no conductance change takes place. Potentiation\/Depression states implement increase\/decrease of synaptic conductances.\n\\begin{figure}\n\\centering\n\\includegraphics[width=\\columnwidth]{master_synapse.png}\n\\caption{MASTISK based implementations of: (a) 1T-1R synaptic circuit and spike scheme, adopted from \\cite{ambrogio2016neuromorphic}(Inset (I) shows corresponding STDP characteristics for different initial synaptic resistance values.) (b) synaptic circuit and spike scheme adopted from \\cite{covi2016analog} (Inset (I) shows resultant voltage across synaptic device (V$_{B}$-V$_{T}$) for different temporal orientation of pre\/post-spikes, Inset (II) shows corresponding STDP characteristics for different initial synaptic resistance values.) (c) Inhibitory connection circuit. Synaptic conductance modulation in AlO$_{x}$\/HfO$_{2}$ based RRAM device due to application of: (d) identical potentiating pulses, (e) identical depressing pulses, adopted from \\cite{woo2016improved}. Synaptic conductance modulation in TiN\/HfO$_{2}$\/Ti\/TiN based RRAM device due to application of (f) varying amplitude potentiating, and (g) varying amplitude depressing pulses, adopted from \\cite{covi2016analog}.}\n\\label{master_synapse}\n\\end{figure}\nThe state of the synaptic circuit can be classified based on the presence or absence of the pre and post-neuronal spikes in any one of the following states: (1) no spikes, (2) only Pre-Spike, (3) only Post-Spike, and (4) both spikes. The functionality of any synaptic architecture can be implemented by establishing the mapping between the states of the synaptic device and the circuit. Two test cases of synaptic circuits and spikes adopted from \\cite{ambrogio2016neuromorphic} and \\cite{covi2016analog} along with their implementation in MASTISK are shown in Fig.\\ref{master_synapse}(a) and (c). For emulation of STDP or any other spatio-temporal rule the exact timing of potentiation and depression relative to spike timing is required. The spike timings and synaptic states are taken care of different parameters. For details on the usage of these parameters, readers are referred to \\cite{mastisk}. The circuit shown in Fig. \\ref{master_synapse} (a) undergoes spike transmission only when there is a pre-spike (V$_{pre}$) (V$_{TB}$ = V$_{post1}$-V$_{node1}$, where V$_{post1}$ is a fixed DC bias when there is no post-spike), potentiation when the portion marked V$^{+}$ in V$_{post1}$ coincides with V$_{pre}$ (V$_{TB}$ = V$^{+}$-V$_{node1}$ \\textgreater V$_{th}$) and depression when V$^{-}$ coincides with V$_{pre}$ (V$_{TB}$ = V$^{-}$-V$_{node1}$ \\textless -V$_{th}$). The equation for current through device during spike transmission is captured by \\textit{ckt\\_params.ex\\_eqs} and can be defined by the user. For simplicity we have assumed V$_{node1}$ = V$_{node2}$, when MOSFET is turned ON during pre-spike. However for more accurate circuit modeling one can define the relation between V$_{node1}$ and V$_{node2}$ in \\textit{ckt\\_params.ex\\_eqs}. The device behavior during potentiation and depression are captured by obtaining experimental conductance modulation data of the device as shown in Fig. \\ref{master_synapse} (b). The conductance states are stored in the arrays: \\textit{device\\_params.synapse\\_levels\\_ltp} and \\textit{device\\_params.synapse\\_levels\\_ltd}. However the conductance modulation characteristics shown in Fig. \\ref{master_synapse} (b) are only good for synaptic circuits where at any point of time, conductance change is caused by identical pulses as in Fig. \\ref{master_synapse} (a). In the circuit shown in Fig. \\ref{master_synapse} (c), the resultant voltage across the device (V$_{TB}$ = V$_{post1}$ - V$_{pre}$) leads to conductance change when it crosses a threshold (V$_{TH}$). The magnitude of V$_{TB}$ that crosses (V$_{TH}$) keeps changing with the temporal orientation of the spikes w.r.t each other. In such case, experimental data showing conductance variation with number of applied pulses for different pulse widths or amplitudes is required (Fig. \\ref{master_synapse} (d)). The expression for resultant voltage (V$_{TB}$) can be set by user using \\textit{ckt\\_params.v\\_app}, and the experimental conductance values can be accommodated in \\textit{ckt\\_params.v\\_example} in form of a 2D matrix, where each row signifies a different pulse amplitude or width.\n\n\\begin{figure}[!t]\n\\centering\n\\includegraphics[scale=0.38]{neuron_block.png}\n\\caption{Generic nanodevice neuron architecture implemented in MASTISK.}\n\\label{neuron}\n\\end{figure}\n\n\\begin{figure}[!t]\n\\centering\n\\includegraphics[scale=0.20]{neuron_spikes_PCM.png}\n\\caption{Neuron frequency vs input pulse width adopted from \\cite{tuma2016stochastic}.}\n\\label{firing_freq}\n\\end{figure}\n\n\\subsection{Neuron device architecture modeling}\nNumerous devices have been proposed for implementing LIF or IF neurons in SNNs \\cite{dutta2017leaky,tuma2016stochastic,sengupta2016magnetic,jaiswal2017proposal}. These devices have a state-variable that gets integrated with incoming signals. In order to simulate neuron functionalities with these devices, generalized architecture of neuron circuit implemented in MASTISK is shown in Fig. \\ref{neuron}. V$_{node\\_ex}$ and V$_{node\\_inh}$ denote the voltages of the nodes where the summed signals from excitatory and inhibitory synapses enter respectively. The signal is then passed through a current-voltage\/voltage-current converter or a buffer circuit. The user can specify the equation that best describes the dynamic power dissipated in their circuit with \\textit{neuron\\_conv\\_dyn\\_power}. The incoming signals may need to be converted to either fixed-amplitude, variable-width signal or vice versa. This width-amplitude characteristic can be defined using the function \\textit{neuron.pulse\\_convert()}. Neuron modeling in MASTISK is achieved by taking the neuron firing frequency variation with the strength of integrating signal as an input. Fig. \\ref{firing_freq} shows frequency modulation with applied pulse width of PCM based IF neuron adopted from \\cite{tuma2016stochastic}. This data can be used to extract the time constant (\\textit{neuron.tau}) and threshold of the neuron device (\\textit{neuron.thres}). LIF or IF equation governing the neuron state-variable updation is set by \\textit{neuron.state\\_eqs}. The state variable evaluating circuit (generally a voltage divider) generates output which is proportional to the conductance of the device and is evaluated against a threshold. Once the neuron spikes the state-variable is reset by the reset circuit. \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Discussion and Limitations}\n\\label{limit}\nThe accuracy of simulation results depends heavily on the choice of simulator time step (dt). Glitches or noisy spikes can only be observed if the time step of the simulation is smaller in comparison to the time period of the fastest spike. Present version of the proposed simulator does not incorporate direct integration with circuit simulators and as a result parameters such as thermal noise in devices and delays in analog circuits are not taken into consideration. However the modular structure of the simulator makes it possible for interfacing with circuit simulators in future. For example, functions governing spike transmission, dynamic power consumption etc, can be provided by user in form of equations or can be defined by a SPICE circuit simulator. Exporting the entire framework to Python, providing a GUI for input parameters, GPU acceleration to improve simulation time are some of the proposed improvements for future version of the present framework. We also would like to further extend the framework's capability to support other learning algorithms such as MLP, CNN, ELM \\cite{suri2015oxram} and RBM \\cite{suri2015neuromorphic,parmar2018design} in inference mode. \n\n\\section{Conclusion}\n\\label{conc}\nIn this paper we present a new MATLAB based integrated technology benchmarking clock-driven SNN simulator framework called MASTISK. We describe our simulator's different functionalities like network modeling, spike modeling, unique synaptic circuit modeling strategy with both identical and non-identical conductance modulation pulses and parameter tuning using genetic algorithm. Unlike the rest, our simulator can also implement nanodevices along with its appropriate circuitry as neurons in SNNs. We show how the different parameters can be set to achieve the above functionalities and also present two case studies in which a RRAM based synapse and a PCM based neuron based SNN is simulated with actual experimental data and circuit topology extracted from literature. Further detailed extension of this work, including multiple device case-studies, parameter optimization etc. will be published in the proceedings of IJCNN 2018. \n\n\n\n\n\\section*{Acknowledgment}\nThis research activity under the PI Prof. M. Suri is partially supported by the Department of Science \\& Technology (DST), SERB-EMR Government of India and IIT-D FIRP grants.\n\n\n\n\n\\bibliographystyle{IEEEtran}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
{"text":"\\section{Introduction}\n\n\n Chiral effective field theory provides a controllable perturbative approach\nof strongly interacting hadrons at low energies. \nA systematic power counting organizes the chiral effective Lagrangian and \nobservables as a perturbative series \nin the Goldstone boson sector of QCD \\cite{Weinberg:1979kz,Gasser:1983yg}.\nEffective field theories (EFTs) with pions and nucleons proved to be more complicated, however,\n the problem of a consistent power counting \\cite{Gasser:1987rb} can be solved by using \neither the heavy-baryon \napproach \\cite{Jenkins:1990jv,Bernard:1992qa,Bernard:1995dp}\nor by choosing a suitable renormalization scheme in a manifestly Lorentz invariant formulation \n\\cite{Tang:1996ca,Becher:1999he,Gegelia:1999gf,Fuchs:2003qc}.\nDue to the relatively small mass difference between the nucleon and the \n$\\Delta$-resonance and the strong coupling to the pion-nucleon system, the delta \ncan be also included in a systematic way in chiral EFT (see e.g. \nRefs.~\\cite{Hemmert:1997ye,Pascalutsa:2002pi,Bernard:2003xf,Pascalutsa:2006up,Hacker:2005fh}).\n\tA clear drawback of the low-energy EFT approach is that unlike the underlying QCD,\n the Lagrangian contains an infinite number of parameters, the low-energy constants (LECs).\nHowever, only a finite number of them contributes to physical quantities calculated up to \na given order. These parameters are fixed by fitting them to \nexperimental data or can be calculated on the lattice, allowing one to \npredict other quantities. A precise determination of these LECs is an important and \nhighly non-trivial task, especially when the $\\Delta$-resonance is included because\nthere are more LECs for a given process than in the pure $\\pi N$ effective Lagrangian.\n \n\nIn this work we calculate the width of the delta resonance in a systematic expansion \nin terms of the pion mass and the nucleon-delta mass difference (collectively denoted by $q$) \nin the framework of baryon chiral perturbation theory up-to-and-including order $q^5$, \nwhich includes the leading two-loop contributions. This counting is often referred to as the\nsmall scale expansion, see e.g. Ref.~\\cite{Hemmert:1997ye}. We use the obtained results to fix a \ncombination of pion-nucleon-delta couplings appearing in this expression from the experimental \ndata, more precisely, we obtain a correlation between the leading $\\pi N \\Delta$ and $\\pi \\Delta$ \ncouplings.\n \n\nThe dressed propagator of the $\\Delta$-resonance in $d$ space-time dimensions\ncan be written as\n\\begin{eqnarray}\n-i D^{\\mu\\nu}(p)&=& -i \\Biggl [\ng^{\\mu\\nu}-\\frac{\\gamma^\\mu\\gamma^\\nu}{d-1} -\\frac{\np^\\mu\\gamma^\\nu-\\gamma^\\mu p^\\nu}{(d-1) m_{\\Delta}^0}-\\frac{d-2}{(d-1)\n(m^0_{\\Delta})^2}\\,p^\\mu p^\\nu\\Biggr]\\, \\frac{1}{p\\hspace{-.35\nem}\/\\hspace{.1em}-m_\\Delta^0 -\\Sigma_1 - p\\hspace{-.4\nem}\/\\hspace{.1em}\\Sigma_6}\\nonumber\\\\&+& {\\rm pole \\ free \\\nterms}\\,,\\label{dressedDpr}\n\\end{eqnarray}\nwhere $m^0_\\Delta$ is the pole mass of the delta in the chiral limit, and \n$\\Sigma^{\\mu\\nu}$ is the self-energy of the $\\Delta$-resonance. It can be parameterized as\n\\begin{eqnarray}\n\\Sigma^{\\mu\\nu} & = & \\Sigma_1(p^2)\\, g^{\\mu\\nu}+\\Sigma_2(p^2)\\,\n\\gamma^{\\mu}\\gamma^{\\nu}+\\Sigma_3(p^2)\\,p^{\\mu}\\gamma^{\\nu}+\\Sigma_4(p^2)\\,\n\\gamma^{\\mu}p^{\\nu}+\\Sigma_5(p^2)\\,p^{\\mu}p^{\\nu}\n\\nonumber\\\\\n&+& \\Sigma_6(p^2)\\, p\\hspace{-.45em}\/\\hspace{.1em}\ng^{\\mu\\nu}+\\Sigma_7(p^2)\\,p\\hspace{-.45em}\/\\hspace{.1em}\n\\gamma^{\\mu}\\gamma^{\\nu}+\\Sigma_8(p^2)\\,\np\\hspace{-.45em}\/\\hspace{.1em} p^{\\mu}\\gamma^{\\nu}+\\Sigma_9(p^2)\\,\np\\hspace{-.45em}\/\\hspace{.1em}\n\\gamma^{\\mu}p^{\\nu}+\\Sigma_{10}(p^2)\\,p\\hspace{-.45em}\/\\hspace{.1em}\np^\\mu p^\\nu. \\label{DseParametrization}\n\\end{eqnarray}\n\n\\begin{figure}[t]\n\\begin{center}\n\\epsfig{file=Delta_SE_R.eps,scale=0.5}\n\\caption{One and two-loop self-energy diagrams contributing to the width \nof the delta resonance up-to-and-including \nfifth order according to the standard power counting. The dashed and double solid lines \nrepresent the pions and the delta resonances, respectively. \nThe double (solid-dotted) lines in the loops correspond to either nucleons or deltas. \nThe numbers in the circles give the chiral orders of the vertices.}\n\\label{figSE2}\n\\end{center}\n\\end{figure}\n\n\n\\noindent\nThe complex pole position $z$ of the $\\Delta$-propagator can be found by\nsolving the equation\n\\begin{equation}\nz - m_{\\Delta}^0 -\\Sigma_1(z^2)-z\\, \\Sigma_6(z^2) \\equiv z - m_{\\Delta}^0 - \\Sigma(z) =0\\,. \n\\label{poleequation}\n\\end{equation}\nThe pole mass and the width are defined by parameterizing the pole position $z$ as\n\\begin{equation}\nz=m_\\Delta-i\\,\\frac{\\Gamma_\\Delta}{2}\\,.\n\\label{polpar}\n\\end{equation}\n\n\n\\medskip\n\n The one- and two-loop self-energy diagrams contributing to the \nwidth of the delta resonance up to order $q^5$ are shown in Fig.~\\ref{figSE2}, where\nthe counterterm diagrams are not displayed. The underlying effective chiral Lagrangian of pions,\nnucleons and the delta resonances is given in the Appendix. For more details and the\nexplicit discussion of the power counting, relevant for the current calculation of the delta \nwidth at leading two-loop order, we refer to Refs.~\\cite{Yao:2016vbz,Gegelia:2016xcw}. \n\n\n\nWe solve Eq.~(\\ref{poleequation}) perturbatively order by order in the loop expansion. \nFor that purpose we write the self-energy as an expansion in the number of loops\n(which is equivalent to an expansion in $\\hbar$)\\footnote{Note that we retain the powers of\n$\\hbar$ for clarity here, otherwise we use natural units $\\hbar=c=1$.}\n\\begin{equation}\n\\Sigma = \\hbar \\,\\Sigma_{(1)}+\\hbar^2 \\Sigma_{(2)} +{\\cal O}(\\hbar^3)\\, ,\n\\label{SEpert}\n\\end{equation}\nand obtain the following expression for the width (modulo higher order corrections)\n\\begin{eqnarray}\n\\Gamma_\\Delta &=& \\hbar \\, 2 i \\,{\\rm Im} \\left[\\Sigma_{(1)}(m_\\Delta)\\right] \\nonumber\\\\\n&+& \\hbar^2 \\, 2 i \\, \\biggl\\{\n{\\rm Im} \\left[\\Sigma_{(1)}(m_\\Delta)\\right]{\\rm Re} \\left[\\Sigma_{(1)}'(m_\\Delta)\\right]\n+ {\\rm Re} \\left[\\Sigma_{(1)}(m_\\Delta)\\right]{\\rm Im} \\left[\\Sigma_{(1)}'(m_\\Delta)\\right] \\biggr\\} \\nonumber\\\\\n&+& \\hbar^2 \\, 2 i \\, {\\rm Im} \\left[\\Sigma_{(2)}(m_\\Delta)\\right] \n+{\\cal O}(\\hbar^3).\n\\label{widthpert}\n\\end{eqnarray}\n\n\n\nTo calculate the contributions of the one-loop self-energy diagrams to\nthe width, specified in the first two lines of \nEq.~(\\ref{widthpert}), we use the corresponding explicit expressions. For the two-loop \ncontribution, i.e. the terms in the third line, we use the Cutkosky cutting rules, that is we relate \nit to the corresponding decay amplitude $A_{\\Delta\\to\\pi N}$ via\n\\begin{eqnarray}} \\newcommand{\\eea}{\\end{eqnarray}\n\\Gamma_\\Delta =\\frac{\\left[(m_\\Delta+m_N)^2-M_\\pi^2\\right]\\left[ \\left(m_\\Delta^2-m_N^2-M_\\pi^2\\right)^2 -4 M_\\pi^2 m_N^2\\right]^{3\/2}}{{192}\\pi m_\\Delta^5}\\,|A_{\\Delta\\to\\pi N}|^2\\,,\n\\label{widthpiN}\n\\eea\n\\medskip\nwhere we have parameterised the amplitude for the decay $\\Delta_{\\mu}^i(p_i)\\to \\pi^a(q_a) N(p_f)$ as\n\\begin{eqnarray}} \\newcommand{\\eea}{\\end{eqnarray}\n{\\cal A}=\\bar{u}_N(p_f)\\left\\{A_{\\Delta\\to\\pi N}\\,q_a^\\mu\\xi_{ai}^{\\frac{3}{2}}\\right\\}u_\\mu(p_i)\\, .\n\\eea\n\n\n\nThe tree and one-loop diagrams contributing to the $\\Delta\\to \\pi N$ \ndecay up to order $q^3$ are shown in Fig.~\\ref{fig:DNPi1}. See again\nRefs.~\\cite{Yao:2016vbz,Gegelia:2016xcw} for the details on the power counting of the \namplitude and the total width of the resonance.\n\\begin{figure}[h]\n\\begin{center}\n\\epsfig{file=DNPi1.eps,scale=0.55}\n\\caption{Feynman diagrams contributing to the decay $\\Delta\\to N\\pi$ up to leading one-loop order. Dashed, solid and double \nlines represent pions, nucleons and delta resonances, respectively. Numbers in the circles mark the chiral orders of the vertices.}\n\\label{fig:DNPi1}\n\\end{center}\n\\end{figure}\n\n\n\\medskip\n\nCalculating one- and two-loop contributions in the delta width as specified \nabove we observe that by defining a linear combination of $\\pi N\\Delta$ couplings \n\\begin{eqnarray}} \\newcommand{\\eea}{\\end{eqnarray}\\label{eq:redefinition.h}\nh_A = h - \\left(b_3\\Delta_{23}+b_8\\,\\Delta_{123}\\right)\n-\\left(f_1\\Delta_{23}+f_2\\,\\Delta_{123}\\right)\\Delta_{123}+2(2f_4-f_5)M_\\pi^2\\, ,\n\\eea\nwith\n\\begin{eqnarray}} \\newcommand{\\eea}{\\end{eqnarray}\n\\Delta_{123}\\equiv \\frac{M_\\pi^2+m_N^2-m_\\Delta^2}{2 m_N}\\, ,\\qquad\\Delta_{23}\\equiv m_N-m_\\Delta\\, ,\n\\eea\nmodulo higher order terms, the whole explicit dependence on the couplings \n$b_3$, $b_8$, $f_1$, $f_2$, $f_4$ and $f_5$ disappears from the expression of the \ndelta width. This allows us to extract with a good accuracy the numerical value of \n$h_A$ from the experimental value of the delta width for a given value of the\nleading $\\pi \\Delta$ coupling constant $g_1$. Such a correlation between $\\pi N\\Delta$ and\n$\\pi \\Delta$ couplings exists in the large $N_C$ limit but, as far as we know, is observed\nhere first for the real world with $N_C=3$.\n\n\n\n\\begin{figure}[t]\n\\begin{center}\n\\epsfig{file=Fig.eps,scale=0.75}\n\\caption{Value of the pion-nucleon-delta coupling $h_A$ as a function of \nthe pion-delta coupling $g_1$ represented \nby the solid red line and the corresponding band given by the dashed red lines. \nThe central line corresponds to $\\Gamma_\\Delta = 100$~MeV, while the band is obtained \nby varying $\\Gamma_\\Delta$ in the range of $98-102$~MeV \\cite{Agashe:2014kda}. \nThe dot-dashed lines correspond to various values of the delta width indicated by their values \n(in MeV). For comparison, the blue dot with error bars represents the real part of the \ncoupling from Ref.~\\cite{Yao:2016vbz}, the purple dots stand for the values of the \nleading order pion-nucleon-delta coupling obtained in the \nlarge-$N_c$ limit and the horizontal dashed line with cyan band corresponds \nto the value (with error represented by the band) from Ref.~\\cite{Bernard:2012hb}.}\n\\label{hAgraph}\n\\end{center}\n\\end{figure}\n\nWe use the following standard \nvalues of the parameters \\cite{Agashe:2014kda}:\n$g_A = 1.27$, $M_\\pi = 139\\,$MeV, $m_N=939\\,$MeV, $m_\\Delta=1210\\,$MeV, \n$F_\\pi=92.2\\,$MeV \nand obtain for the full decay width of the delta resonance \n\\begin{eqnarray}\n\\Gamma_\\Delta &=& 53.91\\,{h}_A^2+0.87g_1^2 {h}_A^2-3.31g_1 {h}_A^2 -0.99\\,{h}_A^4.\n\\label{GammaF}\n\\end{eqnarray}\nSubstituting $\\Gamma_\\Delta = 100\\pm2\\ {\\rm MeV}$ from the PDG in Eq.~(\\ref{GammaF}), \nwe extract $h_A$ as a function of $g_1$. The obtained result is plotted in Fig.~\\ref{hAgraph}. \nFor comparison we also show the numerical value of the $\\pi N \\Delta$ \ncoupling from Ref.~\\cite{Bernard:2012hb} (extracted at leading one-loop order and thus independent of $g_1$), the one obtained by applying symmetry considerations in the large-$N_C$ \nlimit\\footnote{As the large-$N_C$ considerations do not fix the\nrelative sign between the two couplings, we must display two values of $g_1$ for\na given value $h_A$ here.} \nand the real \npart of the same linear combination of the couplings, as in current work, fitted to the pion-nucleon scattering phase shifts of Ref.~\\cite{Yao:2016vbz}, which uses a different renormalization scheme leading to a complex valued $h_A$. Note also that Ref.~\\cite{Hemmert:1997ye}\nextracts 1.05 as the value of the leading order $\\pi N \\Delta$ coupling in the heavy baryon approach.\n\n\n\n\n\n\\medskip\n\n To summarize, in the current work we have calculated the width of the delta resonance up to leading two-loop order \nin baryon chiral perturbation theory. \nUsing the obtained results we fixed a combination of pion-nucleon-delta couplings, \nwhich also contributes in the pion nucleon scattering process, as a function of the leading pion-delta coupling. \n\n\n\n\n\\acknowledgments\nThis work was supported in part by Georgian Shota Rustaveli National\nScience Foundation (grant FR\/417\/6-100\/14) and by the DFG (CRC~110).\nThe work of UGM was also supported by the Chinese Academy of Sciences (CAS) President's\nInternational Fellowship Initiative (PIFI) (Grant No. 2015VMA076). The work of DS was supported by the Ruhr University Research\nSchool PLUS, funded by Germany's Excellence Initiative [DFG GSC 98\/3].\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
{"text":"\\subsection*{Abstract}\n{%\nWe study the pairing Hamiltonian in a set of non degenerate levels.\nFirst, we review in the path integral framework \nthe spontaneous breaking of the $U(1)$ symmetry occurring in such a system\nfor the degenerate situation.\nThen the behaviors with the coupling constant of the ground state energy\nin the multilevel and in the degenerate case are compared.\nNext we discuss, in the multilevel case, an exact strong coupling expansion \nfor the ground state energy which introduces the moments of the\nsingle particle level distribution. \nThe domain of validity of the expansion, which is known in the \nmacroscopic limit, is explored for finite systems and its implications for the \nenergy of the latter is discussed.\nFinally the seniority and Gaudin excitations\nof the pairing Hamiltonian are addressed and shown to display the same gap\nin leading order.\n}\n\\endabstract\n\n{\\small \n\\noindent\n{\\em PACS:}\\ 21.60.-n, 21.30 Fe, 24.10.Cn;\n{\\em Keywords:}\\ Pairing interaction; Bosonization.}\n\n\\section{Introduction}\n\nThe pairing Hamiltonian\n\\begin{equation}\n\\hat H= \\sum_j \\sum_{m=-j}^j e_{j} \n\\hat \\lambda^{\\dag}_{jm} \\hat \\lambda_{jm} -g \\sum_{i,j} \n\\sqrt{\\Omega_i\\Omega_j} \\hat A^{\\dag}_i \\hat A_j,\n\\label{HP}\n\\end{equation}\nwhere (note that this definition differs from the one of \nRefs.~\\cite{Barbaro:2004nk,Barbaro:2003bb,Barbaro:2002fi} by a normalization\nfactor $1\/\\sqrt{\\Omega_j}$)\n\\begin{equation}\n\\hat A_j = \\frac{1}{\\sqrt{\\Omega_j}} \\sum_{m>0}(-1)^{j-m} \n\\hat \\lambda_{j,-m} \\hat \\lambda_{j,m}\\ ,\n\\label{A}\n\\end{equation}\nrepresents a simplified version of the BCS model of superconductivity since it\nassumes a constant two-body matrix element of the pairing force between all \nthe single particle levels.\n\nWhile BCS is generally applied to a macroscopic system, the\nHamiltonian \\eqref{HP} is also suitable for dealing with finite systems.\nIndeed it has been successfully applied to atomic nuclei where\n$N$ fermions live in $L$ single particle levels, each with pair degeneracy \n$\\Omega_j=j+1\/2$. In this case \nin \\eqref{HP} ${\\hat \\lambda}^{\\dag}$, ${\\hat \\lambda}$ are the usual \ncreation and annihilation fermion operators, \n$m$ is the third component of the angular momentum $j$, \nthe $e_{j}$ are the single-particle energies (assumed to be $m$-independent) and \n$g$ is the strength of the pairing force.\n\nOne recovers BCS from \\eqref{HP} by taking $\\Omega_j=1$\nfor all the $L$ levels and letting $L$ and $N$ become very large\n(as appropriate to a macroscopic system, e.g. electrons in a metal), but keeping\ntheir ratio constant and $g L$ finite.\n\nAs is well-known \nin this limit the ground state of the grandcanonical version of \\eqref{HP}\nreads~\\cite{Haag}\n\\begin{equation}\n|BCS> = \\prod_{k>0} \\left(u_k+v_k \n\\hat \\lambda^{\\dag}_{{\\vec k} \\uparrow } \n\\hat \\lambda^{\\dag}_{-{\\vec k} \\downarrow}\n\\right)|0> ~,\n\\label{BCS}\n\\end{equation}\n$|0>$ being the particle vacuum and\n$u_k$, $v_k$ parameters taken to be real.\nThe operators in \\eqref{BCS} create a pair of particles in time reversal\nstates.\n\nThe state \\eqref{BCS} is not invariant under the group $U(1)$ of the global\ngauge symmetry respected by \\eqref{HP}, hence it is\nassociated with the spontaneous breaking of the particle\nnumber conservation.\n\nWhen applied to a system of a {\\em finite}, fixed number \n$N$ of particles \\eqref{BCS}\nbecomes an approximation, valid, however, to the order $1\/N$.\nThis follows from the finding of Cambiaggio \net al.~\\cite{Cambiaggio:1997vz} that the Hamiltonian \\eqref{HP} corresponds\nto an integrable system: hence it is solvable (classically by quadrature,\naccording to the Liouville theorem).\nIndeed Richardson~\\cite{Rich} was able to find out the system of\nalgebraic equations yielding the exact solution which,\nfor a system made up of $n=N\/2$ ($N$ even) pairs of fermions living in $L$ \nsingle particle\nlevels, reads~\\cite{Barbaro:2003bb}\n\\begin{equation}\n|n> = C_n \\prod_{\\nu=1}^n {\\hat B}^\\dag_\\nu |0>\n\\label{n}\n\\end{equation}\nbeing \n\\begin{equation}\n{\\hat B}_\\nu = \\sum_{j=1}^L \\frac{1}{2 e_j-E_\\nu} \\hat A_j ~.\n\\label{B}\n\\end{equation}\nIn \\eqref{n} $C_n$ is a normalization constant and in \\eqref{B} the quantities\n$E_\\nu$ ($\\nu=1\\cdots n$) are solutions of the following system of $n$\nequations (the Richardson equations)\n\\begin{equation}\n\\frac{1}{g} = \\sum_{j=1}^L \\frac{\\Omega_j}{2 e_j-E_\\nu} - \n\\sum_{\\mu(\\neq \\nu)=1}^n \\frac{2}{E_\\mu-E_\\nu} ~,\n\\label{Rich}\n\\end{equation}\nwhereas the $\\hat A_j$ (defined in \\eqref{A})\nare referred to as hard-core boson operators and satisfy the commutation\nrelations\n\\begin{equation}\n\\left[\\hat A_j,\\hat A_{j'}^\\dag\\right] = \\delta_{jj'}\\left(1-\n\\frac{\\hat N_j}{\\Omega_j}\\right)~,\n\\end{equation}\nbeing\n$\n\\hat N_j = \\sum_{m=-j}^j \\hat \\lambda_{jm}^\\dag\\hat \\lambda_{jm} \n$\nthe particle number operator of the $j$-level.\n\nIn Richardson's framework, the eigenvalues ${\\cal E}$ of \\eqref{HP} (in\nparticular the ground state energy) are obtained according to\n\\begin{equation}\n{\\cal E}(n) = \\sum_{\\nu=1}^n E_\\nu ~.\n\\end{equation}\nA question arises \non the relationship between \\eqref{n} and \\eqref{BCS}\nin the limit of $N$ and $L$ very large, the only situation where, strictly\nspeaking, the phase transition associated with\nthe spontaneous symmetry breaking occurs.\nWe shall later comment on this issue, already addressed by\nRoman et al.~\\cite{Roman:2002dh} who, in analyzing \nthis limit, showed that the system of equations \\eqref{Rich} becomes a non linear \nintegral equation\nand identified the order parameter associated with\nthe spontaneous breaking of particle number conservation. This turns out\nto be the BCS gap, namely\n\\begin{equation}\n\\Delta = g \\sum_{\\mu>0} u_\\mu v_\\mu ~.\n\\label{Delta}\n\\end{equation}\nIn general \\eqref{Rich} can only be dealt with numerically.\nHowever it is trivial to solve it when\nonly one single particle level of energy $\\bar e$ and degeneracy $\\Omega$\nis available to the fermions.\nIn fact the system then reduces to the form \n\\begin{equation}\n\\frac{1}{g} = \\frac{\\Omega}{2{\\bar e}-E_\\nu} - \n\\sum_{\\mu(\\ne\\nu)=1}^n\\frac{2}{E_\\mu-E_\\nu}\n\\label{Rich1La}\n\\end{equation}\nor\n\\begin{equation}\nE_\\nu = 2{\\bar e}-g \\Omega\n-2 g \\sum_{\\mu(\\ne\\nu)=1}^n\\frac{E_\\nu}{E_\\mu-E_\\nu} \n+4 g {\\bar e} \\sum_{\\mu(\\ne\\nu)=1}^n\\frac{1}{E_\\mu-E_\\nu}~,\n\\label{Rich1Lb}\n\\end{equation}\nwhich, upon summation over the index $\\nu$, immediately yields the\nwell-known expression for the ground state energy\n\\begin{equation}\n{\\cal E}=\\sum_{\\nu=1}^n E_\\nu=2{\\bar e}n-g n \\left(\\Omega-n+1\\right) ~.\n\\label{Egs}\n\\end{equation}\nNote that in the above\nthe term quadratic in $n$ has a positive sign, \nreflecting the action of the Pauli principle.\n\nIn the $L=1$ case it is also easy to get \nthe excitation spectrum of the Hamiltonian\n\\eqref{HP} which can only correspond to the breaking of pairs.\nIndeed this occurrence can be formally accounted for by changing $\\Omega$ into \n$\\Omega-2 s$ and $n$ into $n-s$, \n$s$ (the so-called pair seniority) counting the number of broken \npairs.\nOne ends up with the remarkably symmetric expression\n\\begin{equation}\n{\\cal E}_s=\\sum_{\\nu=1}^n E_\\nu=2{\\bar e}n-g n \\left(\\Omega-n+1\\right) \n+g s \\left(\\Omega-s+1\\right) ~,\n\\label{Es}\n\\end{equation}\nwhich yields $2{\\bar e}n$, as it should, for $s=n$, namely when all pairs are\nbroken, thus becoming blind to the pairing interaction.\n\n\\section{The path integral approach in the degenerate case}\n\nThe equations \\eqref{Rich}, while exact, do not transparently convey the\noccurrence of the spontaneous symmetry breaking taking place in the system.\n\nTo shed light on this point it turns out that a path integral approach,\nwhile difficult, is more suited.\nThis scheme has been followed in Ref.~\\cite{Barbaro:2004nk} starting from the \ndiscretized Euclidean action corresponding to the Hamiltonian \\eqref{HP}, which in the\n$L=1$ case reads~\\cite{Negele}\n\\begin{equation}\nS=\\tau\\sum_{t=-N_0\/2}^{N_0\/2} \\left\\{\n-g \\Omega {\\bar A}(t) A(t-1) +\n\\sum_{m=-j}^j \\left[ {\\bar\\lambda_m}(t) \\left( \\nabla^+_t+{\\bar e}\\right)\n\\lambda_m(t-1) \\right] \\right\\} ~,\n\\label{S}\n\\end{equation}\n$\\tau$ being the time spacing, $N_0$ the number of points on the time \nlattice, the energy ${\\bar e}$ is measured with respect to the chemical potential to \nselect a sector of $N$ fermions\nand $\\lambda$, $\\bar\\lambda$ are odd Grassmann variables.\nIn terms of the latter\n\\begin{equation}\nA=\\frac{1}{\\sqrt{\\Omega}}\n\\sum_{m>0} (-1)^{j-m} \\lambda_{-m}\\lambda_{m}~,\n\\end{equation}\n\\begin{equation}\n{\\bar A}=\\frac{1}{\\sqrt{\\Omega}}\n\\sum_{m>0} (-1)^{j-m} {\\bar\\lambda_{m}}{\\bar\\lambda_{-m}}\n\\end{equation}\nand the discretized time derivative is defined as follows\n\\begin{equation}\n\\left( \\nabla^\\pm_t f\\right)(t) = \\pm\\frac{1}{\\tau}\n\\left[f(t\\pm 1)-f(t)\\right] ~.\n\\end{equation}\nNext the action \\eqref{S} is dealt with through the Hubbard-Stratonovitch\ntransformation by introducing a bosonic field $\\eta$ cast in the\npolar representation:\n\\begin{equation}\n\\eta = \\sqrt{\\rho} e^{2 i \\theta}\n\\ ,\\ \\ \\ \\ \\ \\ \\ \\\n{\\bar\\eta} = \\sqrt{\\rho} e^{-2 i \\theta} ~.\n\\label{eta}\n\\end{equation}\nIn the end the following effective action, equivalent to \\eqref{S}\nand describing the same physics of the Hamiltonian \\eqref{HP},\n\\begin{equation}\nS_{\\rm eff} = \\tau\\sum_t g\\rho - {\\rm Tr} \\ln \\left(-q^- q^+ + g^2\\rho\\right)\n\\label{Seff}\n\\end{equation}\nis obtained.\nIn \\eqref{Seff} the trace is taken both on the time and on the quantum number $m$\nand\n\\begin{equation}\nq^\\pm = e^{\\mp i \\theta} \\nabla^{\\pm}_t e^{\\pm i \\theta} \\pm {\\bar e} ~.\n\\end{equation}\nNow in the above we see that the field $\\theta$\n\\begin{itemize}\n\\item[a)] appears in the action only under derivative,\n\\item[b)] lives in the coset space of the broken group $U(1)$ with\nrespect to the unbroken group $Z_2$.\nThis follows from the requirement of making the change of variable\n\\eqref{eta} to be one to one. Indeed for this to occur it must be\n$0\\leq\\theta<\\pi$.\n\\item[c)]Moreover, at variance with the Hamiltonian \\eqref{HP}, the $U(1)$\nsymmetry\nis now non-linearly realized in the invariance of the action \\eqref{Seff}\nunder the substitution\n$\n\\theta\\to\\theta+\\alpha ~,\n$\n $\\alpha$ being time independent.\n\\end{itemize}\nThe above features characterize a Goldstone field (see \\cite{Weinberg}).\n\nThese findings show that in the degenerate case the system lives in a regime \nwhere the $U(1)$ symmetry\nis spontaneously broken not only at the macroscopic, but at the microscopic\nscale as well.\n\nAccording to the Goldstone theorem when a continuous symmetry is broken\na bosonic field appears with a dispersion relation which vanishes in \nthe thermodynamic limit.\n\nOf course\nin a finite system, like a deformed nucleus, where the $O(3)$ symmetry \nis spontaneously broken , the Goldstone boson shows up as a rotational \nband whose frequency is not vanishing, but, however, \ntends to zero as the system becomes larger and larger.\nLikewise in the pairing problem the $U(1)$ symmetry is broken and in the \ndegenerate case the energy of the associated Goldstone mode depends upon \nthe pair number $n$ according to \\eqref{Egs}.\nActually setting (see \\cite{Barbaro:2004nk})\n\\begin{equation}\n\\nu=n-[n_0] ~,\n\\end{equation}\nwhere the square bracket means integer part and\n\\begin{equation}\nn_0 = \\frac{\\Omega+1}{2}-\\frac{\\bar e}{g} \n\\end{equation}\nis the value of $n$ where \\eqref{Egs} reaches its minimum, \none sees that the energy of the Goldstone mode \ngoes like $g$, whereas the energy associated with the variable $s$, hence\ncorresponding to the breaking of pairs, goes like $g\\Omega$, thus lying\nat a much higher energy.\nMicroscopically the Goldstone mode\ncorresponds to the addition or to the removal of a pair of fermions.\n\nIn the path integral framework another point is of relevance.\nIn dealing with $S_{\\rm eff}$ with the saddle point expansion one starts by \nsearching for its minimum at constant fields.\nIt turns then out that {\\em the minimum of the effective action} occurs for \n$\\rho=\\bar\\rho$, being~\\cite{Barbaro:2005tp}\n\\begin{equation}\n{\\bar\\rho} = \\frac{1}{(2 g)^2} \\left[\n(g\\Omega)^2-4{\\bar e}^2\\right] = \\frac{\\Delta^2}{g^2} ~,\n\\label{rhobar}\n\\end{equation}\nwhere $\\Delta$, see \\eqref{Delta}, is the well-known gap characterizing the BCS\ntheory in the $L=1$ case. Thus $\\bar\\rho$, but for a factor which\nrenders it dimensionless, coincides with the gap.\nFurthermore, choosing $\\bar e=0$, \\eqref{rhobar} yields $g\\Omega=2\\Delta$.\n\n\n\n\\section{The Richardson approach in the multilevel case}\n\nWe now study the case when $L$ single particle levels are active.\nOur aim here is \nto ascertain how their presence \nmodifies the previous results obtained in the degenerate case, \nspecifically whether the system still lives in the phase where\nthe $U(1)$ symmetry is spontaneously broken.\nIn this connection it helps, as we shall see,\nto derive an\nexact analytic expansion of the lowest eigenvalue of \\eqref{HP} in the\ninverse powers of the coupling constant $g$. \nThis item has been lately pursued by Yuzbashyan et al.~\\cite{Yuz} in the\nframework of the Richardson equations.\nWe also mention that this \nproblem has been recently tackled in the path \nintegral formalism as well~\\cite{Pal}.\nWe shall do the same here, but\nproviding an expression for the expansion coefficients \nvalid for \\emph{any} single particle level (s.p.l.)\ndistribution and\nmore transparently\nlinked to the moments characterizing the latter.\nMoreover new results concerning\n${\\overline g}$, namely the value of the coupling constant defining the domain \nwhere the expansion holds, will be presented.\n\n\\begin{figure}[t]\n\\vspace{-2cm}\n\\begin{minipage}[b]{0.5\\textwidth}\n\\centering\n\\includegraphics[scale=0.7,clip,angle=0]{fig1a.eps}%\n\\end{minipage}\n\\begin{minipage}[b]{0.5\\textwidth}\n\\centering\n\\includegraphics[scale=0.7,clip,angle=0]{fig1b.eps}%\n\\end{minipage}\n\\begin{minipage}[b]{0.5\\textwidth}\n\\centering\n\\includegraphics[scale=0.7,clip,angle=0]{fig1c.eps}%\n\\end{minipage}\n\\begin{minipage}[b]{0.5\\textwidth}\n\\centering\n\\includegraphics[scale=0.7,clip,angle=0]{fig1d.eps}%\n\\end{minipage}\n\\vspace{-3cm}\n{\\caption{\nThe real part of the exact ground state solutions $E_\\nu$ (thick solid curves) \nof the Richardson's equations (6) for a \nsystem of $n$=2,6,8 and 12 pairs living in a set of \nL=12 equispaced non-degenerate ($\\Omega_k=1$) single particle levels with\nenergies $e_k=k$ ($k=0,\\cdots 11$) are displayed versus \nthe strength $g$ together with their sum ${\\cal E}(n)$\n(thin solid curves); the dotted curves represent the shifted total energy\n${\\cal E}(n)-2\\sum_{k=1}^n e_k$ and the dashed curves the energy of $n$ \npairs living on one level of energy $e=0$ and degeneracy $L$.\nThe horizontal dashed lines represent the unperturbed solutions and \nthe cuspids are due to numerical imprecision close to the critical points.\nThese could be avoided by employing the Rombouts method \\cite{Rom}.\n}\\label{fig1}} \n\\end{figure}\n\n\\subsection{Comparison with the degenerate case}\nWe consider the $g$-dependence of the ground state energy to \nassess how it is affected by the presence of the $L$ single particle levels. \nFor sake of illustration in this Subsection \nwe confine ourselves to explore a model of equispaced\nsingle \nparticle levels with unit energy distance and\nwith unit pair degeneracy ($\\Omega_j=1$). In Fig.~\\ref{fig1} we display the \nRichardson solution for $L=\\Omega=12$, for $n$=2, 6 (half filling), 8, \n12 (full filling). The energy of the degenerate case \\eqref{Egs} is also \nshown as a reference.\nMoreover the $g$-behavior of the energy of each pair is displayed.\n\nIt is clearly apparent in\nFig.~\\ref{fig1} the merging mechanism which allows the escaping of the\n pair energies from the grid of the single particle energies from\nwhere they originate at $g=0$ (see\nRefs.~\\cite{Roman:2002dh,Barbaro:2003bb}).\nIn fact, considering the ground state, the pair energies start out, \nfrom the lowest bare single particle energies, being real at small $g$ \nand then, two by two,\n{\\em merge} at the energy of a single particle level (identified by the\nsuperscript $i$) for critical values of the coupling $g_{\\rm crit}^i$:\nclearly $[n\/2]$ of these exist. \nFor $g>g_{\\rm crit}^i$ the two pair energies associated\nwith the index $i$ become complex conjugate and, what is important,\ntheir common real part behaves almost {\\em linearly in $g$}.\nThus when $g>g_{\\rm crit}^{\\rm max}$ all the pair energies become complex \nwith real parts displaying a behavior close to linear in $g$ and, \nremarkably, they add up to yield a \ndownwards pointing straight line with the same slope as \\eqref{Egs}, \nbut upwardly shifted with respect to this one because of the contribution \nstemming from the single particle energies, which is obviously \n$g$-independent. \nOne can accordingly argue that when $L$ single particle levels are active, \nthe system, at variance with the degenerate case, can live in different \nregimes.\nThese in turn depend upon the filling of the levels.\nHere we consider the half filling situation where \nthe system can live in three different regimes \naccording to whether $g<g_{\\rm crit}^{\\rm min}$ (normal fermionic regime, where the\nwave function of each pair has essentially only one component), \n$g>g_{\\rm crit}^{\\rm max}$ (superfluid regime, where\nthe wave function of each pair is spread out over all the $L$ single particle \nunperturbed levels) \nand $g_{\\rm crit}^{\\rm min}<g<g_{\\rm crit}^{\\rm max}$ (mixed regime).\n\nFrom the figure it also appears that \nthe energy of the degenerate and non-degenerate case are markedly\ndifferent only for $g<g_{\\rm crit}^{\\rm max}$.\nHowever this difference becomes less and \nless pronounced as the ratio $n\/\\Omega$ approaches unity (full filling).\nIndeed for $g<g_{\\rm crit}^{\\rm max}$, but $n$ \nsufficiently large, the non-linear behaviors of the exact $E_\\nu(g)$ \ncancel out leading to a system's total energy linear in $g$ as in \nthe degenerate case. \nThis occurrence relates to the absence of an excitation spectrum \nin the full filling situation in both the degenerate and non-degenerate cases.\n\n\nIt is interesting to check whether the above findings agree with the Anderson\ncriterion~\\cite{Anderson} for establishing how small a superconductor can be.\nThis states that in the superconducting phase it must be\n\\begin{equation}\n\\frac{\\Delta}{d} > 1~,\n\\label{And}\n\\end{equation}\n$d$ being the average distance between the levels of the finite system and\n$\\Delta$ the gap.\nNow by combining the above with \\eqref{rhobar} it follows that a \nsuperconducting regime sets in for\n\\begin{equation}\ng>g_{\\rm crit}^{\\rm max} = \\frac{2}{\\Omega} \\sqrt{{\\bar e}^2+d^2}~,\n\\label{ggcr}\n\\end{equation}\nwhich indeed appears well fulfilled in our numerical analysis\n(which assumes $d=1$, $\\Omega=L$ and $\\bar e$ given by \\eqref{mom}).\nHence \\eqref{And} holds valid in our model.\n\nFinally a comment on the above recalled escaping mechanism, responsible for\nthe setting up of a strongly collective ground state, is in order.\n\nIndeed the way the pair energies evade the grid \nchanges its nature when the number of pairs substantially exceeds the \nhalf filling of the single particle levels. This is\nclearly apparent in the last panel of Fig.~\\ref{fig1} (note that there a larger\nspan of $g$ is\ndisplayed), where the energies of the four upper pairs are seen to grow rather \nthan to decrease with $g$.\n\nIn fact the pair energies are differently affected by the pairing\nforce. In particular the lowest lying pairs feel an attractive interaction\nwhereas the upper lying ones feel a repulsive force. Since the sum of the pair \nenergies is constrained to yield the degenerate result, it is not surprising \nthat when $n$ is large the upper lying \npairs contribute to the total system's energy positively and not negatively.\nIn other words the linear behavior in $g$ of their energies \nhas a positive (and not a negative) slope.\n\n\\subsection{The expansion of the ground state energy}\n\nWe now turn to derive an exact analytic expression for the \n$1\/g$ expansion for the system ground state energy \nwhen $L$ s.p.l. and $n$ pairs are active. \nThis we achieve along the lines illustrated in Ref.~\\cite{Yuz}, but \nwe prefer to express the coefficients of \nthe expansion in terms of the mean energy \n\\begin{equation} \\label{emedia}\n\\bar e=\\frac{1}{\\Omega}\\sum_{\\mu=1}^L \\Omega_\\mu e_\\mu\n\\end{equation}\nand of the moments\n\\begin{equation}\\label{mom}\nm^{(k)} = \\frac{1}{\\Omega} \\sum_{\\mu=1}^L \\Omega_\\mu(e_\\mu-{\\bar e})^k\n\\end{equation}\nof the s.p.l. distribution\n(the first of these, namely when\n$k=2,3$ and $4$, correspond to the variance, skewness and kurtosis).\nIn the above $\\Omega=\\sum_{\\mu=1}^L\\Omega_\\mu$ is the total pair degeneracy. \nThis allows to better grasp the impact of s.p.l. \non the spectrum of the system.\n\nWe obtain \n\\begin{equation}\n{\\cal E}_{gs}(n) =\n2 n {\\bar e} \n-g n (\\Omega-n+1)\n- d \\sum_{j=1}^\\infty \\alpha_j \\left(\\frac{d}{g}\\right)^{j}.\n\\label{expansion}\n\\end{equation}\nIn (\\ref{expansion}) $d$ is the average distance between the single particle levels and\n\\begin{equation}\n\\alpha_j=\\Omega\\sum_{p=0}^j 2^{p+1} \\ a_{p+1}^{j-p} \n\\left\\{\nm^{(p+1)}+\\sum_{l=0}^p {p+1\\choose l} {\\bar e}^{p+1-l} m^{(l)}\n\\right\\}~.\n\\label{alpha}\n\\end{equation}\nThe coefficients $a$ in \\eqref{alpha} are related through the following\nrecurrence relations\n\\begin{equation}\na_{p+1}^{q} = -\\frac{1}{\\Omega-p}\n\\left\\{\n\\frac{a_p^{q}}{d}+\n(1-\\delta_{p0}) \\sum_{k=1}^p \\sum_{s=0}^{q} a_{p+1-k}^{q-s} \\ a_k^s\n+\\Omega (1-\\delta_{q0}) \\sum_{k=1}^{q} 2^k \\ a_{p+k+1}^{q-k}\\ \n\\sum_{l=0}^k {k\\choose l} {\\bar e}^{k-l} m^{(l)}\n\\right\\}~,\n\\label{recurrence}\n\\end{equation}\nwhich allows to evaluate the expansion \\eqref{expansion} at any \ngiven order starting from $a_0^k=n \\delta_{k0}$. \n\nEq.~\\eqref{expansion} represents, when it converges, \nthe Laurent expansion of\n${\\cal E}_{gs}$ around the simple pole it displays at $g=\\infty$.\nThe first two terms of the expansion just correspond to the ground state\nenergy of the degenerate case.\nThe other terms thus account for the role of the s.p.l. distribution: indeed\nthe $n$-th order in $1\/g$ involves the moments of the s.p.l. \ndistribution up to the order $n+1$. \nClearly the smaller $g$ is, the more moments are needed to get a faithful\nrepresentation of ${\\cal E}_{gs}(n)$.\nThis is supported by all the cases we have numerically explored.\n\nKeeping the first five terms only one gets (note that our \nterm $1\/g^2$ differs from the one in Ref.~\\cite{Yuz})\n\\begin{eqnarray}\n{\\cal E}_{gs}(n) &=&\n2 n {\\bar e} \n-g n (\\Omega-n+1)\n-2\\frac{m^{(2)}}{(g\\Omega\/2)}\\times\\frac{n(\\Omega-n)}{\\Omega-1}\n+2\\frac{m^{(3)}}{(g\\Omega\/2)^2}\\times\n\\frac{n(\\Omega-n)(\\Omega-2n)}{(\\Omega-1)(\\Omega-2)}\n\\nonumber\\\\\n&-&\\frac{2}{(g\\Omega\/2)^3}\\times\n\\frac{n(\\Omega-n)}{(\\Omega-1)^2(\\Omega-2)(\\Omega-3)}\n\\left\\{\n\\left[\\Omega^2(\\Omega-1)-n(\\Omega-n)(5\\Omega-6)\\right] m^{(4)} \n\\right.\n\\nonumber\\\\\n&-&\\left.\\left[\\Omega^2(2\\Omega-3)-n(\\Omega-n)\n\\frac{3(\\Omega-1)(3\\Omega-4)-\\Omega}{\\Omega-1}\\right] \\left[m^{(2)}\\right]^2\n\\right\\}.\n\\label{EgsY}\n\\end{eqnarray}\n\nFor one pair \\eqref{EgsY} yields back the result of\nRef.~\\cite{Barbaro:2002fi}, whereas when all the levels are filled\n($n=\\Omega$) it yields back \\eqref{Egs},\ncharacterized by\nthe linear $g$-behavior \ndisplayed in Fig.~\\ref{fig1}. \nMoreover, when the levels are equispaced \nonly even moments, hence odd powers of $1\/g$, enter into the expansion.\n\nIn Fig.~\\ref{fig2}, we compare the $n$-behavior of the ground state energy of \nthe system as obtained in the Richardson's framework and from \\eqref{EgsY}.\nThey are both displayed for a few values of $g$ assuming \n$L=12$ and the same set of equispaced single particle energies as before.\nFrom these results one can gauge the validity of the expansion \\eqref{expansion} \neven when only a few terms of the latter are kept.\nIt is \ngratifying that few moments of the level distribution are enough to get \na good representation of the true solution, providing $g$ is not too close to\nthe boundary of the domain of convergence of the expansion.\n\n\\begin{figure}[hbt]\n\\hspace{-2cm}\n\\begin{minipage}[t]{0.5\\textwidth}\n\\centering\n\\includegraphics[scale=0.7,clip,angle=0]{fig2a.eps}%\n\\end{minipage}\n\\begin{minipage}[t]{0.5\\textwidth}\n\\includegraphics[scale=0.7,clip,angle=0]{fig2b.eps}\n\\end{minipage}\n\\vspace{-1cm}\n{\\caption{Left panel: the exact (solid curves) ground state energy for the same model \nas in Fig.~\\ref{fig1} is displayed versus the pair number $n$ for $g$=1 (upper curve), \n2, 4 and 6 (lower curve), respectively; \nthe dashed curves (visible only in the $g$=1 case) correspond to the \napproximate solution of Eq.~(29). Right panel: the same curves divided by $g$;\nfor very large $g$ the curves tend to the parabola $-n(L-n+1)$.\n}\\label{fig2}} \n\\end{figure}\n\nAccordingly we address the problem of establishing \nthe domain of convergence of the expansion \n\\eqref{expansion}. To explore this issue we first \nconsider the half filling situation in the following\nlimiting cases:\nthe macroscopic limit and when only one pair\nlives in two levels. \nIn the former\nwe assume that all the unperturbed levels have unit pair degeneracy \nand live in \na finite energy band of width $2\\hbar\\omega_0=2dn=d\\Omega$, in such a way that\nwhen $g\\to 0$ the ratio $g\/d$ remains finite.\nWe then obtain for the condensation energy\n\\begin{eqnarray}\n{\\cal E}_{gs}^{cond}(n) &=&{\\cal E}_{gs}(n)-2n\\overline e=\n-n\\hbar\\omega_0\\left\\{\\frac{g}{d}+\\frac{1}{3}\\frac{d}{g}-\\frac{1}{45}\\left(\\frac{d}{g}\\right)^3\n+\\cdots \\right\\}\n\\label{EgsY_TL}\n\\end{eqnarray}\nwhich coincides, when it converges, namely for \n\\begin{equation}\n\\frac{g}{d} > \\frac{\\overline g}{d} = 1\/\\pi~, \n\\end{equation}\nwith the Laurent $d\/g$-expansion \nof the BCS energy\n\\begin{eqnarray}\n{\\cal E}_{BCS}^{cond}(n) &=&\n-n\\hbar\\omega_0\\coth\\left(\\frac{d}{g}\\right)\n\\label{EgsY_BCS}\n\\end{eqnarray} \nin accord with\nthe long known result that the BCS energy cannot be\ncomputed perturbatively for small $g\/d$.\n\nOn the other hand the macroscopic limit of the Richardson solution\nhas been explored in Ref.~\\cite{Roman:2002dh}, where it is shown that\nin this instance\n\\begin{eqnarray}\n \\left(\\frac{g}{d}\\right)^{\\rm max}_{\\rm crit}=\\frac{1}{\\rm{arcsinh}(1)}=1.1345\n\\qquad\\textrm{and}\\qquad\\left(\\frac{g}{d}\\right)^{\\rm min}_{\\rm crit}=0.\n\\label{gmax}\n\\end{eqnarray}\n\n\nThus the domain of validity of the expansion \\eqref{EgsY_TL} \ndoes not directly relate to the critical values of the coupling constant:\nindeed although these correspond to branch points of the pair energies \n\\cite{Rich} these singularities cancel out in the expression\nfor the system's global energy.\n\nTurning to the case of one pair living in two levels\nwith unit degeneracy and having an energy distance $d$ from \nEq.~(\\ref{EgsY}) one gets\n\\begin{equation} \\label{eqa}\n{\\cal E}=-g -g\\left(1+\\frac{1}{2}\\left(\\frac{d}{g}\\right)^2-\\frac{1}{8}\\left(\\frac{d}{g}\\right)^4\n+\\frac{1}{16}\\left(\\frac{d}{g}\\right)^6+\\cdots\\right)\n\\end{equation}\nwhich coincides, when it converges, namely for\n\\begin{equation} \\label{eqb}\n\\frac{g}{d} > \\frac{\\overline g}{d} = 1~, \n\\end{equation}\nwith the well-known result \n\\begin{equation}\\label{eqc}\n{\\cal E}=-g-\\sqrt{d^2+g^2}~.\n\\end{equation}\nOf course the range (\\ref{eqb}) is not related to critical values\nof $g$ which are now non existent. Note that this most simple case \nalready shows that a $\\overline g$ can exist without any $g_{\\rm{crit}}$.\nThis occurrence goes in parallel with the findings of \\cite{Rin} where \nthe same is seen to happen in a much more involved situation.\n\nThus for both systems above examined ($n=1$ and $n=\\infty$ at \nhalf filling) the function ${\\cal E}_{gs}(g)$ displays a singularity\n(either a branch point or a pole)\non the imaginary axis of the complex $g$-plane. By contrast\nthe branch points associated with the $g_{\\rm crit}$ all lie\non the real axis.\n\nTo gain insight on the actual value of $\\frac{\\overline g}{d}$ \nin a generic system we quote \nin Tables $1$ and $2$ the values of \n$\\left(\\frac{g}{d}\\right)_{\\rm crit}^{\\rm min}$,\n$\\left(\\frac{g}{d}\\right)_{\\rm crit}^{\\rm max}$ \nand\nof \n$\\frac{\\overline g}{d}$ for a few values of $n$.\nThe radius of convergence $\\frac{\\overline g}{d}$ \nof the expansion has been computed using\nthe Cauchy-Hadamard criterion\n\\begin{equation}\n\\frac{{\\overline g}}{d} \n= \\lim_{j\\to\\infty}|\\alpha_j|^{1\/j}~.\n\\end{equation}\nTo assess the sensitivity to $j$ of the above in Table $2$\nwe report in the first column the results obtained with $j=21$\nand in the second column those obtained (at least in a few cases)\nwith $j=199$.\n\nFrom our results it follows that \nthe radius of convergence $\\frac{\\overline g}{d}$ of\nthe expansion \\eqref{expansion} for any $n$ appears to occur, \nfor the half-filling case and for equispaced s.p.l. of unit pair\ndegeneracy,\nin the domain ranging from \n$\\left(\\frac{g}{d}\\right)_{\\rm crit}^{\\rm min}$\nto\n$\\left(\\frac{g}{d}\\right)_{\\rm crit}^{\\rm max}$.\nThis interval increases with the pair number $n$, \nas it can be inferred from Table 1, where\n$\\left(\\frac{g}{d}\\right)_{\\rm crit}^{\\rm max}$\n($\\left(\\frac{g}{d}\\right)_{\\rm crit}^{\\rm min}$)\nis seen to increase (decrease) with $n$.\n\nWe conclude that $\\frac{{\\overline g}}{d}$ \n\\begin{itemize}\n\\item[a)] varies in the range $\\frac{1}{\\pi}\\leq\\frac{{\\overline g}}{d}\\leq 1$,\nfor any $n$,\n\\item[b)] decreases with $n$,\n\\item[c)] lies, for a given $n$, in the range\n$\n\\left(\\frac{g}{d}\\right)^{\\rm min}_{\\rm crit}\n\\leq\\frac{{\\overline g}}{d}\\leq \n\\left(\\frac{g}{d}\\right)^{\\rm max}_{\\rm crit}\n$ .\n\\end{itemize}\nNote that $\\frac{{\\overline g}}{d}$\n, even for $n=500$, is still far from $1\/\\pi$\n: hence the macroscopic limit of the pairing Hamiltonian\nappears to be reached very slowly.\n\n\\begin{table}[ht]\n\\begin{center}\n\\begin{tabular}{||c|c|c||}\n\\hline\\hline \\raisebox{0pt}[14pt]{} \\raisebox{-7pt}[16pt]{} \n$n$ & $\\left(\\frac{g}{d}\\right)^{\\rm min}_{\\rm crit}$\n& $\\left(\\frac{g}{d}\\right)^{\\rm max}_{\\rm crit}$\n\\\\\n\\hline\\hline\n 2 & 0.66 & - \\\\ \n 4 & 0.46 & 0.82 \\\\ \n 6 & 0.40 & 0.90 \\\\ \n 8 & 0.35 & 0.94 \\\\ \n 10 & 0.32 & 0.96 \\\\ \n 20 & 0.27 & 1.03 \\\\\n $\\infty$ & 0 & 1.13\n\\\\ \\hline\\hline \n\\end{tabular} \n\\caption{For a few values of the pair number $n$ the largest and smallest\ncritical values of the coupling constant are quoted at half filling.\nThe number of critical points is $n\/2$ and clearly for $n=2$ only one\nof them exists.\n}\n\\label{tab:1}\n\\end{center} \n\\end{table} \n\n\\begin{table}[ht]\n\\begin{center}\n\\begin{tabular}{||c|c|c||}\n\\hline\\hline \\raisebox{0pt}[14pt]{} \\raisebox{-7pt}[16pt]{} \n$n$ & $|\\alpha_{21}|^{1\/21}$& $|\\alpha_{199}|^{1\/199}\n\\\\\n\\hline\\hline\n 1 & 0.82 & 0.96\\\\\n 2 & 0.69 &\\\\ \n 50 & 0.41&\\\\\n 100& 0.39 & 0.65\\\\\n 500& 0.36 & 0.64\\\\ \n $\\infty$& 0.31&$0.318\\simeq1\/\\pi$\n\\\\ \\hline\\hline \n\\end{tabular} \n\\caption{The radius of convergence of the expansion \\eqref{expansion} \nfor various\npair numbers $n$ at half-filling and equispaced single particle levels\nof unit pair degeneracy.\n}\n\\label{tab:2}\n\\end{center} \n\\end{table} \n\n\n\\begin{table}[ht]\n\\begin{center}\n\\begin{tabular}{||c|c||}\n\\hline\\hline \\raisebox{0pt}[14pt]{} \\raisebox{-7pt}[16pt]{} \n$n$ & $|\\alpha_{199}|^{1\/199}$\n\\\\\n\\hline\\hline\n 1 & 0.0096 \\\\\n 50 & 0.0077\\\\ \n 100 & 0.0068\\\\\n 150 & 0.0077\\\\\n 199 & 0.0096\\\\ \n 200 & 0\\\\ \n \\hline\\hline \n\\end{tabular} \n\\caption{Values of $\\frac{\\overline g}{d}$ in the case of two levels \nwith equal pair degeneracy $\\Omega_1=\\Omega_2=100$ for a few values of n.\n}\n\\label{tab:3}\n\\end{center} \n\\end{table} \n\n\nWe now abandon the assumption of s.p.l. of unit pair degeneracy and\naddress the question of the impact on $\\frac{\\overline g}{d}$\nof the pair degeneracy of the s.p.l.. In the simple case of one \npair in two levels the generalization of (\\ref{eqc}) is easily found to read\n\\begin{equation}\\label{A1}\n{\\cal E}=(e_1+e_2)-g\\frac{\\Omega_1+\\Omega_2}{2}-\\sqrt{\n\\left[g\\left(\\frac{\\Omega_1+\\Omega_2}{2}\\right)\\right]^2+d^2+gd(\\Omega_2-\\Omega_1)\n}\n\\end{equation}\nwhich yields \n\\begin{equation}\\label{A2}\n\\frac{\\overline g}{d}=2\\left|\\frac{\\Omega_1-\\Omega_2\\pm 2i\\sqrt{\\Omega_1\\Omega_2}}\n{(\\Omega_1+\\Omega_2)^2}\\right|=\\frac{2}{\\Omega_1+\\Omega_2},\n\\end{equation}\nbeing $d=e_1-e_2$, from where it is seen that the singularity\nmoves away from the imaginary axis when $\\Omega_1\\neq\\Omega_2$.\n\nSticking to the case $\\Omega_1=\\Omega_2=\\Omega$ one gets from\n(\\ref{A2})\n\\begin{equation}\n\\frac{\\overline g}{d}=\\left|\\frac{i}{\\Omega}\\right|\n\\end{equation}\nand from (\\ref{A1}) (choosing $e_1+e_2=0$) \n\\begin{equation}\n{\\cal E}=-g\\Omega-\\sqrt{d^2+g^2\\Omega^2}\n\\end{equation}\nwhich coincides with (\\ref{eqc}) when $\\Omega=1$.\nThus the larger $\\Omega$ is, the closer to the real axis the branch \npoint is.\nThis occurrence has a bearance on the physics occurring on \nthe real axis: in fact not only the energy but also the pair wave function\nis strongly modified by $\\Omega$, in the sense that the larger $\\Omega$ is the sooner in $g$ the two components of \\eqref{B} become equally weighted.\n\nIn the limit $\\Omega\\rightarrow\\infty$ where the pair\nbecomes a true boson the singularity \\emph{lies} on the real axis.\n\nTo assess the impact of the s.p.l. filling on $\\overline g\/d$ \nwe quote in Table 3 the\nnumerical values of the latter\nin the case of two levels with \nequal pair degeneracy $\\Omega_1=\\Omega_2=100$ for a few values of $n$.\nFrom Table 3 one sees that $\\overline g\/d$ \nreaches its minimum at half filling being symmetric around the minimum. \n\nFinally we give a striking example illustrating the crucial role \nplayed by the s.p.l. degeneracy on $\\overline g\/d$. For this purpose \nwe consider the two following cases \n\\begin{itemize}\n\\item [a)] $L=2$, $\\Omega_1=\\Omega_2=100$, $\\Omega_{\\rm Tot}=200$, $d=1$, $n=100$\n\\item [and]\n\\item [b)] $L=200$, $\\Omega_1=\\Omega_2=\\cdots=1$, $\\Omega_{\\rm Tot}=200$, $d=1\/199$, \n$n=100$.\n\\end{itemize}\n\nThen from the explicit computation\nit turns out that in a) $\\overline g\/d=0.0068$ \n(see Table 3) whereas in b) \n$\\overline g\/d=0.65$. Thus even in situations where analytical results \nare hard to achieve, the numerical analysis suggests that \n$\\overline g\/d\\propto1\/\\tilde\\Omega$, being $\\tilde\\Omega$\nthe degeneracy of each s.p.l.. \n \n\nIn concluding this Section we return to the problem of the\nthermodynamic limit previously mentioned pointing out that\ntwo quite different\nvariational solutions, namely the Richardson \\eqref{n} and the BCS \\eqref{BCS} \nprojected on a given particle number $M$ which reads \n\\begin{equation}\nP_M|BCS>=\\left(\\sum_k\\frac{v_k}{u_k}\\hat c^\\dagger_{\\vec k\\uparrow}\n\\hat c^\\dagger_{-\\vec k\\downarrow}\\right)^M,\n\\end{equation}\nyield the same ground state energy. \nWhether this remains true for other observables as well is an issue \nworth to be further explored.\n\n\\section{The spectrum of the pairing Hamiltonian}\nIn this Section we shortly examine the spectrum of the pairing Hamiltonian \nin the non-degenerate situation where (\\ref{HP}) predicts new\nexcited states (referred to as Gaudin excitations) beyond those associated\nwith the breaking of pairs (referred to as seniority excitations), which are\nthe only ones present in the degenerate case.\n\nThese new kind of excited states correspond to the raising of pairs into\nhigher-lying s.p.l. and are classified in terms of the\nGaudin's number $N_G$~\\cite{Gau95}, which yields the number of pair \nenergies staying finite when $g$ goes to infinity because they remain\ntrapped in the grid of the s.p.l.. \nFor fixed $N_G$ (of course $0\\leq N_G\\leq n$), the number of such excited states\nis ${\\Omega\\choose N_G}-{\\Omega\\choose N_G-1}$.\n\nAs an illustration of these excitations\nwe consider the simple example of 3 pairs in 5 equispaced levels of unit \npair degeneracy. In this case there are\n4 states with $N_G=1$ and 5 with $N_G=2$ (note that $N_G=3$ is not allowed).\nIn Fig.~\\ref{fig3} we display the pair energies versus $g$ for the ground state \n(panel $a$) and for the $N_G=1$ states (panels $b$-$e$) together with \nthe associated total energies (panel $f$). \nNote that while for the ground state and for each of\nthe first three excited states only one $g_{\\rm crit}$ exists, for the fourth\n$N_G=1$ excited state (panel $e$) a complicated escaping mechanism takes place,\ngiving rise to three $g_{\\rm crit}$.\nIn the last panel, where the total energy of the above states is plotted, \nit clearly appears that, for $g$ high enough, the states tend to\ngroup according to their Gaudin number in accord with the finding of~\\cite{Yuz}.\n\\begin{figure}[t]\n\\vspace{-2cm}\n\\begin{minipage}[b]{0.5\\textwidth}\n\\centering\n\\includegraphics[scale=0.8,clip,angle=0]{3_5_GS_dashed_frame.eps}%\n\\end{minipage}\n\\begin{minipage}[b]{0.5\\textwidth}\n\\centering\n\\includegraphics[scale=0.8,clip,angle=0]{3_5_S0-2-6_dashed_frame.eps}%\n\\end{minipage}\n\\begin{minipage}[b]{0.5\\textwidth}\n\\centering\n\\includegraphics[scale=0.8,clip,angle=0]{3_5_S0-2-8_dashed_frame.eps}%\n\\end{minipage}\n\\begin{minipage}[b]{0.5\\textwidth}\n\\centering\n\\includegraphics[scale=0.8,clip,angle=0]{3_5_S2-4-6_dashed_frame.eps}%\n\\end{minipage}\n\\begin{minipage}[b]{0.5\\textwidth}\n\\centering\n\\includegraphics[scale=0.8,clip,angle=0]{3_5_S0-4-6_dashed_frame.eps}%\n\\end{minipage}\n\\begin{minipage}[b]{0.5\\textwidth}\n\\centering\n\\includegraphics[scale=0.8,clip,angle=0]{5LivelliCfr_frame.eps}%\n\\end{minipage}\n{\\caption{\nThe real part of the exact solutions $E_\\nu$ of the Richardson's equations (6) for a \nsystem of $n$=3 pairs living in a set of \nL=5 equispaced non-degenerate ($\\Omega_k=1$) single particle levels with\nenergies $e_k=k$ ($k=0,\\cdots 4$) are displayed versus \nthe strength $g$. Panel $a$: ground state; \nPanels $b$-$e$: excited states with Gaudin's number $N_G=1$;\npanel $f$: total energy of the ground state (lower\ncurve) and of the $N_G=1$ states (upper curves).\n}\\label{fig3}} \n\\end{figure}\n\n\nTurning to \nthe seniority excitations of \\eqref{HP},\ntheir number, for levels with unit pair degeneracies, is\n${\\Omega\\choose 2s}{\\Omega-2s\\choose n-s}$ (of course it must be \n$0\\leq s\\leq\\rm{min}\\{n,\\Omega-n\\}$).\n\nAgain for illustration we consider the very simple set of configurations\nassociated with 2 pairs living in 4 equispaced single particle \nlevels with unit energy distance and pair degeneracy. In this case 13\nseniority excited states occur, 12 with $s=1$ and 1 \nwith $s=2$. \nOf the former, owing to the degeneracy\nentailed by the symmetry of \\eqref{HP}, only 6 show up in Fig.~\\ref{fig4}a, \nwhere the $g$-behavior\nof the seniority eigenvalues is displayed.\nThese can be analytically computed since they are obtained\nas a solution of a two-levels problem, the other two levels being blocked\nby the broken pair. \nAssuming, e.g., that the blocked single particle levels correspond to \nthe energies\n$e_1$ and $e_2$ (the other cases obtain with trivial permutations) they\nread\n\\begin{equation}\n{\\cal E}(n=2,s=1) = e_1+e_2+e_3+e_4-g\n\\pm\\sqrt{g^2+(e_3-e_4)^2}~,\n\\end{equation}\nwith obvious meaning of the symbols.\nThe $\\pm$ sign accounts for the splitting of the $s=1$ states into \ntwo families. It turns out that the energy of the lower lying family \ncomes pretty close to the energy of the $N_G=1$, $s=0$ state (compare\nFig.~\\ref{fig4}b), whereas the energy of the higher lying family \ncomes close to the one of the $N_G=2$ state.\nMoreover these are almost degenerate with the $s=2$ state, corresponding\nto a configuration where all the four levels are blocked: hence their\nenergy is simply given by the sum of the unperturbed single particle energies.\n\nGeneralizing the above example \nwe argue that the excited states of the pairing \nHamiltonian are identified by two quantum numbers,\nnamely $N_G$ and $s$. However the associated eigenvalues \ngroup, as $g$ becomes large, into families characterized \nby the number $N_B$ of pairs blind to the pairing interaction: whether \nthey are so because they are broken or because they are trapped\nit does not matter for $g$ sufficiently large.\n\n\nFor the energies of these states,\ninspired by (\\eqref{Es}), we accordingly surmise\nthe following approximate expression\n\\begin{eqnarray}\n{\\cal E}(n,N_B)={\\cal E}_{g.s.}(n)-{\\cal E}_{g.s.}(N_B)+2 N_B {\\bar e}~,\n\\label{EgsYs}\n\\end{eqnarray}\nwhich we test against the exact solution in the $N_B=1$\ncase. Confining ourselves to \\eqref{EgsY} the above becomes\n\\begin{equation}\n{\\cal E}(n,N_B=1)-{\\cal E}(n,N_B=0) =\n2\\left[\n\\frac{g\\Omega}{2}+\\frac{m^{(2)}}{(g\\Omega\/2)}-\\frac{m^{(3)}}{(g\\Omega\/2)^2}\n+\\frac{m^{(4)}-2\\left[m^{(2)}\\right]^2}{(g\\Omega\/2)^3}\n\\right]\n\\label{Es1}\n\\end{equation}\nwhose first term,\nsignificantly, is equal to $2\\Delta$.\n\nNotwithstanding the shortcomings of formula (\\ref{EgsYs})\nyet we have found, as seen in Fig.~\\ref{fig4}c, that\nit provides a good account for the exact results, except at low $g$\nnamely in the domain of the critical values of the coupling\n(in our example these occur at 0.66, 1, 2 and 3 in \nunits of the spacing between the single particle levels).\nIn addition (\\ref{Es1}) shows that to leading order, which is the accuracy\nof formula (\\ref{Es1}), the Gaudin and the seniority \nexcitations\nare both characterized by the same gap $\\Omega g$. \n\n\\begin{figure}[ht]\n\\begin{center}\n\\includegraphics[scale=0.75,clip,angle=0]{fig4.ps}\n\\vspace{-4.cm}\n{\\caption{\na) Exact energies of the ground state (solid), \nseniority $s=1$ (dashed) and $s=2$ (dotted) states\nfor a system of 2 pairs in 4 equispaced single particle levels \nof energies\n0, 1, 2 and 3,\nrespectively, and pair degeneracy 1. The \n$s=2$ state trivially corresponds to $E=e_1+e_2+e_3+e_4=6$.\nb) Exact energies of the $s=0$ states with $N_G=0$ (solid), $N_G=1$ (dashed) \nand $N_G=2$ \n(dotted).\nc) Comparison between the $N_G=1$,\n$s=1$ states (dashed) and the $N_G=1$, $s=0$ (solid) excited states.\nThe crosses correspond to the approximate solution (37) for $N_G=1$.\n}\\label{fig4}} \n\\end{center} \n\\end{figure}\n\n\\section{Conclusions}\nThe problem of the pairing Hamiltonian, extensively treated in a number of papers,\nis still the object of many investigations. \nSuch an Hamiltonian employs a drastically simplified interaction\nand, as a consequence, cannot be viewed as a realistic one. \nIt yields, for example, a gap\ntoo large by a factor $(4e)^{1\/3}$ with respect \nto the accurate predictions of Gorkov ~\\cite{Gorkov} and Heiselberg \n~\\cite{Heiselberg}. Yet the fact of being exactly solvable \nmakes \\eqref{HP} interesting \nand helpful for the researches in a variety of fields. To mention a couple \nof these, we remind the microscopic \nderivation of the Arima-Iachello model and the physics of small \nsuperconducting metallic grains, which has lately attracted much interest.\n\n\nIn this work we have addressed a few aspects of the problem which, in our \nview, still deserve further studies. First we have proved in the path integral \nframework and in the degenerate case the superfluidity of the pairing \nHamiltonian solution in a ``finite'' system. This item was already lately \ntreated by us ~\\cite{Barbaro:2004nk}. \nHere, however, we have added the important \nfinding that the saddle point of the effective action derived in \n~\\cite{Barbaro:2004nk} is \nexactly fixed by the BCS gap.\n\nThen we have compared the $g$-behavior of the ground state energy in the \ndegenerate case with the situation when many single particle levels are \nactive. Although this comparison has been limited to levels with unit pair \ndegeneracy it has allowed us to grasp the impact of the s.p.l. \ndistribution on the system's energy and to get, at least in the half filling\nsituation, an orientation on\nthe regimes in which the system lives.\n\nNext we have derived an expansion of the system ground state energy \n${\\cal E}_{g.s.}(g)$ in the inverse of the coupling constant \n(strong coupling expansion). The formula we have obtained generalizes the one \nof Ref.~\\cite{Yuz} since it is expressed in terms of the statistical moments of the \ns.p.l. distribution and is thus valid for \\emph{any} distribution.\nThis appears to be useful for some recent researches on the density functional \napproach to the pairing Hamiltonian \\cite{Papenbrock}, but it is also helpful\nfor a deeper understanding of the dynamics of a system ruled by (\\ref{HP}).\nIndeed the investigation of the domain of validity of the expansion has lead us\nto conclude that such a domain is set by a singularity of the function ${\\cal E}(g)$\nlocated somewhere in the complex \n$g$-plane. Although we have not been able to find a general expression for this \nsingularity (indeed it depends from the specificity of each case)\nwe have numerically found where its modulus $\\overline g$ lies \nat least for s.p.l. with unit pair degeneracy\nand how its location is affected by the distribution of the levels \n(in particular by their degeneracy)\ndefining the space on which the Hamiltonian (\\ref{HP}) acts.\nThis is of relevance because we conjecture that\nthe physics is more or less affected by the singularity \ndepending upon its proximity to the positive real axis, as it has been\nhinted by the simple $n=1$, $L=2$ case: accordingly we surmise that\nif the singularity is located \nremotely from the real axis one deals with a system which is predominantly \nfermionic, if it is closer to the real axis (as it may be the case when the \ns.p.l. have large pair degeneracies)\none deals with a system in which a process of bosonization is taking place.\n\nFinally we have discussed the excitation spectrum of the pairing Hamiltonian. \nWe have found that the Gaudin and seniority excitations are close to each \nother, at least for small $s$ and $N_G$. We have proposed an approximate analytic \nexpression for these excitations that appears to account satisfactorily\nfor the exact result. Importantly, both the Gaudin and the seniority excitations \ndisplay the same gap in the leading order expressed by our approximation.\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
{"text":"\\section{Introduction}\n\\label{intro} In nonequivariant topology, it is a triviality that\nspaces built of only even dimensional cells will have free\ncohomology, regardless of the chosen coefficient ring. It is just\nas easy to see that every space has free cohomology when the\ncoefficient ring is taken to be $\\Z\/2$. Analogous results are not\nso clear in the equivariant setting.\n\nIn \\cite{FL}, it is shown that the $RO(\\Z\/p)$-graded homology of a\n$\\Z\/p$-space built of only even dimensional cells is free as a\nmodule over the homology of a point, regardless of which Mackey\nfunctor is chosen for coefficients. The goal of this paper is to establish a similar result for the cohomology of $G=\\Z\/2$-spaces without the restriction to cells of even degrees, but with the assumption of using\nconstant $\\underline{\\Z\/2}$ Mackey functor coefficients. Here is\nthe main result:\n\n\\begin{thm*}\nIf $X$ is a connected, locally finite, finite\ndimensional $\\text{Rep}(\\Z\/2)$-complex, then\n$H^{*,*}(X;\\underline{\\Z\/2})$ is free as a\n$H^{*,*}(pt;\\underline{\\Z\/2})$-module.\n\\end{thm*}\n\n\\noindent (The bigrading will be explained in Section \\ref{sec:Prelim}.)\n\nThe projective spaces and Grassmann\nmanifolds associated to representations of $\\Z\/2$ are examples of such $\\text{Rep}(\\Z\/2)$-complexes. In these\nparticular cases, the free generators of the cohomology modules are\nin bijective correspondence with the Schubert cells. The precise degrees of the cohomology generators is typically unknown, much\nlike in \\cite{FL}.\n\nSection \\ref{sec:Prelim} provides some of the background and notation required\nfor the rest of the paper. Most of this information can be found in\n\\cite{Alaska} and \\cite{FL} but is reproduced here for convenience.\nSection \\ref{sec:Freeness} holds the main freeness theorem. As\napplications of the freeness theorem, section \\ref{sec:RPs} exhibits\nsome techniques for calculating the cohomology of\n$\\text{Rep}(G)$-complexes. The importance of such calculations lies\nin their potential applications toward understanding $RO(G)$-graded\nequivariant characteristic classes.\n\nThe work in this paper was originally part of the author's\ndissertation while at the University of Oregon.\n\nThe author is indebted to Dan Dugger for his guidance and\ninnumerable helpful conversations.\n\n\n\n\\section{Preliminaries}\n\\label{sec:Prelim} This section contains some of the basic machinery\nand notations that will be used throughout the paper. In this\nsection, $G$ can be any finite group unless otherwise specified.\n\nGiven a $G$-representation $V$, let $D(V)$ and $S(V)$ denote the\nunit disk and unit sphere, respectively, in $V$ with action induced\nby that on $V$. A \\bf$\\text{Rep}(G)$-complex \\rm is a $G$-space $X$ with a\nfiltration $X^{(n)}$ where $X^{(0)}$ is a disjoint union of\n$G$-orbits and $X^{(n)}$ is obtained from $X^{(n-1)}$ by attaching\ncells of the form $D(V_\\alpha)$ along maps $f_\\alpha \\colon\nS(V_\\alpha) \\ra X^{(n-1)}$ where $V_\\alpha$ is an $n$-dimensional\nreal representation of $G$. The space $X^{(n)}$ is referred to as\nthe \\bf $n$-skeleton \\rm of $X$, and the filtration is referred to as a \\bf cell\nstructure\\rm .\n\nFor the precise definition of a Mackey functor when $G=\\Z\/2$, the\nreader is referred to \\cite{LMM} or \\cite{DuggerKR}. A summary\nof the important aspects of a Mackey functor is given here. The\ndata of a Mackey functor are encoded in a diagram like the one\nbelow.\n\n\\[\\xymatrix{ M(\\Z\/2) \\ar@(ur,ul)[]^{t^*} \\ar@\/^\/[r]^(0.6){i_*} & M(e) \\ar@\/^\/[l]^(0.4){i^*}} \\]\n\n\nThe maps must satisfy the following four conditions.\n\\begin{enumerate}\n\\item $(t^*)^2 = id$\n\\item $t^*i^*=i^*$\n\\item $i_*(t^*)^{-1}=i_*$\n\\item $i^*i_*=id+t^*$\n\\end{enumerate}\n\nAccording to \\cite{Alaska}, each Mackey functor $M$ uniquely\ndetermines an $RO(G)$-graded cohomology theory characterized by\n\\begin{enumerate}\n\\item $H^n(G\/H;M) =\\begin{cases}\nM(G\/H) & \\text{ if } n=0 \\\\\n0 & \\text{otherwise}\\end{cases}$\n\\item The map $H^0(G\/K;M) \\ra H^0(G\/H;M)$ induced by $i \\colon G\/H \\ra G\/K$ is the transfer map $i^*$ in the Mackey functor.\n\\end{enumerate}\n\nA $p$-dimensional real $\\Z\/2$-representation $V$ decomposes as\n$V=(\\R^{1,0})^{p-q} \\oplus (\\R^{1,1})^q =\\R^{p,q}$ where $\\R^{1,0}$\nis the trivial 1-dimensional real representation of $\\Z\/2$ and $\\R^{1,1}$ is the nontrivial\n1-dimensional real representation of $\\Z\/2$. Thus the $RO(\\Z\/2)$-graded theory is\na bigraded theory, one grading measuring dimension and the other\nmeasuring the number of ``twists''. In this case, we write\n$H^{V}(X;M)=H^{p,q}(X;M)$ for the $V^{\\text{th}}$ graded component\nof the $RO(\\Z\/2)$-graded equivariant cohomology of $X$ with\ncoefficients in a Mackey functor $M$.\n\nIn this paper, $G$ will typically be $\\Z\/2$ and the Mackey functor\nwill almost always be constant $M=\\underline{\\Z\/2}$ which has the\nfollowing diagram.\n\n\\[\\xymatrix{ \\Z\/2 \\ar@(ur,ul)[]^{id} \\ar@\/^\/[r]^{0} & \\Z\/2 \\ar@\/^\/[l]^{id}} \\]\n\n\nWith these constant coefficients, the $RO(\\Z\/2)$-graded cohomology\nof a point is given by the picture in Figure \\ref{fig:pt}.\n\n\\begin{figure}[htpb]\n\\centering\n\\begin{picture}(100,100)(-100,-100)\n\\put(-100,-50){\\vector(1,0){100}}\n\\put(-50,-100){\\vector(0,1){100}}\n\n\n\n\\put(-50, -51){\\line(0,1){40}} \\put(-50, -51){\\line(1,1){40}}\n\\put(-50, -91){\\line(0,-1){20}} \\put(-50, -91){\\line(-1,-1){20}}\n\\put(-50, -51){\\circle*{2}}\n\n\n\\multiput(-90,-51)(20,0){5}{\\line(0,1){3}}\n\\multiput(-52,-90)(0,20){5}{\\line(1,0){3}}\n\\put(-49,-57){$\\scriptscriptstyle{0}$}\n\\put(-29,-57){$\\scriptscriptstyle{1}$}\n\\put(-9,-57){$\\scriptscriptstyle{2}$}\n\\put(-69,-57){$\\scriptscriptstyle{-1}$}\n\\put(-89,-57){$\\scriptscriptstyle{-2}$}\n\\put(-57,-47){$\\scriptscriptstyle{0}$}\n\\put(-57,-27){$\\scriptscriptstyle{1}$}\n\\put(-57,-7){$\\scriptscriptstyle{2}$}\n\n\\put(-48,-35){$\\tau$} \\put(-30,-35){$\\rho$} \\put(-10,-15){$\\rho^2$}\n\\put(-30,-17){$\\tau\\rho$} \\put(-58, -90){$\\theta$} \\put(-48,\n-102){$\\frac{\\theta}{\\tau}$} \\put(-78, -102){$\\frac{\\theta}{\\rho}$}\n\n\\put(-58,0){${q}$} \\put(0,-60){${p}$}\n\n\n\\multiput(-50, -30)(20,20){2}{\\circle*{2}} \\multiput(-50,\n-50)(20,20){3}{\\circle*{2}} \\put(-50, -10){\\circle*{2}} \\put(-50,\n-90){\\circle*{2}} \\put(-50, -110){\\circle*{2}} \\put(-70,\n-110){\\circle*{2}}\n\n\n\\end{picture}\n\n\\caption{$H^{*,*}(pt;\\Z\/2)$} \\label{fig:pt}\n\\end{figure}\n\nEvery lattice point in the picture that is inside the indicated\ncones represents a copy of the group $\\Z\/2$. The \\bf top cone \\rm is a\npolynomial algebra on the nonzero elements $\\rho \\in\nH^{1,1}(pt;\\underline{\\Z\/2})$ and $\\tau \\in\nH^{0,1}(pt;\\underline{\\Z\/2})$. The nonzero element $\\theta \\in H^{0,-2}(pt;\\underline{\\Z\/2})$ in the \\bf bottom\ncone \\rm is infinitely divisible by both $\\rho$ and $\\tau$. The\ncohomology of $\\Z\/2$ is easier to describe:\n$H^{*,*}(\\Z\/2;\\underline{\\Z\/2})=\\Z\/2[t, t^{-1}]$ where $t \\in\nH^{0,1}(\\Z\/2;\\underline{\\Z\/2})$. Details can be found in\n\\cite{DuggerKR} and \\cite{Caruso}.\n\nA useful tool is the following exact sequence of \\cite{AM}.\n\n\\begin{lem}[Forgetful Long Exact Sequence]\n\\label{lemma:forget} Let $X$ be a based $\\Z\/2$-space. Then for\nevery $q$ there is a long exact sequence\n\n$$\\xymatrix{\\cdots \\ar[r] & H^{p,q}(X) \\ar[r]^(0.4){\\cdot \\rho} & H^{p+1,q+1}(X) \\ar[r]^(0.55){\\psi} & H^{p+1}_{sing}(X) \\ar[r]^(0.45)\\delta & H^{p+1,q}(X)} \\ra \\cdots$$\n\n\\end{lem}\n\n\\noindent The map $\\cdot \\rho$ is multiplication by $\\rho \\in\nH^{1,1}(pt;\\underline{\\Z\/2})$ and $\\psi$ is the forgetful map to\nnon-equivariant cohomology with $\\Z\/2$ coefficients.\n\n\n\\section{The Freeness Theorem}\n\\label{sec:Freeness}\n\nComputing the $RO(G)$-graded cohomology of a $G$-space $X$ is\ntypically quite a difficult task. However, if $X$ has a filtration\n$X^{(0)} \\subseteq X^{(1)} \\subseteq \\cdots$, then we can take\nadvantage of the long exact sequences in cohomology arising from the cofiber\nsequences $X^{(n)} \\subseteq X^{(n+1)} \\ra X^{(n+1)}\/X^{(n)}$.\nThese long exact sequences paste together as an exact couple in the usual way,\ngiving rise to a spectral sequence associated to the filtration.\n\nIf $X$ is a $G$-CW complex or a $\\text{Rep}(G)$-complex, then $X$\nhas a natural filtration coming from the cell structure. In either\ncase, if $X$ is connected, the quotient spaces $X^{(n+1)}\/X^{(n)}$\nare wedges of $(n+1)$-spheres with action determined by the type of\ncells that were attached. Examples of this sort appear throughout the paper.\n\nFor the remainder of the paper, we\nwill only be interested in the case $G=\\Z\/2$ and always take coefficients to be $\\underline{\\Z\/2}$. These choices will be implicit in our notation.\n\nGiven a filtered $\\Z\/2$ space $X$, for each fixed $q$ there is a long exact sequence\n\n\\[ \\cdots H^{*,q}(X^{(n+1)}\/X^{(n)})\\ra H^{*,q}(X^{(n+1)}) \\ra H^{*,q}(X^{(n)}) \\ra H^{*+1,q}(X^{(n+1)}\/X^{(n)}) \\cdots \\]\n\n\\noindent and so there is one spectral sequence for each integer $q$. The\nspecifics are given in the following proposition.\n\n\\begin{prop} Let $X$ be a filtered $\\Z\/2$-space. Then for each $q\\in\\Z$ there is a spectral sequence with\n$$E_1^{p,n} = H^{p,q}(X^{(n+1)}, X^{(n)})$$\nconverging to $H^{p,q}(X)$.\n\\end{prop}\n\nThe construction of the spectral sequence is completely standard. See, for example, Proposition 5.3 of \\cite{McC}.\n\nIt is convenient to plot the $RO(\\Z\/2)$-graded cohomology in the\nplane with $p$ along the horizontal axis and $q$ along the vertical\naxis, and this turns out to be a nice way to view the cellular spectral\nsequences as well. When doing so, the\ndifferentials on each page of the spectral sequence have bidegree\n$(1,0)$ in the plane, but reach farther up the filtration on each\npage. It is important to keep track of at\nwhat stage of the filtration each group arises. In practice, this can be done by using different colors for group that arise at different stages of the filtration.\n\nIt is often quite difficult to determine the effect of all of the\nattaching maps in the cell attaching long exact sequences. If $X$\nis locally finite, then the cells can be attached one at a time, in\norder of dimension. This simplicity will make it easier to analyze\nthe differentials in the spectral sequence of the `one at a time'\ncellular filtration, even when the precise impact of the attaching maps are not a priori known.\n\n\\begin{lem}\n Let $B$ be a $\\text{Rep}(\\Z\/2)$-complex with free cohomology that is built only of cells of dimension strictly less than $p$. Suppose $X$ is obtained from $B$ by attaching a single $(p,q)$-cell and let $\\nu$ denote the generator for the cohomology of $X\/B \\cong S^{p,q}$. Then after an appropriate change of basis either\n\\begin{enumerate}\n\\item all attaching maps to the top cone of $\\nu$ are zero (that is, $d(a)=0$ for all $a$ with $a\\in H^{*,q_a}(B)$ with $q_a \\geq q-1$),\n\\item the cell attaching `kills' $\\nu$ and a free generator in dimension $(p-1,q)$, or\n\\item all nonzero differentials hit the bottom cone of $\\nu$.\n\\end{enumerate}\n\\end{lem}\n\n\\begin{proof}\nConsider the cellular spectral sequence associated to attaching a single $(p,q)$ cell to $B$. The effects of attaching such a cell can cause the lower dimensional\ngenerators to hit either the `top cone' or the `bottom cone'\nof the newly attached free generator $\\nu$ of degree $(p,q)$.\n\nSuppose first that all nonzero differentials hit the top\ncone. Then any free generator $\\omega_i$ having a nonzero\ndifferential in the spectral sequence must have degree $(p_i,q_i)$\nwhere $p_i=p-1$ and $q_i \\geq q$. For illustrative purposes, the $E_1$ page of the cellular spectral sequence of an example of this type is pictured in\nFigure \\ref{fig:attach1pqcell}. In this example, there are two generators $\\omega_1$ and $\\omega_2$ with bidegree $(p-1,q_1)$ and one generator $\\omega_3$ with bidegree $(p-1,q_i)$.\n\n\\begin{figure}[htpb]\n\n\\centering\n\n\\begin{picture}(330,230)(-150,-110)\n\\put(-110,-50){\\vector(1,0){235}}\n\\put(-50,-100){\\vector(0,1){200}}\n\n\n\n\\put(-8, 30){\\line(0,1){70}} \\put(-8, 30){\\line(1,1){70}} \\put(-8,\n-10){\\line(0,-1){80}} \\put(-8, -10){\\line(-1,-1){80}} \\put(-8,\n30){\\circle*{2}}\n\\put(-23,25){$\\omega_3$}\n\n\n\\put(-10, 9){\\line(0,1){90}} \\put(-10, 9){\\line(1,1){90}} \\put(-10,\n-31){\\line(0,-1){60}} \\put(-10, -31){\\line(-1,-1){60}} \\put(-10,\n9){\\circle*{2}}\n\n\\put(-40,5){$\\omega_1,\\omega_2$}\n\n\\put(-12, 9){\\line(0,1){90}} \\put(-12, 9){\\line(1,1){90}} \\put(-12,\n-31){\\line(0,-1){60}} \\put(-12, -31){\\line(-1,-1){60}} \\put(-12,\n9){\\circle*{2}}\n\n\n\\put(10, -31){\\line(0,1){130}} \\put(10, -31){\\line(1,1){130}}\n\\put(10, -71){\\line(0,-1){20}} \\put(10, -71){\\line(-1,-1){20}}\n\\put(10, -31){\\circle*{2}}\n\n\\put(3, -35){$\\nu$}\n\n\n\\multiput(-90,-51)(20,0){11}{\\line(0,1){3}}\n\\multiput(-52,-90)(0,20){10}{\\line(1,0){3}}\n\n\n\\put(14,-57){$p$} \\put(-60,30){$q_i$} \\put(-60,10){$q_1$}\n\\put(-60,-30){$q$}\n\n\\end{picture}\n\n\\caption{The $E_1$ page of the cellular spectral sequence attaching\na single $(p,q)$-cell to $B$.} \\label{fig:attach1pqcell}\n\n\\end{figure}\n\n\nHere, only the generator associated to the $(p,q)$-cell and the\ngenerators with nonzero differentials are shown. Each of the\n$\\omega_i$ satisfies $d(\\omega_i)=\\tau^{n_i} \\nu$ for integers\n$n_i$. Relabeling if necessary, we can arrange so that the\n$\\omega_i$ satisfy $n_1 \\leq n_2 \\leq \\cdots$.\n\nLet $A=\\langle\\omega_i\\rangle$, the $H^{*,*}(pt)$-span of the $\\omega_i$'s. A change of basis can be performed\non $A$, after which we may assume $d(\\omega_1)=\\tau^{n_1}\\nu$ and\n$d(\\omega_i)=0$ for $i>1$. Indeed,\n$\\{\\tau^{n_i-n_1}\\omega_1+\\omega_i\\}$ is a basis for $A$ and\n$d(\\tau^{n_i-n_1}\\omega_1+\\omega_i)=\\tau^{n_1}\\nu$ if $i=1$ and is\nzero otherwise. (In effect, the attaching map can `slide' off of all the $\\omega_i$\nexcept for the one for which $q_i$ is minimal.)\n\nIf $\\omega_1$ happens to be in dimension $(p-1,q)$, then the newly\nattached cell `kills' $\\omega_1$ and $\\nu$.\nOtherwise the nonzero portion of the spectral sequence is illustrated in Figure \\ref{fig:changebasis}.\n\n\\begin{figure}[htpb]\n\n\\centering\n\n\\begin{picture}(330,230)(-150,-110)\n\\put(-110,-50){\\vector(1,0){235}}\n\\put(-50,-100){\\vector(0,1){200}}\n\n\n\n\n\n\\put(-10, 9){\\line(0,1){90}} \\put(-10, 9){\\line(1,1){90}}\n\\put(-10,-31){\\line(0,-1){60}} \\put(-10, -31){\\line(-1,-1){60}}\n\\put(-10, 9){\\circle*{2}} \\put(-17, 5){$\\omega$}\n\n\n\n\\put(10, -31){\\line(0,1){130}} \\put(10, -31){\\line(1,1){130}}\n\\put(10, -71){\\line(0,-1){20}} \\put(10, -71){\\line(-1,-1){20}}\n\\put(10, -31){\\circle*{2}}\n\n\\put(4, -35){$\\nu$}\n\n\n\\multiput(-90,-51)(20,0){11}{\\line(0,1){3}}\n\\multiput(-52,-90)(0,20){10}{\\line(1,0){3}}\n\n\n\\put(14,-57){$p$} \\put(-60,10){$q_1$} \\put(-60,-30){$q$}\n\n\\end{picture}\n\\caption{The nonzero portion of the same spectral sequence, after a\nchange of basis.} \\label{fig:changebasis}\n\\end{figure}\n\n\nAfter taking cohomology, the spectral sequence collapses, as in Figure \\ref{fig:E2page}.\n\n\\begin{figure}[htpb]\n\n\\centering\n\n\\begin{picture}(330,230)(-150,-110)\n\\put(-110,-50){\\vector(1,0){235}}\n\\put(-50,-100){\\vector(0,1){200}}\n\n\n\n\n\n\\put(-10, -30){\\line(0,-1){60}} \\put(-10, -30){\\line(-1,-1){60}}\n\\put(-10, -70){\\circle*{2}} \\put(-10, -50){\\circle*{2}}\n\n\\put(-8,-45){$\\omega_1 \\frac{\\theta}{\\tau^m}$}\n\n\\put(10, -30){\\line(0,1){20}} \\put(10,-10){\\line(1,1){110}} \\put(10,\n-30){\\line(1,1){130}} \\put(10, -70){\\line(0,-1){20}} \\put(10,\n-70){\\line(-1,-1){20}} \\put(10, -30){\\circle*{2}}\n\n\\put(14, -35){$\\nu$}\n\n\n\\multiput(-90,-51)(20,0){11}{\\line(0,1){3}}\n\\multiput(-52,-90)(0,20){10}{\\line(1,0){3}}\n\n\n\\put(14,-57){$p$} \\put(-60,10){$q_1$} \\put(-60,-30){$q$}\n\n\\end{picture}\n\\caption{The $E_2=E_\\infty$ page of the above spectral sequence.}\n\\label{fig:E2page}\n\\end{figure}\n\n\nThere is a class $\\omega_1 \\frac{\\theta}{\\tau^m}$ that, potentially,\ncould satisfy $\\rho \\cdot \\omega_1 \\frac{\\theta}{\\tau^m}=\\nu$.\nHowever, for degree reasons, $\\rho \\cdot \\omega_1\n\\frac{\\theta}{\\tau^{m+1}}=0$ and since $\\rho$ and $\\tau$ commute, $\\rho \\cdot \\omega_1 \\frac{\\theta}{\\tau^m}=0$. This\nmeans $\\nu$ determines a nonzero class in $H^{*,*}(X)$ that is not\nin the image of $\\cdot \\rho$. If $B$ is based, then $X$ is based,\nand so, by the forgetful long exact sequence, $\\nu$ determines a\nnonzero class in non-equivariant cohomology. Then since $\\tau$ maps to $1$\nin non-equivariant cohomology, $\\tau^n \\nu$ is nonzero for all $n$. But,\nas the picture indicates, $\\tau^n \\nu$ is zero for large enough\n$n$. This contradiction implies that there could not have been any\nnonzero differentials hitting the top cone of $\\nu$.\n\nThis argument is independent of whether there are any differentials\nhitting the bottom cone, and so there simply cannot be any nonzero\ndifferentials on the top cone.\\qed\n\\end{proof}\n\n\nDifferentials hitting the bottom cone can cause a shifting in degree of the cohomology generators.\n\n\n\\begin{thm} Suppose $X$ is a $\\text{Rep}(\\Z\/2)$-complex formed by attaching a single $(p,q)$-cell to a space $B$. Suppose also that $\\tilde{H}^{*,*}(B)$ is a free $H^{*,*}(pt)$-module with a single generator $\\omega$ of dimension strictly less than $p$. Then $H^{*,*}(X)$ is a free $H^{*,*}(pt)$-module. In particular, one of the following must hold:\n\\begin{enumerate}\n\\item $H^{*,*}(X) \\cong H^{*,*}(pt)$.\n\n\\item $H^{*,*}(X) \\cong H^{*,*}(B)\\oplus \\Sigma^\\nu H^{*,*}(pt)$, where the degree of $\\nu$ is $(p,q)$.\n\n\\item $H^{*,*}(X)$ is free with two generators $a$ and $b$.\n\\end{enumerate}\n\nIn (3) above, the degrees of the generators $a$ and $b$ are\n$(p-n-1,q-n-1)$ and $(p,q-m-1)$ where\n$d(\\omega)=\\frac{\\theta}{\\rho^n\\tau^m}\\nu$.\n\n\\end{thm}\n\\begin{proof} Under these hypotheses, there is a cofiber sequence of the form $B \\stackrel{i}\\inc X \\stackrel{j}\\fib S^{p,q}$. Denote by $\\nu$ the generator of $H^{*,*}(S^{p,q})$.\n\nIf $d(\\omega)=\\nu$ then $(1)$ holds and $H^{*,*}(X)$ is free. If\n$d(\\omega)=0$, then $(2)$ holds and again $H^{*,*}(X)$ is free. The\nremaining case is $d(\\omega)\\neq0$. By the previous lemma, $d(\\omega)$ is in the bottom cone of $\\nu$. That is $d(\\omega)=\\frac{\\theta}{\\rho^n\\tau^m}\\nu$ for some $n$ and\n$m$. Recall that $\\nu$ has dimension $(p,q)$ and so $\\omega$ has\ndimension $(p-n-1, q-n-m-2)$. The $E_1$ page of the cellular\nspectral sequence is given in Figure \\ref{fig:E1bottom}.\n\n\\begin{figure}[htpb]\n\n\\centering\n\n\\begin{picture}(330,230)(-150,-110)\n\\put(-110,-50){\\vector(1,0){235}}\n\\put(-50,-100){\\vector(0,1){200}}\n\n\n\\put(-30, -30){\\line(0,1){130}} \\put(-30, -30){\\line(1,1){130}}\n\\put(-30, -70){\\line(0,-1){20}} \\put(-30, -70){\\line(-1,-1){20}}\n\\put(-30, -30){\\circle*{2}} \\put(-40,-30){$\\omega$}\n\n\\put(10, 71){\\line(0,1){30}} \\put(10, 71){\\line(1,1){30}} \\put(10,\n30){\\line(0,-1){120}} \\put(10, 30){\\line(-1,-1){120}} \\put(10,\n71){\\circle*{2}} \\put(0,70){$\\nu$} \\put(-10, -30){\\circle*{2}}\n\\put(-20,-42){$\\frac{\\theta}{\\rho^n\\tau^m}\\nu$}\n\n\\multiput(-90,-51)(20,0){11}{\\line(0,1){3}}\n\\multiput(-52,-90)(0,20){10}{\\line(1,0){3}}\n\n\\put(14,-57){$p$} \\put(-60,70){$q$}\n\n\\end{picture}\n\\caption{The $E_1$ page of the cellular spectral sequence with a\nsingle nonzero differential hitting the bottom cone of an attached\n$(p,q)$-cell.} \\label{fig:E1bottom}\n\\end{figure}\n\n\nAfter taking cohomology, the spectral sequence collapses, and what\nremains is pictured in Figure \\ref{fig:E2bottom}.\n\n\\begin{figure}[htpb]\n\n\\centering\n\n\\begin{picture}(330,230)(-150,-110)\n\\put(-110,-50){\\vector(1,0){235}}\n\\put(-50,-100){\\vector(0,1){200}}\n\n\n\\put(-30, 30){\\line(0,1){70}} \\put(-30, 30){\\line(1,1){70}}\n\\put(-30, 30){\\circle*{2}} \\put(-40,30){$a$} \\put(10,\n10){\\line(1,1){90}} \\put(10, 10){\\circle*{2}} \\put(0, 10){$b$}\n\\put(10, 30){\\line(1,1){70}} \\put(10, 30){\\circle*{2}} \\put(0,\n30){$b_1$} \\put(10, 50){\\line(1,1){50}} \\put(10, 50){\\circle*{2}}\n\\put(0, 50){$b_2$} \\put(-30, -70){\\line(0,-1){20}} \\put(-30,\n-70){\\line(-1,-1){20}}\n\n\n\\put(10, 71){\\line(0,1){30}} \\put(10, 71){\\line(1,1){30}} \\put(10,\n71){\\circle*{2}} \\put(-3, 71){$b_m$} \\put(10, -30){\\line(0,-1){60}}\n\\put(10, -30){\\circle*{2}} \\put(-30, -10){\\line(-1,-1){80}}\n\\put(-30, -10){\\circle*{2}} \\put(-30, -30){\\line(-1,-1){60}}\n\\put(-30, -30){\\circle*{2}} \\put(-30, -50){\\line(-1,-1){40}}\n\\put(-30, -50){\\circle*{2}} \\put(-10, -48){\\line(-1,-1){40}}\n\\put(-10, -48){\\line(0,-1){40}} \\put(-10, -48){\\circle*{2}}\n\n\\multiput(-90,-51)(20,0){11}{\\line(0,1){3}}\n\\multiput(-52,-90)(0,20){10}{\\line(1,0){3}}\n\n\\put(14,-57){$p$} \\put(-60,70){$q$}\n\n\\end{picture}\n\\caption{The $E_2=E_\\infty$ page of the cellular spectral sequence\nwith a single nonzero differential hitting the bottom cone of an\nattached $(p,q)$-cell.} \\label{fig:E2bottom}\n\\end{figure}\n\n\nLet $a$ be the generator in degree $(p-n-1, q-n-1)$ and $b$ the generator in dimension\n$(p,q-m-1)$. For degree reasons, $b$ is not in the image\nof $\\cdot \\rho$ and so determines a nonzero class in non-equivariant\ncohomology. Thus, $\\tau^i b$ is nonzero for all $i$, and so we have\nthat $b_i=\\tau^i b$. In particular, $\\rho^{n+1} a$ and $\\tau^m b$\ngenerate $H^{p,q}(X)$. Consider the portion of the long exact\nsequence associated to the cofiber sequence $B \\stackrel{i}\\inc X\n\\stackrel{j}\\fib S^{p,q}$ given below:\n\n$$\\xymatrix{\\cdots \\ar[r] & H^{p,q}(S^{p,q}) \\ar[r]^{j^*} & H^{p,q}(X) \\ar[r]^{i^*} & H^{p,q}(B) \\ar[r] & 0}$$\n\nSince $i^*(\\rho^{n+1} a)=i^*(\\tau^m b)=\\rho^{n+1}\\tau^m\\omega$,\nexactness implies that $j^*(\\nu)=\\rho^{n+1} a + \\tau^m b$. Also\n$j^*$ is an $H^{*,*}(pt)$-module homomorphism, and so\n$j^*(\\frac{\\theta}{\\rho^{n+1}}\\nu)=\\theta a$ and\n$j^*(\\frac{\\theta}{\\tau^m}\\nu)=\\theta b$. In particular, we can\ncreate a map $f$ from a free module with generators $\\alpha$ and\n$\\beta$ in degrees $(p-n-1,q-n-1)$ and $(p,q-m-1)$ respectively\nto $\\tilde{H}^{p,q}(X)$ with $f(\\alpha)=a$ and $f(\\beta)=b$. This\n$f$ is an isomorphism. \n\n\n\\end{proof}\n\n\\begin{thm}[Freeness Theorem]\n\\label{thm:freeness} If $X$ is a connected, locally finite, finite\ndimensional $\\text{Rep}(\\Z\/2)$-complex, then\n$H^{*,*}(X;\\underline{\\Z\/2})$ is free as a\n$H^{*,*}(pt;\\underline{\\Z\/2})$-module.\n\\end{thm}\n\n\\begin{proof}\n\nSince $X$ is locally finite, the cells can be attached one at a\ntime. Order the cells $\\alpha_1, \\alpha_2, \\dots$ so that their\ndegrees satisfy $p_i \\leq p_j$ if $i\\leq j$ and $q_i \\leq q_j$ if\n$p_i=p_j$ and $i \\leq j$. We can proceed by induction over the\nspaces in the `one-at-a-time' cell filtration $X^{(0)} \\subseteq \\cdots \\subseteq X^{(n)}\n\\subseteq \\cdots \\subseteq X$, with the base case obvious since $X$\nis connected.\n\nFirst, suppose that $H^{*,*}(X^{(n)})$ is a free\n$H^{*,*}(pt)$-module and that $X^{(n+1)}$ is obtained by attaching a\nsingle $(p,q)$-cell and that $X^{(n)}$ has no $p$-cells. Denote by\n$\\nu$ the free generator of $H^{*,*}(X^{(n+1)}\/X^{(n)}) \\cong\nH^{*,*}(S^{p,q})$. Consider the spectral sequence of the filtration\n$X^{(n)} \\subseteq X^{(n+1)}$. An example is pictured below in Figure\n\\ref{fig:free1} to aid in the discussion.\n\n\n\\begin{figure}[htpb]\n\n\\centering\n\n\\begin{picture}(330,230)(-150,-110)\n\n\n\\put(-10, -10){\\line(0,1){110}} \\put(-10, -10){\\line(1,1){110}}\n\\put(-10, -50){\\line(0,-1){40}} \\put(-10, -50){\\line(-1,-1){40}}\n\\put(-10, -10){\\circle*{2}}\n\n\\put(-8, 50){\\line(0,1){50}} \\put(-8, 50){\\line(1,1){50}} \\put(-8,\n10){\\line(0,-1){100}} \\put(-8, 10){\\line(-1,-1){100}} \\put(-8,\n50){\\circle*{2}}\n\n\\put(-10, 70){\\line(0,1){30}} \\put(-10, 70){\\line(1,1){30}}\n\\put(-10, 30){\\line(0,-1){120}} \\put(-10, 30){\\line(-1,-1){120}}\n\\put(-10, 70){\\circle*{2}}\n\n\n\\put(-12, -30){\\line(0,1){130}} \\put(-12, -30){\\line(1,1){130}}\n\\put(-12, -70){\\line(0,-1){20}} \\put(-12, -70){\\line(-1,-1){20}}\n\\put(-12, -30){\\circle*{2}}\n\n\\put(-12, -50){\\line(0,1){150}} \\put(-12, -50){\\line(1,1){150}}\n\\put(-12, -50){\\circle*{2}}\n\n\\put(-28, -50){\\line(0,1){150}} \\put(-28, -50){\\line(1,1){150}}\n\\put(-28, -50){\\circle*{2}}\n\n\\put(-48, -70){\\line(0,1){150}} \\put(-48, -70){\\line(1,1){150}}\n\\put(-48, -70){\\circle*{2}}\n\n\\put(-90, -90){\\line(0,1){170}} \\put(-90, -90){\\line(1,1){170}}\n\\put(-90, -90){\\circle*{2}}\n\n\n\\put(10, 31){\\line(0,1){70}} \\put(10, 31){\\line(1,1){70}} \\put(10,\n-10){\\line(0,-1){80}} \\put(10, -10){\\line(-1,-1){80}} \\put(10,\n31){\\circle*{2}} \\put(17,30){$\\nu$} \\put(15,-12){$\\theta\\nu$}\n\n\n\\put(-23,70){$\\alpha'$}\n\\put(-23,50){$\\alpha$}\n\\put(-24,-10){$\\omega'''$}\n\\put(-23,-30){$\\omega''$}\n\\put(-23,-50){$\\omega_n$}\n\\put(-43,-50){$\\omega'$}\n\\put(-63,-70){$\\omega_1$}\n\n\n\n\\end{picture}\n\n\\caption{The spectral sequence of a filtration for attaching a\nsingle $(p,q)$-cell to a space with free cohomology.}\n\\label{fig:free1}\n\\end{figure}\n\nAs before, a change of basis allows us to focus on a subset $\\omega_1, \\dots,\n\\omega_n$ of the free generators of $H^{*,*}(X^{(n)})$ whose\ndifferentials hit the bottom cone of $\\nu$ and that satisfy\n\\begin{enumerate}\n\\item $d(\\omega_i) \\neq 0$ for all $i$,\n\\item $|\\omega_i| > |\\omega_j|$ when $i > j$,\n\\item $|\\omega_i^G| > |\\omega_j^G|$ when $i > j$,\n\\end{enumerate}\n\n\\noindent and all other basis elements have zero differentials to\nthe bottom cone of $\\nu$. This is similar to what is referred to in\n\\cite{FL} as a ramp of length $n$. Also, we can change the basis\nagain so that there is only one free generator, $\\alpha$, of\n$H^{*,*}(X^{(n)})$ with a nonzero differential to the top cone of\n$\\nu$. Then, after this change of basis, the nonzero portion of the\nspectral sequence of the filtration looks like the one in Figure\n\\ref{fig:free2}\n\\begin{figure}[htpb]\n\n\\centering\n\n\\begin{picture}(330,230)(-150,-110)\n\n\n\\put(-8, 50){\\line(0,1){50}} \\put(-8, 50){\\line(1,1){50}} \\put(-8,\n10){\\line(0,-1){100}} \\put(-8, 10){\\line(-1,-1){100}} \\put(-8,\n50){\\circle*{2}} \\put(-6,45){$\\alpha$}\n\n\\put(-12, -50){\\line(0,1){150}} \\put(-12, -50){\\line(1,1){150}}\n\\put(-12, -50){\\circle*{2}} \\put(-20,-60){$\\omega_{n}$}\n\n\n\n\\put(-48, -70){\\line(0,1){150}} \\put(-48, -70){\\line(1,1){150}}\n\\put(-48, -70){\\circle*{2}} \\put(-55,-80){$\\omega_{n-1}$}\n\n\n\\put(-90, -90){\\line(0,1){170}} \\put(-90, -90){\\line(1,1){170}}\n\\put(-90, -90){\\circle*{2}}\n\n\n\\put(10, 31){\\line(0,1){70}} \\put(10, 31){\\line(1,1){70}} \\put(10,\n-10){\\line(0,-1){80}} \\put(10, -10){\\line(-1,-1){80}} \\put(10,\n31){\\circle*{2}} \\put(17,30){$\\nu$} \\put(15,-12){$\\theta\\nu$}\n\n\n\n\\end{picture}\n\n\\caption{The nonzero portion of the above spectral sequence, after a\nchange of basis.} \\label{fig:free2}\n\\end{figure}\n\nAs above, $\\alpha$ cannot support a nonzero differential, and we can see that each of the\n$\\omega_i$'s will shift up in $q$-degree and $\\nu$ will shift down in $q$-degree. That is, the $\\omega_i$'s and $\\nu$ each give rise to free generators in the cohomology of $H^{*,*}(X^{(n+1)})$, but in different bidegree than their predecessors. Thus, $H^{*,*}(X^{(n+1)})$ is again free.\n\nNow suppose that $X^{(n+1)}$ is obtained by attaching a $(p,q)$-cell\n$\\nu'$ and that $X^{(n)}$ has a single $p$-cell $\\nu$ already. Then\nby the previous case, the generator for $\\nu$ was either shifted\ndown, killed off, or was left alone at the previous stage. In any\ncase, because of our choice of ordering of the cells, the generator\nfor $\\nu$ cannot support a differential to the generator for $\\nu'$.\nThus, the only nonzero differentials to $\\nu'$ are from strictly\nlower dimensional cells. Thus, we are reduced again to the previous\ncase and $H^{*,*}(X^{(n+1)})$ is free. By induction, $H^{*,*}(X)$\nis free. \n\n\\end{proof}\n\n\n\n\\section{Real Projective Spaces and Grassmann Manifolds}\n\\label{sec:RPs}\n\nIn this section, $G=\\Z\/2$ exclusively, and the coefficient Mackey will always be $M = \\underline{\\Z\/2}$ and will be suppressed from the notation.\n\nSince each representation $\\R^{p,q}$ has a linear $\\Z\/2$-action,\nthere is an induced action of $\\Z\/2$ on $G_n(\\R^{p,q})$, the \\bf\nGrassmann manifold \\rm of $n$-dimensional linear subspaces of\n$\\R^{p,q}$. These Grassmann manifolds play a central role in the\nclassification of equivariant vector bundles, and so it is important\nto understand their cohomology. As a special case we have the real\nprojective spaces $\\bP(\\R^{p,q})=G_1(\\R^{p,q})$.\n\nThe usual Schubert cell decomposition endows the Grassmann manifolds\nwith a $\\text{Rep}(\\Z\/2)$-cell structure. However, the number of\ntwists in each cell is dependent upon the flag of subrepresentations\nof $\\R^{p,q}$ that is chosen. A \\bf flag symbol \\rm $\\varphi$ is a sequence\nof integers $\\varphi = (\\varphi_1, \\dots, \\varphi_q)$ satisfying\n$1\\leq \\varphi_1 < \\dots < \\varphi_q \\leq q$. A flag symbol $\\varphi$ determines a flag of subrepresenations $V_0=0 \\subset V_1 \\subset \\cdots\n\\subset V_p=\\R^{p,q}$ satisfying\n$V_{\\varphi_i}\/V_{\\varphi_i-1}=\\R^{1,1}$ for all $i=1, \\dots, q$,\nand all other quotients of consecutive terms are $\\R^{1,0}$. For concreteness, we also require that $V_{i}$ is obtained from $V_{i-1}$ by adjoining a coordinate basis\nvector. For example, there is a\nflag in $\\R^{5,3}$ determined by the flag symbol $\\varphi=(1,3,4)$\nof the form $\\R^{0,0} \\subset \\R^{1,1} \\subset \\R^{2,1} \\subset\n\\R^{3,2} \\subset \\R^{4,3} \\subset \\R^{5,3}$.\n\nA \\bf Schubert symbol \\rm $\\sigma = (\\sigma_1, \\dots , \\sigma_n)$ is a sequence of integers such that $1\\leq\\sigma_1 <\n\\sigma_2 < \\dots < \\sigma_n\\leq p$. Given a Schubert symbol $\\sigma$\nand a flag symbol $\\varphi$, let $e(\\sigma,\\varphi)$ be the set of\nplanes $\\ell \\in G_n(\\R^{p,q})$ for which $\\dim(\\ell \\cap\nV_{\\sigma_i}) = 1+\\dim(\\ell \\cap V_{\\sigma_i-1})$, where $V_0\n\\subset \\cdots \\subset V_n$ is the flag determined by $\\varphi$.\nThen $e(\\sigma,\\varphi)$ is the interior of a cell $D(W)$ for some\nrepresentation $W$. The dimension of the cell is determined by the\nSchubert symbol $\\sigma$ just as in nonequivariant topology, but the\nnumber of twists depends on both $\\sigma$ and the flag symbol\n$\\varphi$.\n\nFor example, consider $G_2(\\R^{5,3})$, $\\sigma = (3,5)$, and\n$\\varphi = (1,3,4)$. Then $e(\\sigma,\\varphi)$ consists of planes\n$\\ell$ which have a basis with echelon form given by the matrix\nbelow.\n\n$$\n\\begin{array}{cc}\n& \\begin{array}{ccccc} \\phantom{(}- & + & - & - & +\\phantom{)}\\end{array}\\\\\n& \\left( \\begin{array}{ccccc} \\ast & \\ast & 1 & 0 & 0 \\\\\n\\ast & \\ast & 0 & \\ast & 1\\end{array} \\right)\n\n\\end{array}\n$$\n\nHere, the action of $\\Z\/2$ on the columns, as determined by\n$\\varphi$, has been indicated by inserting the appropriate signs\nabove the matrix. After acting, this becomes the following.\n\n$$\n\\begin{array}{cc}\n& \\begin{array}{ccccc} \\phantom{(}- & + & - & - & +\\phantom{)}\\end{array}\\\\\n& \\left( \\begin{array}{ccccc} -\\ast & \\ast & -1 & 0 & 0 \\\\\n-\\ast & \\ast & 0 & -\\ast & 1\\end{array} \\right)\n\n\\end{array}\n$$\n\n\nWe require the last nonzero entry of each row to be 1, and so we\nscale the fisrt row by $-1$.\n\n$$\n\\begin{array}{cc}\n& \\begin{array}{ccccc} \\phantom{(}- & + & - & - & +\\phantom{)}\\end{array}\\\\\n& \\left( \\begin{array}{ccccc} \\ast & -\\ast & 1 & 0 & 0 \\\\\n-\\ast & \\ast & 0 & -\\ast & 1\\end{array} \\right)\n\n\\end{array}\n$$\n\nThere are five coordinates which can be any real numbers, three\nof which the $\\Z\/2$ action of multiplication by -1, so this is a\n$(5,3)$-cell. Through a similar process, we can obtain a cell\nstructure for $G_n(\\R^{p,q})$ given any flag $\\varphi$. The type of\ncell determined by the Schubert symbol $\\sigma$ and the flag\n$\\varphi$ is given by the following proposition. Here,\n$\\underline{\\sigma_i} = \\{1, \\dots, \\sigma_i\\}$ and $\\sigma(i) =\n\\{\\sigma_1, \\dots, \\sigma_i\\}$.\n\n\\begin{prop}\n\\label{prop:schubertcells} Let $\\sigma = (\\sigma_1, \\dots,\n\\sigma_n)$ be a Schubert symbol and $\\varphi = (\\varphi_1, \\dots,\n\\varphi_q)$ be a flag symbol for $\\R^{p,q}$. The cell\n$e(\\sigma,\\varphi)$ of $G_n(\\R^{p,q})$ is of dimension $(a,b)$ where\n$a=\\sum_{i=1}^n (\\sigma_i-i)$ and $b=\\sum_{\\sigma_i \\in \\varphi}\n|\\underline{\\sigma_i}\\setminus (\\varphi \\cup\n\\sigma(i))|+\\sum_{\\sigma_i \\notin \\varphi} |(\\underline{\\sigma_i}\n\\cap \\varphi)\\setminus \\sigma(i)|$.\n\n\\end{prop}\n\\begin{proof}\nThe formula for $a$ is exactly the same as in the nonequivariant\ncase. The one for $b$ follows since the number of twisted\ncoordinates in each row is exactly the number of $\\ast$ coordinates\nfor which the action is opposite to that on the coordinate\ncontaining the 1 in that echelon row.\n\\end{proof}\n\n\\begin{cor} Real and complex projective spaces and Grassmann manifolds have free $RO(\\Z\/2)$-graded cohomology with $\\underline{\\Z\/2}$ coefficients.\n\\end{cor}\n\n\\begin{prop} If $V \\subseteq V'$ is an inclusion of representations and $\\varphi \\subseteq \\varphi'$ is an extension of flag symbols for $V$ and $V'$, then there is a cellular inclusion $G_n(V) \\inc G_n(V')$.\n\n\\end{prop}\n\n\nThe following theorem guarantees that the cohomology of Grassmann manifolds have cohomology generators in bijective correspondence with the Schubert cells.\n\n\\begin{thm} $H^{*,*}(G_n(\\R^{u,v}))$ is a free $H^{*,*}(pt)$-module with generators in bijective correspondence with the Schubert cells. \\label{thm:cellbij}\n\\end{thm}\n\\begin{proof}\nSince $G_n(\\R^{u,v})$ has a $\\text{Rep}(\\Z\/2)$-complex structure, we know $H^{*,*}(G_n(\\R^{u,v}))$ is free by the freeness theorem, Theorem \\ref{thm:freeness}. Let $\\{\\omega_1, \\dots,\n\\omega_k\\}$ be a set of free generators. Then $k \\leq m$ where $m$\nis the number of Schubert cells.\n\nThese spaces are based, so we can appeal to the forgetful long exact\nsequence Lemma \\ref{lemma:forget}. By freeness and finite dimensionality,\nthe multiplication by $\\rho$ map\nis an injection for large enough $q$. Thus the forgetful map to\nnon-equivariant cohomology is surjective. Since\n$H^{*}_{sing}(G_n(\\R^{u,v}))$ is free with generators $a_1, \\dots\na_m$ in bijective correspondence with the Schubert cells,\n$H^{*,*}(G_n(\\R^{u,v}))$ has a set of elements, $\\{\\alpha_1, \\dots,\n\\alpha_m\\}$, with $\\psi(\\alpha_i)=a_i$. We can uniquely express\neach $\\alpha_i$ as $\\alpha_i =\\sum_{j=1}^k \\rho^{e_{ij}}\n\\tau^{f_{ij}}\\omega_j$. We can ignore any terms that have $\\rho$ in\nthem since $\\psi(\\rho)=0$. This gives a new set of elements,\n$\\bar{\\alpha}_i = \\sum_{j=1}^k \\epsilon_{ij} \\tau^{f_{ij}}\\omega_j$,\nwhere $\\epsilon_{ij}=0$ or $1$ and $\\psi(\\bar{\\alpha}_i)=a_i$. Since\n$\\psi(\\tau)=1$, we have that $\\sum_{j=1}^k\n\\epsilon_{ij}\\psi(\\omega_j)=a_i$. Since linear combinations of the\nlinearly independent $\\omega_j$'s map to the linearly independent\n$a_i$'s, there are at least as many $\\omega_j$'s as there are\n$a_i$'s. That is, $k \\geq m$. \n\\end{proof}\n\nThe above theorem is enough to determine the additive structure of the $RO(\\Z\/2)$-graded cohomology of the real projective spaces.\n\nRecall that ${\\cat U}=(\\R^{2,1})^\\infty$ is a complete universe in the sense of \\cite{Alaska}. Denote by\n$\\R\\bP^\\infty_{tw}=\\bP({\\cat U})$, the space of lines in the\ncomplete universe ${\\cat U}$.\n\nDenote by $\\R\\bP^{n}_{tw}=\\bP(\\R^{n+1,\\left\\lfloor \\frac{n+1}{2}\n\\right\\rfloor})$, the equivariant space of lines in\n$\\R^{n+1,\\left\\lfloor \\frac{n+1}{2} \\right\\rfloor}$. For example,\n$\\R\\bP^3_{tw} = \\bP(\\R^{4,2})$, $\\R\\bP^4_{tw} = \\bP(\\R^{5,2})$, and\n$\\R\\bP^1_{tw} = S^{1,1}$. There are natural cellular inclusions\n$\\R\\bP^n_{tw} \\inc \\R\\bP^{n+1}_{tw}$, the colimit of which is $\\R\\bP^\\infty_{tw}$.\n\n\n\n\n\\begin{lem} $\\R \\bP^{n}_{tw}$ has a $\\text{Rep}(\\Z\/2)$-structure with cells in dimension $(0,0)$, $(1,1)$, $(2,1)$, $(3,2)$, $(4,2)$, $\\dots, (n,\\left\\lceil \\frac{n}{2} \\right\\rceil)$.\n\\end{lem}\n\\begin{proof} This follows from Proposition \\ref{prop:schubertcells} using the flag symbol $\\varphi=(2, 4, 6, \\dots)$. \n\\end{proof}\n\n\\begin{lem}\n$\\R \\bP^\\infty_{tw}$ has a cell\nstructure with a single cell in dimension $(n, \\left\\lceil\n\\frac{n}{2} \\right\\rceil)$, for all $n \\in \\N$.\n\\end{lem}\n\\begin{proof}\nThe inclusions $\\R\\bP^{1}_{tw} \\inc \\R\\bP^{2}_{tw} \\inc\n\\cdots$ are cellular and their colimit is $\\R\\bP^\\infty_{tw}$. \n\\end{proof}\n\n\n\\begin{prop}\n\nAs a $H^{*,*}(pt)$-module, $H^{*,*}(\\R \\bP^{n}_{tw})$ is free with a\nsingle generator in each degree $(k, \\left\\lceil \\frac{k}{2}\n\\right\\rceil)$ for $k= 0, 1, \\dots, n$.\n\n\\end{prop}\n\\begin{proof}\nAny nonzero differentials in the cellular spectral sequence associated to the cell structure using the flag symbol $\\varphi = (2,4,6,\\dots)$ would decrease the number of cohomology generators below the number of cells. (See Figures \\ref{fig:nodd} and \\ref{fig:neven}.) By Theorem \\ref{thm:cellbij} this is not the case, and so the cohomology generators have degrees matching the dimensions of the cells. \n\\end{proof}\n\n\n\\begin{figure}[htbp]\n\\centering\n\\begin{picture}(330,230)(-150,-110)\n\n\\put(-110,-50){\\vector(1,0){235}}\n\\put(-50,-100){\\vector(0,1){200}}\n\n\n\n\\put(-50, -51){\\line(0,1){130}} \\put(-50, -51){\\line(1,1){130}}\n\\put(-50, -91){\\line(0,-1){20}} \\put(-50, -91){\\line(-1,-1){20}}\n\\put(-50, -51){\\circle*{2}}\n\n\\put(-28, -31){\\line(0,1){110}} \\put(-28, -31){\\line(1,1){110}}\n\\put(-28, -71){\\line(0,-1){40}} \\put(-28, -71){\\line(-1,-1){40}}\n\\put(-28, -31){\\circle*{2}} \\put(-26,-35){$a_{1,1}$}\n\n\\put(-10, -31){\\line(0,1){110}} \\put(-10, -31){\\line(1,1){110}}\n\\put(-10, -71){\\line(0,-1){40}} \\put(-10, -71){\\line(-1,-1){40}}\n\\put(-10, -31){\\circle*{2}} \\put(-6,-35){$a_{2,1}$}\n\n\\put(12, -11){\\line(0,1){90}} \\put(12, -11){\\line(1,1){90}} \\put(12,\n-51){\\line(0,-1){60}} \\put(12, -51){\\line(-1,-1){60}} \\put(12,\n-11){\\circle*{2}}\n\n\\put(32, -11){\\line(0,1){90}} \\put(32, -11){\\line(1,1){70}} \\put(32,\n-51){\\line(0,-1){60}} \\put(32, -51){\\line(-1,-1){60}} \\put(32,\n-11){\\circle*{2}}\n\n\\put(52, 10){\\line(0,1){70}} \\put(52, 10){\\line(1,1){50}} \\put(52,\n-30){\\line(0,-1){80}} \\put(52, -30){\\line(-1,-1){80}} \\put(52,\n10){\\circle*{2}} \\put(55,7){$a_{n,(n-1)\/2}$}\n\n\n\\multiput(-90,-51)(20,0){11}{\\line(0,1){3}}\n\\multiput(-52,-90)(0,20){10}{\\line(1,0){3}}\n\\put(-46,-57){$\\scriptscriptstyle{0}$}\n\\put(-26,-57){$\\scriptscriptstyle{1}$}\n\\put(-6,-57){$\\scriptscriptstyle{2}$}\n\\put(14,-57){$\\scriptscriptstyle{3}$}\n\\put(34,-57){$\\scriptscriptstyle{4}$}\n\\put(-66,-57){$\\scriptscriptstyle{-1}$}\n\\put(-86,-57){$\\scriptscriptstyle{-2}$}\n\\put(-57,-47){$\\scriptscriptstyle{0}$}\n\\put(-57,-27){$\\scriptscriptstyle{1}$}\n\\put(-57,-7){$\\scriptscriptstyle{2}$}\n\\put(-57,13){$\\scriptscriptstyle{3}$}\n\n\n\n\\put(-58,95){${q}$} \\put(115,-60){${p}$}\n\\end{picture}\n\n\\caption{The $E_1$ page of the cellular spectral sequence for\n$\\R\\bP^{n}_{tw}$ for $n$ odd.} \\label{fig:nodd}\n\\end{figure}\n\n\n\\begin{figure}[htbp]\n\\centering\n\\begin{picture}(330,230)(-150,-110)\n\\put(-110,-50){\\vector(1,0){235}}\n\\put(-50,-100){\\vector(0,1){200}}\n\n\n\n\\put(-50, -51){\\line(0,1){130}} \\put(-50, -51){\\line(1,1){130}}\n\\put(-50, -91){\\line(0,-1){20}} \\put(-50, -91){\\line(-1,-1){20}}\n\\put(-50, -51){\\circle*{2}}\n\n\\put(-28, -31){\\line(0,1){110}} \\put(-28, -31){\\line(1,1){110}}\n\\put(-28, -71){\\line(0,-1){40}} \\put(-28, -71){\\line(-1,-1){40}}\n\\put(-28, -31){\\circle*{2}} \\put(-26,-35){$a_{1,1}$}\n\n\\put(-10, -31){\\line(0,1){110}} \\put(-10, -31){\\line(1,1){110}}\n\\put(-10, -71){\\line(0,-1){40}} \\put(-10, -71){\\line(-1,-1){40}}\n\\put(-10, -31){\\circle*{2}} \\put(-6,-35){$a_{2,1}$}\n\n\\put(12, -11){\\line(0,1){90}} \\put(12, -11){\\line(1,1){90}} \\put(12,\n-51){\\line(0,-1){60}} \\put(12, -51){\\line(-1,-1){60}} \\put(12,\n-11){\\circle*{2}}\n\n\\put(32, -11){\\line(0,1){90}} \\put(32, -11){\\line(1,1){70}} \\put(32,\n-51){\\line(0,-1){60}} \\put(32, -51){\\line(-1,-1){60}} \\put(32,\n-11){\\circle*{2}}\n\n\\put(52, 10){\\line(0,1){70}} \\put(52, 10){\\line(1,1){50}} \\put(52,\n-30){\\line(0,-1){80}} \\put(52, -30){\\line(-1,-1){80}} \\put(52,\n10){\\circle*{2}}\n\n\\put(70, 10){\\line(0,1){70}} \\put(70, 10){\\line(1,1){30}} \\put(70,\n-30){\\line(0,-1){80}} \\put(70, -30){\\line(-1,-1){80}} \\put(70,\n10){\\circle*{2}} \\put(73,7){$a_{n,n\/2}$}\n\n\\multiput(-90,-51)(20,0){11}{\\line(0,1){3}}\n\\multiput(-52,-90)(0,20){10}{\\line(1,0){3}}\n\\put(-46,-57){$\\scriptscriptstyle{0}$}\n\\put(-26,-57){$\\scriptscriptstyle{1}$}\n\\put(-6,-57){$\\scriptscriptstyle{2}$}\n\\put(14,-57){$\\scriptscriptstyle{3}$}\n\\put(34,-57){$\\scriptscriptstyle{4}$}\n\\put(-66,-57){$\\scriptscriptstyle{-1}$}\n\\put(-86,-57){$\\scriptscriptstyle{-2}$}\n\\put(-57,-47){$\\scriptscriptstyle{0}$}\n\\put(-57,-27){$\\scriptscriptstyle{1}$}\n\\put(-57,-7){$\\scriptscriptstyle{2}$}\n\\put(-57,13){$\\scriptscriptstyle{3}$}\n\n\n\n\\put(-58,95){${q}$} \\put(115,-60){${p}$}\n\\end{picture}\n\\caption{The $E_1$ page of the cellular spectral sequence for\n$\\R\\bP^{n}_{tw}$ for $n$ even.} \\label{fig:neven}\n\\end{figure}\n\n\n\n\n\\begin{prop}\n\nAs a $H^{*,*}(pt)$-module, $H^{*,*}(\\R \\bP^\\infty_{tw})$ is free\nwith a single generator in each degree $(n, \\left\\lceil \\frac{n}{2}\n\\right\\rceil)$, for all $n \\in \\N$.\n\n\\end{prop}\n\n\\begin{proof}\n\n$\\R\\bP^\\infty_{tw}$ is the colimit of the above projective spaces.\nThus, any non-zero differential for $\\R\\bP^\\infty_{tw}$ would induce\na non-zero differential at some finite stage. By the above proposition, this is not the case. \n\n\\end{proof}\n\n\\begin{lem}\n\nAs a $H^{*,*}(pt)$-module, $H^{*,*}(S^{1,1})$ is free with a single\ngenerator $a$ in degree $(1,1)$. As a ring, $H^{*,*}(S^{1,1}) \\cong\nH^{*,*}(pt)[a]\/(a^2 = \\rho a)$.\n\n\\end{lem}\n\\begin{proof}\n\nThe statement about the module structure is immediate since $S^{1,1}\n\\cong \\R\\bP^1_{tw}$.\n\nSince $S^{1,1}$ is a $K(\\Z(1),1)$, we can consider $a \\in\n[S^{1,1},S^{1,1}]$ as the class of the identity and $\\rho \\in [pt,\nS^{1,1}]$ as the inclusion. Then $a^2$ is the composite\n\n$\\xymatrix{a^2 \\colon S^{1,1} \\ar[r]^(0.45)\\Delta & S^{1,1} \\Smash\nS^{1,1} \\ar[r]^(0.6){a \\Smash a} & S^{2,2} \\ar[r] & K(\\Z\/2(2),2)}$.\n\n\\noindent Similarly, $\\rho a$ is the composite\n\n$\\xymatrix{\\rho a \\colon S^{1,1} \\ar[r] & S^{0,0} \\Smash S^{1,1}\n\\ar[r]^(0.6){\\rho \\Smash a} & S^{2,2} \\ar[r] & K(\\Z\/2(2),2)}$.\n\n\\noindent The claim is that these two maps are homotopic.\nConsidering the spheres involved as one point compactifications of\nthe corresponding representations, the map $a^2$ is inclusion of\n$(\\R^{1,1})^+$ as the diagonal in $(\\R^{2,2})^+$ and $\\rho a$ is\ninclusion of $(\\R^{1,1})^+$ as the vertical axis. There is then an\nequivariant homotopy $H\\colon (\\R^{1,1})^+ \\times I \\ra\n(\\R^{2,2})^+$ between these two maps given by $H(x,t) = (t x, x)$.\n\n\n\\end{proof}\n\n\nFrom here, we are poised to compute the ring structure of the $RO(\\Z\/2)$-graded cohomology of each real projective space.\n\n\n\\begin{thm}\n\\label{thm:rpinfty} $H^{*,*}(\\R\\bP^\\infty_{tw})=H^{*,*}(pt)[a,\nb]\/(a^2=\\rho a +\\tau b)$, where $\\deg(a)=(1,1)$ and $\\deg(b)=(2,1)$.\n\\end{thm}\n\\begin{proof}\nIt remains to compute the multiplicative\nstructure of the cohomology ring. Denote by $a = a_{(1, 1)}$, and $b=a_{(2,1)}$.\nBy Lemma \\ref{lemma:forget}, the forgetful map $\\psi\\colon H^{*,*}(\\R\n\\bP^\\infty_{tw}) \\ra\nH_{sing}^*(\\R\\bP^\\infty)$ maps $\\psi(a) = z$ and $\\psi(b) = z^2$\nwhere $z \\in H^1_{sing}(\\R\\bP^\\infty)$ is the ring generator for\nnon-equivariant cohomology. Since $\\psi$ is a homomorphism of rings,\n$\\psi(ab)=z^3 \\neq 0$, and so the product $ab$ is nonzero in $H^{*,*}(\\R\n\\bP^\\infty_{tw})$.\nObserve that $\\rho b$ is also in degree $(3,2)$ in $H^{*,*}(\\R\n\\bP^\\infty_{tw})$, but\n$\\psi(\\rho b) = 0$ since $\\psi(\\rho)=0$. Thus $ab$ and $\\rho b$\ngenerate $H^{*,*}(\\R\n\\bP^\\infty_{tw})$ in degree $(3, 2)$. Also, $\\psi(b^2)=z^4$, and so\n$b^2$ in nonzero in $H^{*,*}(\\R\n\\bP^\\infty_{tw})$. This means that $b^2$ is the unique\nnonzero element of $H^{*,*}(\\R\n\\bP^\\infty_{tw})$ in degree $(4,2)$. Inductively, it can be\nshown that if $n$ is even the unique nonzero element of $R$ in\ndegree $(n,\\frac{n}{2})$ is $b^{n\/2}$ and that if $n$ is odd, then\n$ab^{(n-1)\/2}$ is linearly independent from $\\rho b^{(n-1)\/2}$.\n\nNow, $a^2 \\in H^{2,2}(\\R \\bP^\\infty_{tw})$ and so is a linear\ncombination of $\\rho a$ and $\\tau b$. Since $\\psi(a^2)=z^2$, there\nmust be a $\\tau b$ term in the expression for $a^2$. Also, upon\nrestriction to $\\R\\bP^1_{tw}=S^{1,1}$, $a^2$ restricts to $a^2=\\rho\na$. Thus, $a^2=\\rho a + \\tau b \\in H^{*,*}(\\R\n\\bP^\\infty_{tw})$.\n\n\n\n\\end{proof}\n\n\\begin{thm} Let $n > 2$. If $n$ is even, then $H^{*,*}(\\bP(\\R^{n,\\frac{n}{2}})) = H^{*,*}(pt)[a_{1,1},b_{2,1}]\/ \\sim$ where the generating relations are $a^2 = \\rho a + \\tau b$ and $b^k =0$ for $k \\geq \\frac{n}{2}$. If $n$ is odd, then $H^{*,*}(\\bP(\\R^{n,\\frac{n-1}{2}})) = H^{*,*}(pt)[a_{1,1},b_{2,1}]\/ \\sim$ where the generating relations are $a^2 = \\rho a + \\tau b$, $b^k =0$ for $k \\geq \\frac{n+1}{2}$, and $a\\cdot b^{(n-1)\/2}=0$.\n\n\\end{thm}\n\n\\begin{proof}\nOnly the multiplicative structure needs to be checked since the\ncohomology is free and the generators given above are in the correct\ndegrees. Considering the restriction of the corresponding\nclasses $a$ and $b$ in $H^{*,*}(\\R\\bP^\\infty_{tw})$, the relation\n$a^2 = \\rho a + \\tau b$ is immediate. The relations $b^k =0$ for $k\n> \\frac{n}{2}$ when $n$ is even and $b^k =0$ for $k \\geq\n\\frac{n+1}{2}$ when $n$ is odd follow for degree reasons. Also,\nsince the class $ab^{(n-1)\/2} \\in H^{*,*}(\\R\\bP^\\infty_{tw})$ is a free\ngenerator, it restricts to zero in\n$H^{*,*}(\\bP(\\R^{n,\\frac{n-1}{2}}))$. Thus $ab^{(n-1)\/2} = 0 \\in\nH^{*,*}(\\bP(\\R^{n,\\frac{n-1}{2}}))$.\n\n\\end{proof}\n\n\nWe can also compute the cohomology of projective spaces associated\nto arbitrary representations. The following easy lemma will be\nuseful. In particular, it allows us to only consider the projective\nspaces associated to representations $V \\cong \\R^{p,q}$ where $q\\leq\np\/2$.\n\n\\begin{lem}\n\\label{lemma:easyRP} $\\bP(\\R^{p,q}) \\cong \\bP(\\R^{p,p-q})$.\n\n\n\\end{lem}\n\\begin{proof}\nConsider a basis of $\\R^{p,q}$ in which the first $q$ coordinates\nhave the nontrivial action, and a basis of $\\R^{p,p-q}$ in which the\nfirst $q$ coordinates are fixed by the action. Then the map $f\n\\colon \\bP(\\R^{p,q}) \\ra \\bP(\\R^{p,p-q})$ that sends the span of\n$(x_1, \\dots , x_p)$ to the span of $(x_1, \\dots , x_p)$ is\nequivariant. It is clearly a homeomorphism. \n\\end{proof}\n\n\\begin{lem}If $q \\leq p\/2$, then $\\bP(\\R^{p,q})$ has a cell structure with a single cell in each dimension $(0,0)$, $(1,1)$, $(2,1)$, $(3,2)$, $(4,2), \\dots,$ $(2q-1,q)$, $(2q,q)$, $\\dots,$ $(p-1,q)$.\n\\end{lem}\n\nFor example, $\\bP(\\R^{4,1})$ has a single cell in each dimension\n$(0,0)$, $(1,1)$, $(2,1)$, and $(3,1)$.\n\n\\begin{proof}\nThe result follows by Proposition \\ref{prop:schubertcells} using the flag symbol $\\varphi=(2,4,\\dots,2q)$. \n\n\\end{proof}\n\n\\begin{lem}\n\\label{lemma:RPmodule} As a $H^{*,*}(pt)$-module,\n$H^{*,*}(\\bP(\\R^{p,q}))$ is free with a single generator in\ndegrees $(0,0)$, $(1,1)$, $(2,1)$, $(3,2)$, $(4,2), \\dots,$\n$(2q,q)$, $(2q+1,q), \\dots,$ $(p-1,q)$.\n\n\\end{lem}\n\n\\begin{proof}\n\nUsing the cell structure in the previous lemma, Theorem \\ref{thm:cellbij} implies there can be no nonzero differentials in the cellular spectral sequence.\n\n\\end{proof}\n\n\nThe ring structure of the other projective spaces can be computed by considering the restriction of\n$H^{*,*}(\\R\\bP^\\infty_{tw})$ to $H^{*,*}(\\bP(\\R^{p,q}))$.\n\\begin{prop}\n\\label{prop:ringRP} $H^{*,*}(\\bP(\\R^{p,q}))$ is a truncated\npolynomial algebra over $H^{*,*}(pt)$ on generators in degrees\n$(1,1)$, $(2,1)$, $(2q+1,q)$, $(2q+2,q), \\dots,$ $(p-1,q)$, subject\nto the relations determined by the restriction of\n\n\n\n\n\n\n$H^{*,*}(\\R\\bP^\\infty_{tw})$ to $H^{*,*}(\\bP(\\R^{p,q}))$.\n\n\\end{prop}\n\nFor example, consider $\\bP(\\R^{4,1})$. By the above proposition, $H^{*,*}(\\bP(\\R^{4,1}))$ is generated by classes\n$a_{1,1}$, $b_{2,1}$, and $c_{3,1}$. The classes $a$\nand $b$ in $H^{*,*}(\\R\\bP^\\infty_{tw})$ restrict to $a$ and $b$\nrespectively, so $a^2 = \\rho a + \\tau b$ in\n$H^{*,*}(\\bP(\\R^{4,1}))$. Now, $ab$ has degree $(3,2)$ and so $ab =\n? \\rho b + ? \\tau c$. However, the product $ab$ in\n$H^{*,*}(\\R\\bP^\\infty_{tw})$ restricts to the class $\\tau c$. Since\nrestriction is a map of rings, $ab = \\tau c$ in\n$H^{*,*}(\\bP(\\R^{4,1}))$. Similar considerations show that $bc =0$\nand $c^2=0$. Thus $H^{*,*}(\\bP(\\R^{4,1}))=H^{*,*}(pt)[a_{1,1},\nb_{2, 1}, c_{3,1}]\/\\sim$, where the generating relations are $a^2 =\n\\rho a + \\tau b$, $ab = \\tau c$, $bc=0$, and $c^2=0$.\n\nIn some cases, the Freeness Theorem is enough to determine the\nadditive structure of the $RO(\\Z\/2)$-graded cohomology of Grassmann\nmanifolds.\n\n\\begin{prop} $G_2(\\R^{p,1})$ has a $\\text{Rep}(\\Z\/2)$-complex structure so that\n$H^{*,*}(G_n(\\R^{p,1}))$ is a free $H^{*,*}(pt)$-module on\ngenerators whose degree match the dimensions of the cells.\n\\end{prop}\n\n\\begin{proof}\nUsing the flag symbol $\\varphi=(2)$, every cell, except the\n$(0,0)$-cell, has either one or two twists. The cells are in\nbidegrees so that there can be no dimension shifting in the cellular\nspectral sequence. The result now follows by Theorem\n\\ref{thm:cellbij}. \n\\end{proof}\n\nFor example, $H^{*,*}(G_2(\\R^{4,1});\\underline{\\Z\/2})$ is a free\n$H^{*,*}(pt;\\underline{\\Z\/2})$-module with generators in degrees\n$(0,0)$, $(1,1)$, $(2,1)$, $(2,1)$, $(3,1)$, and $(4,2)$ (see Figure\n\\ref{fig:g2r41}).\n\n\n\n\\begin{figure}[htbp]\n\n\\centering\n\n\\begin{picture}(330,230)(-150,-110)\n\\put(-110,-50){\\vector(1,0){235}}\n\\put(-50,-100){\\vector(0,1){200}}\n\n\n\n\\put(-50, -51){\\line(0,1){70}} \\put(-50, -51){\\line(1,1){150}}\n\\put(-50, -91){\\line(0,-1){20}} \\put(-50, -91){\\line(-1,-1){20}}\n\\put(-50, -51){\\circle*{2}}\n\n\\put(-28, -31){\\line(0,1){130}} \\put(-28, -31){\\line(1,1){130}}\n\\put(-28, -71){\\line(0,-1){40}} \\put(-28, -71){\\line(-1,-1){40}}\n\\put(-28, -31){\\circle*{2}}\n\n\\put(-10, -31){\\line(0,1){130}} \\put(-10, -31){\\line(1,1){110}}\n\\put(-10, -71){\\line(0,-1){40}} \\put(-10, -71){\\line(-1,-1){40}}\n\\put(-10, -31){\\circle*{2}}\n\n\\put(-8, -31){\\line(0,1){130}} \\put(-8, -31){\\line(1,1){110}}\n\\put(-8, -71){\\line(0,-1){40}} \\put(-8, -71){\\line(-1,-1){40}}\n\\put(-8, -31){\\circle*{2}}\n\n\\put(10, -31){\\line(0,1){130}} \\put(10, -31){\\line(1,1){90}}\n\\put(10, -71){\\line(0,-1){40}} \\put(10, -71){\\line(-1,-1){40}}\n\\put(10, -31){\\circle*{2}}\n\n\\put(32, -11){\\line(0,1){110}} \\put(32, -11){\\line(1,1){70}}\n\\put(32, -51){\\line(0,-1){60}} \\put(32, -51){\\line(-1,-1){60}}\n\\put(32, -11){\\circle*{2}}\n\n\n\\multiput(-90,-51)(20,0){11}{\\line(0,1){3}}\n\\multiput(-52,-90)(0,20){10}{\\line(1,0){3}}\n\\put(-46,-57){$\\scriptscriptstyle{0}$}\n\\put(-26,-57){$\\scriptscriptstyle{1}$}\n\\put(-6,-57){$\\scriptscriptstyle{2}$}\n\\put(14,-57){$\\scriptscriptstyle{3}$}\n\\put(34,-57){$\\scriptscriptstyle{4}$}\n\\put(-66,-57){$\\scriptscriptstyle{-1}$}\n\\put(-86,-57){$\\scriptscriptstyle{-2}$}\n\\put(-57,-47){$\\scriptscriptstyle{0}$}\n\\put(-57,-27){$\\scriptscriptstyle{1}$}\n\\put(-57,-7){$\\scriptscriptstyle{2}$}\n\\put(-57,13){$\\scriptscriptstyle{3}$}\n\n\n\n\\put(-58,95){${q}$} \\put(125,-60){${p}$}\n\\end{picture}\n\\caption{$H^{*,*}(G_2(\\R^{4,1}))$} \\label{fig:g2r41}\n\\end{figure}\n\n\nInterestingly, there are situations where there must be nonzero\ndifferentials in the cellular spectral sequences.\n\n\nAs another example, consider now $X=G_2(\\R^{4,2})$. Consider the\nthree flag symbols $\\varphi_1=(2,3)$, $\\varphi_2=(2,4)$, and\n$\\varphi_3=(3,4)$. The spectral sequences associated to the cell\nstructures with these flag symbols have $E_1$ term given in Figures\n\\ref{fig:g2r42phi23}, \\ref{fig:g2r42phi24}, and \\ref{fig:g2r42phi34}\nrepsectively.\n\n\\begin{figure}[htpb]\n\n\\centering\n\n\\begin{picture}(330,230)(-150,-110)\n\\put(-110,-50){\\vector(1,0){235}}\n\\put(-50,-100){\\vector(0,1){200}}\n\n\\put(-50, -51){\\line(0,1){130}} \\put(-50, -51){\\line(1,1){130}}\n\\put(-50, -91){\\line(0,-1){20}} \\put(-50, -91){\\line(-1,-1){20}}\n\\put(-50, -51){\\circle*{2}}\n\n\\put(-28, -51){\\line(0,1){130}} \\put(-28, -51){\\line(1,1){130}}\n\\put(-28, -91){\\line(0,-1){20}} \\put(-28, -91){\\line(-1,-1){20}}\n\\put(-28, -51){\\circle*{2}}\n\n\\put(-10, -11){\\line(0,1){90}} \\put(-10, -11){\\line(1,1){90}}\n\\put(-10, -51){\\line(0,-1){60}} \\put(-10, -51){\\line(-1,-1){60}}\n\\put(-10, -11){\\circle*{2}}\n\n\\put(-8, -11){\\line(0,1){90}} \\put(-8, -11){\\line(1,1){90}} \\put(-8,\n-51){\\line(0,-1){60}} \\put(-8, -51){\\line(-1,-1){60}} \\put(-8,\n-11){\\circle*{2}}\n\n\\put(10, -11){\\line(0,1){90}} \\put(10, -11){\\line(1,1){90}} \\put(10,\n-51){\\line(0,-1){60}} \\put(10, -51){\\line(-1,-1){60}} \\put(10,\n-11){\\circle*{2}}\n\n\\put(32, -11){\\line(0,1){90}} \\put(32, -11){\\line(1,1){90}} \\put(32,\n-51){\\line(0,-1){60}} \\put(32, -51){\\line(-1,-1){60}} \\put(32,\n-11){\\circle*{2}}\n\n\n\n\\multiput(-90,-51)(20,0){11}{\\line(0,1){3}}\n\\multiput(-52,-90)(0,20){10}{\\line(1,0){3}}\n\\put(-46,-57){$\\scriptscriptstyle{0}$}\n\\put(-26,-57){$\\scriptscriptstyle{1}$}\n\\put(-6,-57){$\\scriptscriptstyle{2}$}\n\\put(14,-57){$\\scriptscriptstyle{3}$}\n\\put(34,-57){$\\scriptscriptstyle{4}$}\n\\put(-66,-57){$\\scriptscriptstyle{-1}$}\n\\put(-86,-57){$\\scriptscriptstyle{-2}$}\n\\put(-57,-47){$\\scriptscriptstyle{0}$}\n\\put(-57,-27){$\\scriptscriptstyle{1}$}\n\\put(-57,-7){$\\scriptscriptstyle{2}$}\n\\put(-57,13){$\\scriptscriptstyle{3}$}\n\n\n\\put(-58,95){${q}$} \\put(125,-60){${p}$}\n\\end{picture}\n\n\\caption{The $E_1$ page of the cellular spectral sequence for\n$G_2(\\R^{4,2})$ using $\\varphi_1=(2,3)$.} \\label{fig:g2r42phi23}\n\n\\end{figure}\n\n\n\\begin{figure}[htpb]\n\n\\centering\n\n\\begin{picture}(330,230)(-150,-110)\n\\put(-110,-50){\\vector(1,0){235}}\n\\put(-50,-100){\\vector(0,1){200}}\n\n\\put(-50, -51){\\line(0,1){130}} \\put(-50, -51){\\line(1,1){130}}\n\\put(-50, -91){\\line(0,-1){20}} \\put(-50, -91){\\line(-1,-1){20}}\n\\put(-50, -51){\\circle*{2}}\n\n\\put(-28, -31){\\line(0,1){110}} \\put(-28, -31){\\line(1,1){110}}\n\\put(-28, -71){\\line(0,-1){40}} \\put(-28, -71){\\line(-1,-1){40}}\n\\put(-28, -31){\\circle*{2}}\n\n\\put(-10, -31){\\line(0,1){110}} \\put(-10, -31){\\line(1,1){110}}\n\\put(-10, -71){\\line(0,-1){40}} \\put(-10, -71){\\line(-1,-1){40}}\n\\put(-10, -31){\\circle*{2}}\n\n\\put(-8, -31){\\line(0,1){110}} \\put(-8, -31){\\line(1,1){110}}\n\\put(-8, -71){\\line(0,-1){40}} \\put(-8, -71){\\line(-1,-1){40}}\n\\put(-8, -31){\\circle*{2}}\n\n\\put(10, 10){\\line(0,1){70}} \\put(10, 10){\\line(1,1){70}} \\put(10,\n-31){\\line(0,-1){80}} \\put(10, -31){\\line(-1,-1){80}} \\put(10,\n10){\\circle*{2}}\n\n\\put(32, -11){\\line(0,1){90}} \\put(32, -11){\\line(1,1){90}} \\put(32,\n-51){\\line(0,-1){60}} \\put(32, -51){\\line(-1,-1){60}} \\put(32,\n-11){\\circle*{2}}\n\n\n\n\\multiput(-90,-51)(20,0){11}{\\line(0,1){3}}\n\\multiput(-52,-90)(0,20){10}{\\line(1,0){3}}\n\\put(-46,-57){$\\scriptscriptstyle{0}$}\n\\put(-26,-57){$\\scriptscriptstyle{1}$}\n\\put(-6,-57){$\\scriptscriptstyle{2}$}\n\\put(14,-57){$\\scriptscriptstyle{3}$}\n\\put(34,-57){$\\scriptscriptstyle{4}$}\n\\put(-66,-57){$\\scriptscriptstyle{-1}$}\n\\put(-86,-57){$\\scriptscriptstyle{-2}$}\n\\put(-57,-47){$\\scriptscriptstyle{0}$}\n\\put(-57,-27){$\\scriptscriptstyle{1}$}\n\\put(-57,-7){$\\scriptscriptstyle{2}$}\n\\put(-57,13){$\\scriptscriptstyle{3}$}\n\n\n\\put(-58,95){${q}$} \\put(125,-60){${p}$}\n\\end{picture}\n\\caption{The $E_1$ page of the cellular spectral sequence for\n$G_2(\\R^{4,2})$ using $\\varphi_2=(2,4)$.} \\label{fig:g2r42phi24}\n\n\\end{figure}\n\n\\begin{figure}[htpb]\n\n\\centering\n\n\\begin{picture}(330,230)(-150,-110)\n\\put(-110,-50){\\vector(1,0){235}}\n\\put(-50,-100){\\vector(0,1){200}}\n\n\\put(-50, -51){\\line(0,1){130}} \\put(-50, -51){\\line(1,1){130}}\n\\put(-50, -91){\\line(0,-1){20}} \\put(-50, -91){\\line(-1,-1){20}}\n\\put(-50, -51){\\circle*{2}}\n\n\\put(-28, -31){\\line(0,1){110}} \\put(-28, -31){\\line(1,1){110}}\n\\put(-28, -71){\\line(0,-1){40}} \\put(-28, -71){\\line(-1,-1){40}}\n\\put(-28, -31){\\circle*{2}}\n\n\\put(-10, -31){\\line(0,1){110}} \\put(-10, -31){\\line(1,1){110}}\n\\put(-10, -71){\\line(0,-1){40}} \\put(-10, -71){\\line(-1,-1){40}}\n\\put(-10, -31){\\circle*{2}}\n\n\\put(-8, -31){\\line(0,1){110}} \\put(-8, -31){\\line(1,1){110}}\n\\put(-8, -71){\\line(0,-1){40}} \\put(-8, -71){\\line(-1,-1){40}}\n\\put(-8, -31){\\circle*{2}}\n\n\\put(10, -31){\\line(0,1){110}} \\put(10, -31){\\line(1,1){110}}\n\\put(10, -71){\\line(0,-1){40}} \\put(10, -71){\\line(-1,-1){40}}\n\\put(10, -31){\\circle*{2}}\n\n\\put(32, 29){\\line(0,1){50}} \\put(32, 29){\\line(1,1){50}} \\put(32,\n-11){\\line(0,-1){100}} \\put(32, -11){\\line(-1,-1){100}} \\put(32,\n29){\\circle*{2}}\n\n\n\n\\multiput(-90,-51)(20,0){11}{\\line(0,1){3}}\n\\multiput(-52,-90)(0,20){10}{\\line(1,0){3}}\n\\put(-46,-57){$\\scriptscriptstyle{0}$}\n\\put(-26,-57){$\\scriptscriptstyle{1}$}\n\\put(-6,-57){$\\scriptscriptstyle{2}$}\n\\put(14,-57){$\\scriptscriptstyle{3}$}\n\\put(34,-57){$\\scriptscriptstyle{4}$}\n\\put(-66,-57){$\\scriptscriptstyle{-1}$}\n\\put(-86,-57){$\\scriptscriptstyle{-2}$}\n\\put(-57,-47){$\\scriptscriptstyle{0}$}\n\\put(-57,-27){$\\scriptscriptstyle{1}$}\n\\put(-57,-7){$\\scriptscriptstyle{2}$}\n\\put(-57,13){$\\scriptscriptstyle{3}$}\n\n\n\\put(-58,95){${q}$} \\put(125,-60){${p}$}\n\\end{picture}\n\\caption{The $E_1$ page of the cellular spectral sequence for\n$G_2(\\R^{4,2})$ using $\\varphi_3=(3,4)$.} \\label{fig:g2r42phi34}\n\n\\end{figure}\n\nThe cohomology of $X$ can be deduced by comparing these three\ncellular spectral sequences. We can see from the picture for\n$\\varphi_2$ that $H^{1,0}(X)=0$, and so the differential leaving the\n$(1,0)$ generator in the $\\varphi_1$ spectral sequence is non-zero.\nThus, $H^{1,1}(X)=\\Z\/2$, $H^{2,1}(X)=\\Z\/2$ and $H^{2,0}(X)=\\Z\/2$.\nIn particular, there is a free generator in degree $(1,1)$ and there\nis a nontrivial differential leaving the $(2,1)$ generators of the\nspectral sequence for $\\varphi_2$. After a change of basis, if\nnecessary, the differential can be adjusted so that it is zero on\none of the $(2,1)$ generators and the other generator maps\nnontrivially. Now from $\\varphi_1$ we see that $H^{4,1}(X)=0$, and\nso there is a nontrivial differential leaving the $(3,1)$ generator\nin the $\\varphi_3$ spectral sequence. This means that the $(4,2)$\ngenerator in the $\\varphi_1$ and $\\varphi_2$ spectral sequences must\nsurvive. Thus, all differentials in the $\\varphi_2$ spectral\nsequence are known. They are all zero, except for the one leaving\nthe two $(2,1)$ generators, which behaves as described above. That\nspectral sequence collapses almost immediately to give the\ncohomology of $G_2(\\R^{4,2})$ pictured in Figure \\ref{fig:g2r42}.\n\n\\begin{figure}[htpb]\n\n\\centering\n\n\\begin{picture}(330,230)(-150,-110)\n\\put(-110,-50){\\vector(1,0){235}}\n\\put(-50,-100){\\vector(0,1){200}}\n\n\\put(-50, -51){\\line(0,1){130}} \\put(-50, -51){\\line(1,1){130}}\n\\put(-50, -91){\\line(0,-1){20}} \\put(-50, -91){\\line(-1,-1){20}}\n\\put(-50, -51){\\circle*{2}}\n\n\\put(-28, -31){\\line(0,1){110}} \\put(-28, -31){\\line(1,1){110}}\n\\put(-28, -71){\\line(0,-1){40}} \\put(-28, -71){\\line(-1,-1){40}}\n\\put(-28, -31){\\circle*{2}}\n\n\\put(-10, -31){\\line(0,1){110}} \\put(-10, -31){\\line(1,1){110}}\n\\put(-10, -71){\\line(0,-1){40}} \\put(-10, -71){\\line(-1,-1){40}}\n\\put(-10, -31){\\circle*{2}}\n\n\\put(-8, -13){\\line(0,1){90}} \\put(12, -11){\\line(1,1){90}} \\put(-8,\n-71){\\line(0,-1){40}} \\put(-8, -71){\\line(-1,-1){40}} \\put(-8,\n-13){\\circle*{2}} \\put(12, -11){\\circle*{2}}\n\n\\put(10, 10){\\line(0,1){70}} \\put(10, 10){\\line(1,1){70}} \\put(10,\n-51){\\line(0,-1){60}} \\put(-10, -51){\\line(-1,-1){40}} \\put(10,\n10){\\circle*{2}} \\put(10, -51){\\circle*{2}} \\put(-10,\n-51){\\circle*{2}}\n\n\\put(32, -11){\\line(0,1){90}} \\put(32, -11){\\line(1,1){70}} \\put(32,\n-51){\\line(0,-1){60}} \\put(32, -51){\\line(-1,-1){60}} \\put(32,\n-11){\\circle*{2}}\n\n\n\n\\multiput(-90,-51)(20,0){11}{\\line(0,1){3}}\n\\multiput(-52,-90)(0,20){10}{\\line(1,0){3}}\n\\put(-46,-57){$\\scriptscriptstyle{0}$}\n\\put(-26,-57){$\\scriptscriptstyle{1}$}\n\\put(-6,-57){$\\scriptscriptstyle{2}$}\n\\put(14,-57){$\\scriptscriptstyle{3}$}\n\\put(34,-57){$\\scriptscriptstyle{4}$}\n\\put(-66,-57){$\\scriptscriptstyle{-1}$}\n\\put(-86,-57){$\\scriptscriptstyle{-2}$}\n\\put(-57,-47){$\\scriptscriptstyle{0}$}\n\\put(-57,-27){$\\scriptscriptstyle{1}$}\n\\put(-57,-7){$\\scriptscriptstyle{2}$}\n\\put(-57,13){$\\scriptscriptstyle{3}$}\n\n\n\\put(-58,95){${q}$} \\put(125,-60){${p}$}\n\\end{picture}\n\\caption{$H^{*,*}(G_2(\\R^{4,2}))$} \\label{fig:g2r42}\n\\end{figure}\n\nBy the Freeness Theorem \\ref{thm:freeness}, we know that\n$H^{*,*}(G_2(\\R^{4,2}))$ is free. Counting the $\\Z\/2$ vector space\ndimensions in each bidegree reveals that the degrees are the same as\nthose of a free $H^{*,*}(pt)$-module with generators in degrees\n$(1,1)$, $(2,1)$, $(2,2)$, ($3,2)$, and $(4,2)$. This is the only\nfree $H^{*,*}(pt)$-module with these $\\Z\/2$ dimensions, and so we\nhave the following computation.\n\n\\begin{prop}$H^{*,*}(G_2(\\R^{4,2}))$ is a free $H^{*,*}(pt)$-module with generators in degrees $(1,1)$, $(2,1)$, $(2,2)$, $(3,2)$, and $(4,2)$.\n\\end{prop}\n\nThat is, $H^{*,*}(G_2(\\R^{4,2}))$ has free generators as displayed\nin Figure \\ref{fig:cohomg2r42}.\n\n\\begin{figure}[htpb]\n\n\\centering\n\n\\begin{picture}(330,230)(-150,-110)\n\\put(-110,-50){\\vector(1,0){235}}\n\\put(-50,-100){\\vector(0,1){200}}\n\n\\put(-50, -51){\\line(0,1){130}} \\put(-50, -51){\\line(1,1){130}}\n\\put(-50, -91){\\line(0,-1){20}} \\put(-50, -91){\\line(-1,-1){20}}\n\\put(-50, -51){\\circle*{2}}\n\n\\put(-28, -31){\\line(0,1){110}} \\put(-28, -31){\\line(1,1){110}}\n\\put(-28, -71){\\line(0,-1){40}} \\put(-28, -71){\\line(-1,-1){40}}\n\\put(-28, -31){\\circle*{2}}\n\n\\put(-10, -31){\\line(0,1){110}} \\put(-10, -31){\\line(1,1){110}}\n\\put(-10, -71){\\line(0,-1){40}} \\put(-10, -71){\\line(-1,-1){40}}\n\\put(-10, -31){\\circle*{2}}\n\n\\put(-8, -11){\\line(0,1){90}} \\put(-8, -11){\\line(1,1){90}} \\put(-8,\n-51){\\line(0,-1){60}} \\put(-8, -51){\\line(-1,-1){60}} \\put(-8,\n-11){\\circle*{2}}\n\n\\put(12, -11){\\line(0,1){90}} \\put(12, -11){\\line(1,1){90}} \\put(12,\n-51){\\line(0,-1){60}} \\put(12, -51){\\line(-1,-1){60}} \\put(12,\n-11){\\circle*{2}}\n\n\\put(30, -11){\\line(0,1){90}} \\put(30, -11){\\line(1,1){70}} \\put(30,\n-51){\\line(0,-1){60}} \\put(30, -51){\\line(-1,-1){60}} \\put(30,\n-11){\\circle*{2}}\n\n\n\n\\multiput(-90,-51)(20,0){11}{\\line(0,1){3}}\n\\multiput(-52,-90)(0,20){10}{\\line(1,0){3}}\n\\put(-46,-57){$\\scriptscriptstyle{0}$}\n\\put(-26,-57){$\\scriptscriptstyle{1}$}\n\\put(-6,-57){$\\scriptscriptstyle{2}$}\n\\put(14,-57){$\\scriptscriptstyle{3}$}\n\\put(34,-57){$\\scriptscriptstyle{4}$}\n\\put(-66,-57){$\\scriptscriptstyle{-1}$}\n\\put(-86,-57){$\\scriptscriptstyle{-2}$}\n\\put(-57,-47){$\\scriptscriptstyle{0}$}\n\\put(-57,-27){$\\scriptscriptstyle{1}$}\n\\put(-57,-7){$\\scriptscriptstyle{2}$}\n\\put(-57,13){$\\scriptscriptstyle{3}$}\n\n\n\\put(-58,95){${q}$} \\put(125,-60){${p}$}\n\\end{picture}\n\n\\caption{$H^{*,*}(G_2(\\R^{4,2}))$ with free generators shown.}\n\\label{fig:cohomg2r42}\n\\end{figure}\n\nIt should be noted that in the case of $G_2(\\R^{4,1})$, with the\nproper choice of flag symbols, the cell structure is such that the\ndifferentials are all zero, and so the cohomology is free with\ngenerators in the same degrees as the dimensions of the cells. This\nis \\bf{not} \\rm the case with $G_2(\\R^{4,2})$. Regardless of the\nchoice of flag symbol, there are some nonzero differentials which\ncause some degree shifting of the cohomology generators.\n\nUnfortunately, we cannot play this game indefinitely. For the\nGrassmann manifolds $G_n (\\R^{p,q})$ with $n$ and $q$ small enough,\nsay $n\\leq 2$ and $q\\leq 2$, the above techniques can be used to\nobtain the additive structure of $H^{*,*}(G_n (\\R^{p,q}))$. However,\nthere are examples where the precise degrees of the cohomology\ngenerators cannot be determined by comparing the cellular spectral\nsequences for various flag symbols. A serious inquiry into the\ngeometry of the attaching maps in these cell structures may reveal\nmore information.\n\\newpage\n\\bibliographystyle{alpha}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
{"text":"\\section{Introduction} \n\\setcounter{equation}{0}\nOn a compact $\\Spin$ surface, Th. Friedrich and E.C. Kim proved that any eigenvalue $\\lambda$ of the Dirac operator satisfies the equality \\cite[Thm. 4.5]{fk}:\n\\begin{equation}\\label{eq:Fk}\n\\lambda^2=\\frac{\\pi\\chi(M)}{Area(M)}+\\frac{1}{Area(M)}\\int_M|T^\\psi|^2v_g, \n\\end{equation}\nwhere $\\chi(M)$ is the Euler-Poincar\\'e characteristic of $M$ and $T^\\psi$ is the field of quadratic forms called the Energy-Momentum tensor. It is given on the complement set of zeroes of the eigenspinor $\\psi$ by \n$$ T^\\psi (X, Y) = g(\\ell^\\psi (X), Y) = \\frac 12 \\Re(X\\cdot\\nabla_Y\\psi + Y\\cdot\\nabla_X\\psi,\\frac{\\psi}{\\vert\\psi\\vert^2}),$$\nfor every $X, Y \\in \\Gamma(TM)$. Here $\\ell^\\psi$ is the field of symmetric endomorphisms associated with the field of quadratic forms $T^\\psi$. We should point out that since $\\psi$ is an eigenspinor, the zero set is discret \\cite{barrr}.\nThe proof of Equality (\\ref{eq:Fk}) relies mainly on a local expression of the covariant derivative of $\\psi$ and the use of the Schr\\\"odinger-Lichnerowicz formula. This equality has many direct consequences.\nFirst, since the trace of $\\ell^\\psi$ is equal to $\\lambda$, we have by the Cauchy-Schwarz inequality that $\\vert \\ell^\\psi\\vert^2 \\geqslant \\frac{(tr(\\ell^\\psi))^2}{n} = \\frac{\\lambda^2}{2},$ where $tr$ denotes the trace of $\\ell^\\psi$. Hence, Equality (\\ref{eq:Fk}) implies the B\\\"{a}r inequality \\cite{bar2} given by\n\\begin{eqnarray}\n\\lambda^2 \\geqslant \\lambda_1^2 := \\frac{2\\pi\\chi(M)}{Area(M)}.\n \\label{barspin}\n\\end{eqnarray}\nMoreover, from Equality (\\ref{eq:Fk}), Th. Friedrich and E.C. Kim \\cite{fk} deduced\n that $\\int_M {\\rm det}(T^\\psi)v_g=\\pi\\chi(M)$, which gives an information on the Energy-Momentum tensor without knowing the eigenspinor nor the eigenvalue. Finally, for any closed surface $M$ in $\\mathbb{R}^3$ of constant mean curvature $H$, the restriction to $M$ of a parallel spinor on $\\mathbb{R}^3$ is a generalized Killing spinor of eigenvalue $-H$ with Energy-Momentum tensor equal to the Weingarten tensor $II$ (up to the factor $-\\frac{1}{2})$ \\cite{m1} and we have by \\eqref{eq:Fk}\n$$H^2=\\frac{\\pi\\chi(M)}{Area(M)}+\\frac{1}{4Area(M)}\\int_M|II|^2v_g.$$\nIndeed, given any surface $M$ carrying such a spinor field, Th. Friedrich \\cite{fr3} showed \nthat the Energy-Momentum tensor associated with this spinor satisfies the Gauss-Codazzi equations and hence $M$ is locally immersed into $\\mathbb{R}^3$.\\\\\\\\\nHaving a $\\Spinc$ structure on manifolds is a weaker condition than having a $\\Spin$ structure because every $\\Spin$ manifold has a trivial $\\Spinc$ structure. Additionally, any compact surface or any product of a compact surface with $\\mathbb{R}$ has a $\\Spinc$ structure carrying particular spinors. In the same spirit as in \\cite{13}, when using a suitable conformal change, the second author \\cite{r1} established a B\\\"{a}r-type inequality for the eigenvalues of the Dirac operator on a compact surface endowed with any $\\Spinc$ structure. In fact, any eigenvalue $\\lambda$ of the Dirac operator satisfies\n\\begin{eqnarray}\n\\lambda^2 \\geqslant \\lambda_1^2 := \\frac{2\\pi \\chi(M)}{Area(M)} - \\frac{1}{Area(M)} \\int_M \\vert\\Omega\\vert v_g,\n\\label{barspinc}\n\\end{eqnarray}\nwhere $i\\Omega$ is the curvature form of the connection on the line bundle given by the $\\Spinc$ structure.\nEquality is achieved if and only if the eigenspinor $\\psi$ associated with the first eigenvalue $\\lambda_1$ is a Killing $\\Spinc$ spinor, i.e., for every $X\\in \\Gamma(TM)$ the eigenspinor $\\psi$ satisfies\n\\begin{eqnarray}\n\\left\\{\n\\begin{array}{l}\n\\nabla_X\\psi = -\\frac{\\lambda_1}{2}X\\cdot\\psi,\\\\\n\\Omega\\cdot\\psi =i \\vert\\Omega\\vert\\psi.\n\\end{array}\n\\label{lamba}\n\\right.\n\\end{eqnarray}\nHere $X\\cdot\\psi$ denotes the Clifford multiplication and $\\nabla$ the spinorial Levi-Civita connection \\cite{fr1}.\\\\\\\\\nStudying the Energy-Momentum tensor on a compact Riemannian $\\Spin$ or $\\Spinc$\nmanifolds has been done by many authors, since it is related to several geometric situations.\nIndeed, on compact $\\Spin$ manifolds, J.P. Bourguignon and P. Gauduchon \\cite{BG92}\nproved that the Energy-Momentum tensor appears naturally in the study of the variations\nof the spectrum of the Dirac operator. Th. Friedrich and E.C. Kim \\cite{fk2} obtained\nthe Einstein-Dirac equation as the Euler-Lagrange equation of a certain functional. The second author extented \nthese last two results to $\\Spinc$ manifolds \\cite{r2}. Even if it is not a\ncomputable geometric invariant, the Energy-Momentum tensor is, up to a constant, the\nsecond fundamental form of an isometric immersion into a $\\Spin$ or $\\Spinc$ manifold carrying a\nparallel spinor \\cite{m1, r2}. For a better understanding of the tensor $q^\\phi$\n associated with a spinor field $\\phi$, the first author \\cite{habib1}\n studied Riemannian flows and proved\nthat, if the normal bundle carries a parallel spinor $\\psi$, the tensor $q^\\phi$ associated with $\\phi$ (the restriction of $\\psi$ to the flow) \n is the O'Neill tensor of the flow.\\\\\\\\\nIn this paper, we give a formula corresponding to \\eqref{eq:Fk} for any eigenspinor $\\psi$ of the square of the Dirac operator on compact surfaces endowed with any $\\Spinc$ structure (see Theorem \\ref{procarrdir}). It is motivated by the following two facts: First, when we consider eigenvalues of the square of the Dirac operator, another tensor field is of interest. It is the skew-symmetric tensor field $Q^\\psi$ given by \n$$Q^\\psi(X,Y)= g(q^\\psi (X), Y) = \\frac{1}{2}\\Re(X\\cdot\\nabla_Y\\psi-Y\\cdot\\nabla_X\\psi,\\frac{\\psi}{|\\psi|^2}),$$ \nfor all vector fields $X,Y\\in \\Gamma(TM).$ This tensor was studied by the first author in the context of Riemannian flows \\cite{habib1}. Second, we consider any compact surface $M$ immersed in $\\cercle^2\\times \\mathbb{R}$ where $\\cercle^2$ is the round sphere equipped with a metric of curvature one. The $\\Spinc$ structure on $\\cercle^2\\times \\mathbb{R},$ induced from the canonical one on $\\cercle^2$ and the $\\Spin$ struture on $\\mathbb{R},$ admits a parallel spinor \\cite{mo}. The restriction to $M$ of this $\\Spinc$ structure is still a $\\Spinc$ structure with a generalized Killing spinor \\cite{r2}.\\\\\\\\\nIn Section \\ref{pre}, we recall some basic facts on $\\Spinc$ structures and the restrictions of these structures to hypersurfaces. In Section \\ref{sect:3} and after giving a formula corresponding to \\eqref{eq:Fk} for any eigenspinor $\\psi$ of the square of the Dirac operator, we deduce a formula for the integral of the determinant of $T^\\psi+ Q^\\psi$ and we establish a new proof of the B\\\"{a}r-type inequality (\\ref{barspinc}). In Section \\ref{sec4}, we consider the $3$-dimensional case and treat examples of hypersurfaces in $\\CC \\PP^2$. In the last section, we come back to the question of a spinorial characterisation of surfaces in $\\cercle^2\\times \\mathbb{R}$. Here we use a different approach than the one in \\cite{r}. In fact, we prove that given any surface $M$ carrying a generalized Killing spinor associated with a particular $\\Spinc$ structure, the Energy-Momentum tensor satisfies the four compatibility equations stated by B. Daniel \\cite{daniel}. Thus there exists a local immersion of $M$ into $\\cercle^2\\times \\mathbb{R}$.\n\\section{Preliminaries}\\label{pre}\n\\setcounter{equation}{0}\nIn this section, we begin with some preliminaries concerning $\\Spinc$ structures and\nthe Dirac operator. Details can be found in \\cite{lm}, \\cite{montiel}, \\cite{fr1}, \\cite{r1} and\n\\cite{r2}.\\\\ \\\\\n{\\bf The Dirac operator on $\\Spinc$ manifolds:} Let $(M^n, g)$ be a Riemannian manifold of dimension $n\\geqslant2$ without\nboundary. We denote by ${\\rm SO}M$ the \n${\\rm SO}_n$-principal bundle over $M$ of positively oriented orthonormal frames. A\n$\\Spinc$ structure of $M$ is a $\\Spin_n^c$-principal bundle $(\\Spinc M,\\pi,M)$\n and an $\\cercle^1$-principal bundle $(\\cercle^1 M ,\\pi,M)$ together with a double\ncovering given by $\\theta: \\Spinc M \\longrightarrow {\\rm SO}M\\times_{M}\\cercle^1 M$ such\nthat $\\theta (ua) = \\theta (u)\\xi(a),$\nfor every $u \\in \\Spinc M$ and $a \\in \\Spin_n^c$, where $\\xi$ is the $2$-fold\ncovering of $\\Spin_n ^c$ over ${\\rm SO}_n\\times \\cercle^1$. \nLet $\\Sigma M := \\Spinc M \\times_{\\rho_n} \\Sigma_n$ be the associated spinor bundle\nwhere $\\Sigma_n = \\CC^{2^{[\\frac n2]}}$ and $\\rho_n : \\Spin_n^c\n\\longrightarrow \\End(\\Sigma_{n})$ denotes the complex spinor representation. A section of\n$\\Sigma M$ will be called a spinor field. The spinor bundle $\\Sigma M$ is equipped with a\nnatural Hermitian scalar product denoted by $(.,.)$. We\ndefine an $L^2$-scalar product\n$<\\psi,\\phi> = \\int_M (\\psi,\\phi) v_g,$\nfor any spinors $\\psi$ and $\\phi$. \\\\ Additionally, any connection 1-form $A: T(\\cercle^1 M)\\longrightarrow i\\RR$ on\n$\\cercle^1 M$ and the connection 1-form \n$\\omega^M$ on ${\\rm SO} M$, induce a connection\non the principal bundle ${\\rm SO} M\\times_{M} \\cercle^1 M$, and hence \na covariant derivative $\\nabla$ on $\\Gamma(\\Sigma M)$ \\cite{fr1,r2}. The curvature\nof $A$ is an imaginary valued 2-form denoted by $F_A= dA$, i.e., $F_A = i\\Omega$,\nwhere $\\Omega$ is a real valued 2-form on $\\cercle^1 M$. We know\n that $\\Omega$ can be viewed as a real valued 2-form on $M$ $\\cite{fr1, kn}$. In\nthis case $i\\Omega$ is the curvature form of the associated line bundle $L$. It is\nthe complex line bundle associated with the $\\cercle^1$-principal bundle via the\nstandard representation of the unit circle.\nFor every spinor $\\psi$, the Dirac operator is locally defined \nby $$D\\psi =\\sum_{i=1}^n e_i \\cdot \\nabla_{e_i} \\psi,$$\nwhere $(e_1,\\ldots,e_n)$ is a local oriented orthonormal tangent frame and ``$\\cdot$'' denotes the Clifford multiplication. The Dirac\noperator is an elliptic, self-adjoint operator with respect to the $L^2$-scalar\nproduct and verifies, for any spinor field $\\psi$, the Schr\\\"odinger-Lichnerowicz formula \n\\begin{eqnarray}\nD^2\\psi=\\nabla^*\\nabla\\psi+\\frac{1}{4}S\\psi+\\frac{i}{2}\\Omega\\cdot\\psi\n\\label{bochner}\n\\end{eqnarray}\nwhere $\\Omega\\cdot$ is the extension of the Clifford multiplication to differential\nforms given by \n$(e_i ^* \\wedge e_j ^*)\\cdot\\psi = e_i\\cdot e_j \\cdot\\psi$. For any spinor $\\psi \\in \\Gamma(\\Sigma M)$, we have \\cite{mo2}\n\\begin{eqnarray}\n (i\\Omega\\cdot\\psi,\\psi)\\ \\geqslant -\\frac{c_n}{2} \\vert \\Omega \\vert_g\\vert\n\\psi\\vert^2,\n\\label{cs}\n\\end{eqnarray}\nwhere $\\vert \\Omega \\vert_g$ is the norm of $\\Omega$, with respect to $g$ given by\n$\\vert \\Omega \\vert_g^2=\\sum_{i<j} (\\Omega_{ij} )^2$\nin any orthonormal local frame and $c_n = 2[\\frac\nn2]^\\frac12$. Moreover, equality holds in (\\ref{cs}) if and only\nif $\\Omega\\cdot\\psi = i \\frac {c_n}{2}\\vert\\Omega\\vert_g \\psi$.\\\\\nEvery $\\Spin$ manifold has a trivial $\\Spinc$ structure \\cite{fr1, bm}. In fact, we\nchoose the trivial line bundle with the trivial connection whose curvature $i\\Omega$\nis zero. Also every K\\\"ahler manifold $M$ of complex dimension $m$ has a canonical $\\Spinc$ structure. Let $\\ltimes$ by the K\\\"{a}hler form defined by the complex structure $J$, i.e. $\\ltimes (X, Y)= g(JX, Y)$ for all vector fields $X,Y\\in \\Gamma(TM).$ The complexified cotangent bundle \n$$T^*M\\otimes \\CC = \\Lambda^{1,0} M \\oplus \\Lambda^{0,1}M$$\ndecomposes into the $\\pm i$-eigenbundles of the complex linear extension of the complex structure. Thus, the spinor bundle of the canonical $\\Spinc$ structure is given by $$\\Sigma M = \\Lambda^{0,*} M =\\oplus_{r=0}^m \\Lambda^{0,r}M,$$\nwhere $\\Lambda^{0,r}M = \\Lambda^r(\\Lambda^{0,1}M)$ is the bundle of $r$-forms of type $(0, 1)$. The line bundle of this canonical $\\Spinc$ structure is given by $L = (K_M)^{-1}= \\Lambda^m (\\Lambda^{0,1}M)$, where $K_M$ is the canonical bundle of $M$ \\cite{fr1, bm}. This line bundle $L$ has a canonical holomorphic connection induced from the Levi-Civita connection whose curvature form is given by $i\\Omega = -i\\rho$, where $\\rho$ is the Ricci form given by $\\rho(X, Y) = \\Ric(JX, Y)$. \nWe point out that the canonical $\\Spinc$ structure on every K\\\"{a}hler manifold carries a parallel spinor \\cite{fr1, mo}.\\\\\\\\ \n{\\bf Spin$^c$ hypersurfaces and the Gauss formula:} Let $\\cZ$ be an oriented ($n+1$)-dimensional Riemannian $\\Spinc$ manifold and $M\n\\subset \\cZ$ be an oriented hypersurface. The manifold $M$ inherits a $\\Spinc$\nstructure induced from the one on $\\cZ$, and we have \\cite{r2}\n$$ \\Sigma M\\simeq \\left\\{\n\\begin{array}{l}\n\\Sigma \\cZ_{|_M} \\ \\ \\ \\ \\ \\ \\text{\\ \\ \\ if\\ $n$ is even,} \\\\\\\\\n \\SZp_{|_M} \\ \\text{\\ \\ \\ \\ \\ \\ if\\ $n$ is odd.}\n\\end{array}\n\\right.\n$$\nMoreover Clifford multiplication by a vector field $X$, tangent to $M$, is given by \n\\begin{eqnarray}\nX\\bullet\\phi = (X\\cdot\\nu\\cdot \\psi)_{|_M},\n\\label{Clifford}\n\\end{eqnarray}\nwhere $\\psi \\in \\Gamma(\\Sigma \\cZ)$ (or $\\psi \\in \\Gamma(\\SZp)$ if $n$ is odd),\n$\\phi$ is the restriction of $\\psi$ to $M$, ``$\\cdot$'' is the Clifford\nmultiplication on $\\cZ$, ``$\\bullet$'' that on $M$ and $\\nu$ is the unit normal\nvector. The connection 1-form defined on the restricted $\\SSS^1$-principal bundle $(\\PSM :=\n\\PSZ_{|_M},\\pi,M)$, is given by $A= {A^\\cZ}_{|_M} : T(\\PSM) = T(\\PSZ)_{|_M}\n\\longrightarrow i\\RR.$ Then the curvature 2-form $i\\Omega$ on the\n$\\SSS^1$-principal bundle $\\PSM$ is given by $i\\Omega= {i\\Omega^\\cZ}_{|_M}$,\nwhich can be viewed as an imaginary 2-form on $M$ and hence as the curvature form of\nthe line bundle $L^M$, the restriction of the line bundle $L^\\cZ$ to $M$. For every\n$\\psi \\in \\Gamma(\\SZ)$ ($\\psi \\in \\Gamma(\\SZp)$ if $n$ is odd), the real 2-forms\n$\\Omega$ and $\\Omega^\\cZ$ are related by \\cite{r2}\n\\begin{eqnarray}\n(\\Omega^\\cZ \\cdot\\psi)_{|_M} = \\Omega\\bullet\\phi -\n(\\nu\\lrcorner\\Omega^\\cZ)\\bullet\\phi.\n\\label{glucose}\n\\end{eqnarray}\nWe denote by $\\nabla^{\\Sigma \\cZ}$ the spinorial Levi-Civita connection on $\\Sigma\n\\cZ$ and by $\\nabla$ that on $\\Sigma M$. For all $X\\in \\Gamma(TM)$, we have the spinorial Gauss formula \\cite{r2}:\n\\begin{equation}\n(\\nabla^{\\Sigma \\cZ}_X\\psi)_{|_M} = \\nabla_X \\phi + \\half II(X)\\bullet\\phi,\n\\label{spingauss}\n\\end{equation}\nwhere $II$ denotes the Weingarten map of the hypersurface. Moreover, Let $D^\\cZ$ and $D^M$ be the Dirac operators on $\\cZ$ and $M$, after\ndenoting by the same symbol any spinor and its restriction to $M$, we have\n\\begin{equation}\n\\nu\\cdot D^\\cZ\\phi = \\DDM\\phi +\\frac{n}{2}H\\phi -\\nabla^{\\Sigma\\cZ}_{\\nu}\\phi,\n\\label{diracgauss}\n\\end{equation}\nwhere $H = \\frac1n \\tr(II)$ denotes the mean curvature and $\\DDM = D^M$ if $n$ is even and $\\DDM=D^M \\oplus(-D^M)$ if $n$ is odd.\n\\section{The $2$-dimensional case}\\label{sect:3}\n\\setcounter{equation}{0}\nIn this section, we consider compact surfaces endowed with any $\\Spinc$ structure. We have \n\\begin{thm}\\label{procarrdir}\n Let $(M^2, g)$ be a Riemannian manifold and $\\psi$ an eigenspinor of the\nsquare of the Dirac operator $D^2$ with eigenvalue $\\lambda^2$ associated with any $\\Spinc$ structure. Then we have \n$$ \\lambda^2 = \\frac{S}{4} + {\\vert{T}^{\\psi} \\vert^2} + \\vert{Q}^{\\psi} \\vert^2 +\\Delta f+|Y|^2-2Y(f)+(\\frac i2 \\Omega\\cdot\\psi, \\frac{\\psi}{\\vert\\psi\\vert^2}), $$ \nwhere $f$ is the real-valued function defined by $f=\\frac{1}{2}{\\rm ln}|\\psi|^2$\nand $Y$ is a vector field on $TM$ given by $g(Y,Z)=\n\\frac{1}{\\vert\\psi\\vert^2} \\Re(D\\psi, Z\\cdot\\psi)$ for any $Z \\in \\Gamma(TM)$.\n\\label{thm1}\n\\end{thm}\n{\\bf Proof.} Let $\\{e_1, e_2\\}$ be an orthonormal frame of $TM$. Since the spinor bundle $\\Sigma M$ is of real dimension $4$, the set $\\{\\frac{\\psi}{|\\psi|}, \\frac{e_1\\cdot\\psi}{|\\psi|},\\frac{e_2\\cdot\\psi}{|\\psi|},\\frac{e_1\\cdot e_2\\cdot\\psi}{|\\psi|}\\}$ is orthonormal with respect to the real product $\\Re(\\cdot, \\cdot)$. The covariant derivative of $\\psi$ can be expressed in this frame as \n\\begin{eqnarray} \n \\nabla_X\\psi = \\delta(X) \\psi + \\alpha(X)\\cdot\\psi + \\beta(X) e_1\\cdot\ne_2\\cdot\\psi,\n\\label{equa1}\n\\end{eqnarray}\nfor all vector fields $X,$ where $\\delta$ and $\\beta$ are $1$-forms and $\\alpha$ is a $(1,1)$-tensor field. Moreover $\\beta$, $\\delta$ and $\\alpha$ are uniquely\ndetermined by the spinor $\\psi$. In fact, taking the scalar product of (\\ref{equa1}) respectively with $\\psi, e_1\\cdot\\psi, e_2\\cdot\\psi, e_1\\cdot e_2\\cdot\\psi$, we get $\\delta =\\frac{d(\\vert\\psi\\vert^2)}{2\\vert\\psi\\vert^2}$ and\n$$\\alpha (X)=- \\ell^\\psi(X)+q^\\psi(X)\\quad\\text{and}\\quad\\beta(X)=\\frac{1}{\\vert\\psi\\vert^2} \\Re(\\nabla_X\\psi, e_1\\cdot e_2\\cdot\\psi).$$ \nUsing (\\ref{bochner}), it follows that\n$$\\lambda^2 = \\frac {\\Delta ( \\vert \\psi \\vert^2 )} { 2 \\vert \\psi \\vert^2 } +\n\\vert \\alpha \\vert^2 + \\vert \\beta\\vert^2 + \\vert \\delta\\vert^2 +\\frac 14 S + (\\frac i2 \\Omega\\cdot\\psi, \\frac{\\psi}{\\vert\\psi\\vert^2}).$$\nNow it remains to compute the term $\\vert\\beta\\vert^2$. We have\n\\begin{eqnarray*}\n\\vert\\beta\\vert^2 &=& \\frac{1}{\\vert\\psi\\vert^4}\\Re(\\nabla_{e_1}\\psi, e_1\\cdot e_2\n\\cdot\\psi)^2+\\frac{1}{\\vert\\psi\\vert^4}\\Re(\\nabla_{e_2}\\psi, e_1\\cdot e_2\n\\cdot\\psi)^2\\\\\n&=& \\frac{1}{\\vert\\psi\\vert^4} \\Re(D\\psi - e_2 \\cdot\\nabla_{e_2}\\psi, e_2 \\cdot\\psi)^2\n+ \\frac{1}{\\vert\\psi\\vert^4} \\Re(D\\psi - e_1 \\cdot\\nabla_{e_1}\\psi, e_1 \\cdot\\psi)^2 \\\\\n&=& g(Y, e_1)^2 + g(Y, e_2)^2 + \\frac{\\vert d(\\vert\\psi\\vert^2)\\vert^2}{4\n\\vert\\psi\\vert^4} - g(Y, \\frac{d(\\vert\\psi\\vert^2)}{\\vert\\psi\\vert^2})\\\\\n&=& |Y|^2-2Y(f) + \\frac{\\vert d(\\vert\\psi\\vert^2)\\vert^2}{4\n\\vert\\psi\\vert^4},\n\\end{eqnarray*}\nwhich gives the result by using the fact that $\\Delta f=\\frac {\\Delta ( \\vert \\psi \\vert^2)} {2\\vert \\psi \\vert^2}+\\frac{\\vert d(\\vert\\psi\\vert^2)\\vert^2}{2\n\\vert\\psi\\vert^4}.$\n\\hfill$\\square$\n\\begin{remark} \\label{rempropre} Under the same conditions as Theorem \\ref{thm1}, if $\\psi$ is an eigenspinor of $D$ with eigenvalue $\\lambda$, we get\n$$ \\lambda^2 = \\frac{S}{4} + {\\vert{T}^{\\psi} \\vert^2} +\\Delta f+(\\frac i2 \\Omega\\cdot\\psi, \\frac{\\psi}{\\vert\\psi\\vert^2}).$$ \nIn fact, in this case $Y=0$ and \n\\begin{eqnarray}\n0 &=& \\Re(D\\psi,e_1\\cdot e_2\\cdot\\psi)=\\Re(e_1\\cdot\\nabla_{e_1}\\psi+e_2\\cdot\\nabla_{e_2}\\psi,e_1\\cdot e_2\\cdot\\psi)\\nonumber \\\\\n&=&\\Re(-e_2\\cdot\\nabla_{e_1}\\psi+e_1\\cdot\\nabla_{e_2}\\psi,\\psi) =2Q^\\psi(e_1,e_2)|\\psi|^2.\n\\label{qpsi}\n\\end{eqnarray} \nThis was proven by Friedrich and Kim in \\cite{fk} for a $\\Spin$ structure on $M$.\n\\end{remark}\nIn the following, we will give an estimate for the integral $\\displaystyle\\int_M {\\rm det}(T^\\psi+Q^\\psi) v_g$ in terms of geometric quantities, which has the advantage that it does not depend on the eigenvalue $\\lambda$ nor on the eigenspinor $\\psi$. This is a generalization of the result of Friedrich and Kim in \\cite{fk} for $\\Spin$ structures.\n\\begin{thm} \nLet $M$ be a compact surface and $\\psi$ any eigenspinor of $D^2$ associated with eigenvalue $\\lambda^2$. Then we have \n\\begin{eqnarray}\n\\int_M {\\rm det}(T^\\psi+Q^\\psi)v_g \\geq \\frac{\\pi\\chi(M)}{2}- \\frac 14 \\int_M \\vert\\Omega\\vert v_g.\n\\label{dett}\n\\end{eqnarray}\nEquality in \\eqref{dett} holds if and only if either $\\Omega$ is zero or has constant sign. \n\\label{D2}\n\\end{thm} \n{\\bf Proof.} As in the previous theorem, the spinor $D\\psi$ can be expressed in the orthonormal frame of the spinor bundle. Thus the norm of $D\\psi$ is equal to\n\\begin{eqnarray}\n|D\\psi|^2&=&\\frac{1}{|\\psi|^2}\\Re(D\\psi,\\psi)^2+\\frac{1}{|\\psi|^2}\\sum_{i=1}^2 \\Re(D\\psi,e_i\\cdot\\psi)^2+\\frac{1}{|\\psi|^2}\\Re(D\\psi,e_1\\cdot e_2\\cdot\\psi)^2\\nonumber\\\\\n&=&({\\tr}\\,T^\\psi)^2|\\psi|^2+|Y|^2|\\psi|^2+\\frac{1}{|\\psi|^2}\\Re(D\\psi,e_1\\cdot e_2\\cdot\\psi)^2,\n\\label{eq:Dirac}\n\\end{eqnarray}\nwhere we recall that the trace of $T^\\psi$ is equal to $-\\frac{1}{|\\psi|^2}\\Re(D\\psi,\\psi).$ On the other hand, by (\\ref{qpsi}) we have that\n$\\frac{1}{|\\psi|^2}\\Re(D\\psi,e_1\\cdot e_2\\cdot\\psi)^2=2|Q^\\psi|^2|\\psi|^2.$ Thus Equation \\eqref{eq:Dirac} reduces to \n$$\\frac{|D\\psi|^2}{|\\psi|^2}=({\\tr}\\, T^\\psi)^2+|Y|^2+2|Q^\\psi|^2.$$\nNow with the use of the equality $\\Re(D^2\\psi,\\psi)=|D\\psi|^2-{\\rm div}\\xi,$ where $\\xi$ is the vector field given by $\\xi=|\\psi|^2Y,$ we get \n\\begin{equation}\n\\lambda^2+\\frac{1}{|\\psi|^2}{\\rm div}\\xi=({\\tr}\\, T^\\psi)^2+|Y|^2+2|Q^\\psi|^2.\n\\label{eq:Dirac2}\n\\end{equation} \nAn easy computation leads to $\\frac{1}{|\\psi|^2}{\\rm div}\\xi={\\rm div} Y+2Y(f)$ where we recall that $f=\\frac{1}{2}{\\rm ln}(|\\psi|^2).$ Hence substituting this formula into \\eqref{eq:Dirac2} and using Theorem \\ref{thm1} yields \n$$\\frac{S}{4} + (\\frac i2 \\Omega\\cdot\\psi, \\frac{\\psi}{\\vert\\psi\\vert^2}) + \\Delta f+{\\rm div} Y= ({\\tr} T^\\psi)^2+|Q^\\psi|^2-|T^\\psi|^2=2{\\rm det}(T^\\psi+Q^\\psi).$$\nFinally integrating over $M$ and using the Gauss-Bonnet formula, we deduce the required result with the help of Equation (\\ref{cs}). Equality holds if and only if $\\Omega\\cdot\\psi = i\\vert\\Omega\\vert \\psi$. In the orthonormal frame $\\{e_1, e_2\\}$, the $2$-form $\\Omega$ can be written $\\Omega= \\Omega_{12} \\ e_1 \\wedge e_2$, where $\\Omega_{12}$ is a function defined on $M$. Using the decomposition of $\\psi$ into positive and negative spinors $\\psi^+$ and $\\psi^-$, we find that the equality is attained if and only if\n$$\\Omega_{12} \\ e_1 \\cdot e_2 \\cdot\\psi^+ + \\Omega_{12}\\ e_1 \\cdot e_2 \\cdot\\psi^- = i \\vert\\Omega_{12} \\vert \\psi^+ +i \\vert\\Omega_{12} \\vert \\psi^-,$$\nwhich is equivalent to say that,\n$$\\Omega_{12} \\psi^+ = -\\vert\\Omega_{12}\\vert \\psi^+\\ \\ \\ \\ \\text{and}\\ \\ \\ \\ \\Omega_{12} \\psi^- = \\vert\\Omega_{12}\\vert \\psi^-.$$\nNow if $\\psi^+ \\neq 0$ and $\\psi^- \\neq 0$, we get $\\Omega = 0$. Otherwise, it has constant sign. In the last case, we get that $\\int_M|\\Omega|v_g=2\\pi\\chi(M),$ which means that the l.h.s. of this equality is a topological invariant.\n\\hfill$\\square$\\\\ \\\\\n\nNext, we will give another proof of the B\\\"ar-type inequality (\\ref{barspinc}) for the eigenvalues of any $\\Spinc$ Dirac operator. The following theorem was proved by the second author in \\cite{r1} using conformal deformation of the spinorial Levi-Civita connection. \n\\begin{thm}\n Let $M$ be a compact surface. For any $\\Spinc$ structure on $M$, any eigenvalue $\\lambda$ of the Dirac operator $D$ to which is attached an eigenspinor $\\psi$ satisfies\n\\begin{eqnarray}\n\\lambda^2 \\geqslant \\frac{2\\pi \\chi(M)}{Area(M)} - \\frac{1}{Area(M)} \\int_M \\vert\\Omega\\vert v_g.\n\\label{inequalitybar}\n\\end{eqnarray}\nEquality holds if and only if the eigenspinor $\\psi$ is a\n $\\Spinc$ Killing spinor, i.e., it satisfies $\\Omega\\cdot\\psi = i \\vert\\Omega\\vert\\psi$ and $\\nabla_X \\psi = -\\frac{\\lambda}{2} X\\cdot\\psi$ for any $X \\in \\Gamma(TM)$.\n\\label{thmspinc}\n\\end{thm}\n{\\bf Proof.} With the help of Remark \\eqref{rempropre}, we have that\n\\begin{eqnarray}\n \\lambda^2 = \\frac{S}{4} + \\vert T^\\psi\\vert^2 + \\bigtriangleup f +(\\frac i2 \\Omega\\cdot\\psi, \\frac{\\psi}{\\vert\\psi\\vert^2}).\n\\label{barspinccc}\n\\end{eqnarray}\nSubstituting the Cauchy-Schwarz inequality, i.e. $\\vert T^\\psi\\vert^2 \\geqslant \\frac{\\lambda^2}{2}$ and the estimate (\\ref{cs}) into Equality (\\ref{barspinccc}), we easily deduce the result after integrating over $M$. Now the equality in \\eqref{inequalitybar} holds if and only if the eigenspinor $\\psi$ satisfies $\\Omega\\cdot\\psi = i \\vert\\Omega\\vert\\psi$ and $\\vert T^\\psi\\vert^2= \\frac{\\lambda^2}{2}$. Thus, the second equality is equivalent to say that $\\ell^\\psi (X) = \\frac{\\lambda}{2}X$ for all $X\\in \\Gamma(TM)$. \nFinally, a straightforward computation of the spinorial curvature of the spinor field $\\psi$ gives in a local frame $\\{e_1,e_2\\}$ after using the fact $\\beta=-(*\\delta)$ that\n\\begin{eqnarray*}\n\\frac{1}{2}R_{1212}\\ e_1\\cdot e_2\\cdot\\psi&=&\\Big(\\frac{\\lambda^2}{2}+e_1(\\delta(e_1))+e_2(\\delta(e_2))\\Big)e_2\\cdot e_1\\cdot\\psi-\\lambda\\delta(e_2) e_1\\cdot\\psi\\\\&&+\\lambda \\delta(e_1)e_2\\cdot\\psi+\\Big(e_1(\\delta(e_2))-e_2(\\delta(e_1))\\Big)\\psi.\n\\end{eqnarray*}\nThus the scalar product with $e_1\\cdot\\psi$ and $e_2\\cdot\\psi$ implies that $\\delta=0$. Finally, $\\beta=0$ and the eigenspinor $\\psi$ is a $\\Spinc$ Killing spinor.\n\\hfill$\\square$\\\\ \\\\\nNow, we will give some examples where equality holds in (\\ref{inequalitybar}) or in \\eqref{dett}. Some applications of Theorem \\ref{thm1} are also given.\\\\ \\\\\n{\\bf Examples:}\n\\begin{enumerate}\n \\item Let $\\cercle^2$ be the round sphere equipped with the standard metric of curvature one. As a K\\\"ahler manifold, we endow the sphere with the canonical $\\Spinc$ structure of curvature form equal to $i\\Omega=-i\\ltimes$, where $\\ltimes$ is the K\\\"{a}hler $2$-form. Hence, we have $\\vert\\Omega\\vert = \\vert\\ltimes\\vert =1$. Furthermore, we mentionned that for the canonical $\\Spinc$ structure, the sphere carries parallel spinors, i.e., an eigenspinor associated with the eigenvalue $0$ of the Dirac operator $D$. Thus equality holds in (\\ref{inequalitybar}). On the other hand, the equality in \\eqref{dett} also holds, since the sign of the curvature form $\\Omega$ is constant. \n\\item Let $f: M \\rightarrow \\cercle^3$ be an isometric immersion of a surface\n$M^2$ into the sphere equipped with its unique $\\Spin$ structure and assume that the mean curvature $H$ is\nconstant. The restriction of a Killing spinor on $\\cercle^3$ to the surface $M$ defines a spinor field\n$\\phi$ solution of the following equation \\cite{gj}\n\\begin{eqnarray}\n \\nabla_X \\phi = -\\frac 12 II(X) \\bullet \\phi + \\frac 12 J(X)\\bullet\\phi,\n\\label{generalized}\n\\end{eqnarray}\nwhere $II$ denotes the second fundamental form of the surface and $J$ is the complex\nstructure of $M$ given by\nthe rotation of angle $\\frac{\\pi}{2}$ on $TM$. It is\neasy to check that $\\phi$ is an eigenspinor for $D^2$ associated with the\neigenvalue $H^2 +1$. Moreover $D \\phi = H \\phi + e_1\\cdot e_2\\cdot \\phi$, so that $Y=0$. Moreover the tensor $T^\\phi=\\frac{1}{2}II$ and $Q^\\phi=\\frac{1}{2}J$. Hence by Theorem \\ref{thm1}, and since the norm of $\\phi$ is\nconstant, we obtain\n$$H^2 + \\frac12 = \\frac14 S +\\frac{1}{4}|II|^2.$$\n\\item On two-dimensional manifolds, we can define another Dirac operator associated with the complex\nstructure $J$ given by $\\widetilde D =Je_1 \\cdot \\nabla_{e_1} +Je_2 \\cdot \\nabla_{e_2} = e_2\\cdot \\nabla_{e_1}- e_1 \\cdot\\nabla_{e_2}$. Since $\\widetilde D$ satisfies $D^2 = (\\widetilde D)^2$, all the above results are also true for the eigenvalues of $\\widetilde D$.\n\n\\item Let $M^2$ be a surface immersed in $\\cercle^2 \\times \\RR$. The product of the canonical $\\Spinc$ structure on $\\cercle^2$ and the unique $\\Spin$ structure on $\\RR$ define a $\\Spinc$ structure on $\\cercle^2\\times \\RR$ carrying parallel spinors \\cite {mo}. Moreover, by the Schr\\\"{o}dinger-Lichnerowicz formula, any parallel spinor $\\psi$ satisfies $\\Omega^{\\cercle^2\\times \\mathbb{R}}\\cdot\\psi=i\\psi$, where $\\Omega^{\\cercle^2\\times \\mathbb{R}}$ is the curvature form of the auxiliary line bundle. Let $\\nu$ be a unit normal vector field of the surface. We then write $\\partial t=T+f\\nu$ where $T$ is a vector field on $TM$ with $||T||^2+f^2=1$. On the other hand, the vector field $T$ splits into $T=\\nu_1+h\\partial t$ where $\\nu_1$ is a vector field on the sphere. The scalar product of the first equation by $T$ and the second one by $\\partial t$ gives $||T||^2=h$ which means that $h=1-f^2$. Hence the normal vector field $\\nu$ can be written as $\\nu=f\\partial t-\\frac{1}{f}\\nu_1.$ As we mentionned before, the $\\Spinc$ structure on $\\cercle^2 \\times \\RR$ induces a $\\Spinc$ structure on $M$ with induced auxiliary line bundle. Next, we want to prove that the curvature form of the auxiliary line bundle of $M$ is equal to $i\\Omega(e_1,e_2)=-if$, where $\\{e_1,e_2\\}$ denotes a local orthonormal frame on $TM$. Since the spinor $\\psi$ is parallel, we have by \\cite{mo} that for all $X\\in T(\\cercle^2 \\times \\RR)$ the equality ${\\rm Ric}^{\\cercle^2 \\times \\RR} X\\cdot\\psi=i(X\\lrcorner\\Omega^{\\cercle^2 \\times \\RR})\\cdot\\psi$. Therefore, we compute\n\\begin{eqnarray*}\n(\\nu\\lrcorner \\Omega^{\\cercle^2 \\times \\RR})\\bullet\\phi&=&(\\nu\\lrcorner \\Omega^{\\cercle^2 \\times \\RR})\\cdot\\nu\\cdot\\psi|_M=i\\nu\\cdot{\\rm Ric}^{\\cercle^2 \\times \\RR}\\ \\nu\\cdot\\psi|_M\\\\\n&=&-\\frac{1}{f}i\\nu\\cdot\\nu_1\\cdot\\psi|_M=i\\nu\\cdot (\\nu-f\\partial t).\\psi|_M\\\\\n&=&(-i\\psi-if\\nu\\cdot\\partial t\\cdot\\psi)|_M. \n\\end{eqnarray*}\nHence by Equation \\eqref{glucose}, we get that $\\Omega\\bullet\\phi=-i (f\\nu\\cdot\\partial t\\cdot\\psi)|_M.$ The scalar product of the last equality with $e_1\\cdot e_2\\cdot\\psi$ gives\n$$\\Omega(e_1,e_2)|\\phi|^2=-f\\Re(i\\nu\\cdot\\partial t\\cdot\\psi,e_1\\cdot e_2\\cdot\\psi)|_M= -f\\Re(i\\partial t\\cdot\\psi,\\psi)|_M.$$\nWe now compute the term $i\\partial t\\cdot\\psi$. For this, let $\\{e'_1, J e'_1\\}$ be a local orthonormal frame of the sphere $\\cercle^2$. The complex volume form acts as the identity on the spinor bundle of $\\cercle^2 \\times \\RR$, hence $\\partial t \\cdot\\psi = e'_1\\cdot Je'_1\\cdot\\psi$. But we have\n$$\\Omega^{\\cercle^2 \\times \\RR}\\cdot\\psi = -\\rho\\cdot\\psi = -\\ltimes\\cdot\\psi = -e'_1\\cdot Je'_1\\cdot\\psi.$$\nTherefore, $i\\partial t\\cdot\\psi = \\psi$. Thus we get $\\Omega(e_1,e_2)=-f$. Finally, \n$$(i\\Omega\\bullet\\phi,\\phi) = f \\Re(\\nu\\cdot\\partial t\\cdot\\psi,\\psi)|_M=-fg(\\nu,\\partial t)\\vert\\phi\\vert^2=-f^2\\vert\\phi\\vert^2.$$ Hence Equality in Theorem \\ref{thm1} is just \n$$H^2=\\frac{S}{4}+\\frac{1}{4}|II|^2-\\frac{1}{2}f^2.$$\n\\end{enumerate}\n\n\n\n\n\n\n\n\n\n\n\n\n \n\n\n\\section{The 3-dimensional case}\\label{sec4} \n\\setcounter{equation}{0}\nIn this section, we will treat the $3$-dimensional case.\n\\begin{thm}\n Let $(M^3,g)$ be an oriented Riemannian manifold. For any $\\Spinc$ structure on $M$, any eigenvalue\n$\\lambda$ of the Dirac operator to which is attached an eigenspinor $\\psi$ satisfies\n$$\\lambda^2 \\leqslant \\frac{1}{\\vol(M, g)} \\int_M (\\vert T^\\psi\\vert^2 +\\frac S4\n+\\frac{\\vert\\Omega\\vert}{2})v_g.$$\nEquality holds if and only if the norm of $\\psi$ is constant and $\\Omega\\cdot\\psi\n= i\\vert\\Omega\\vert \\psi$. \n\\label{thm4}\n\\end{thm}\n\n{\\bf Proof.} As in the proof of Theorem \\ref{thm1}, the set $\\{\\frac{\\psi}{|\\psi|}, \\frac{e_1\\cdot\\psi}{|\\psi|},\\frac{e_2\\cdot\\psi}{|\\psi|},\\frac{e_3\\cdot\\psi}{|\\psi|}\\}$ is orthonormal with respect to the real product $\\Re(\\cdot, \\cdot)$. The covariant derivative of $\\psi$ can be expressed in this frame as \n\\begin{eqnarray} \n \\nabla_X\\psi = \\eta(X) \\psi + \\ell(X)\\cdot\\psi,\n\\label{eq1}\n\\end{eqnarray}\nfor all vector fields $X,$ where $\\eta$ is a $1$-form and $\\ell$ is a $(1,1)$-tensor field. Moreover $\\eta =\\frac{d(\\vert\\psi\\vert^2)}{2\\vert\\psi\\vert^2}$ and $\\ell(X)=- \\ell^\\psi(X)$. Using (\\ref{bochner}), it follows that\n\\begin{eqnarray*}\n\\lambda^2 &=& \\frac {\\Delta ( \\vert \\psi \\vert^2 )} { 2 \\vert \\psi \\vert^2 } +\n\\vert T^\\psi \\vert^2 + \\frac{\\vert d(\\vert\\psi\\vert^2)\\vert^2}{4\\vert\\psi\\vert^4} +\\frac 14 S + (\\frac i2 \\Omega\\cdot\\psi, \\frac{\\psi}{\\vert\\psi\\vert^2})\\\\\n&=& \\Delta f - \\frac{\\vert d(\\vert\\psi\\vert^2)\\vert^2}{2\\vert\\psi\\vert^4} \n+\\vert T^\\psi \\vert^2 +\\frac 14 S + (\\frac i2 \\Omega\\cdot\\psi, \\frac{\\psi}{\\vert\\psi\\vert^2}).\n \\end{eqnarray*}\nBy the Cauchy-Schwarz inequality, we have $\\frac 12 (i\\Omega \\cdot\\psi,\\frac{\\psi}{\\vert\\psi\\vert^2}) \\leqslant \\frac 12 \\vert\\Omega\\vert$. Integrating over $M$ and using the fact that $\\vert d(\\vert\\psi\\vert^2)\\vert^2\\geqslant 0$, we get the result.\n\\hfill$\\square$\n\n\n\n\\begin{example}\nLet $M^3$ be a 3-dimensional Riemannian manifold immersed in $\\CC \\PP^2$ with\nconstant mean curvature $H$. Since $\\CC \\PP^2$ is a K\\\"{a}hler manifold, we endow it\nwith the canonical $\\Spinc$ structure whose line bundle has\ncurvature equal to $-3i\\ltimes$. Moreover, by the Schr\\\"{o}dinger-Lichnerowicz formula we have that any parallel spinor $\\psi$ satisfies $\\Omega^{\\CC \\PP^2} \\cdot\\psi = 6i \\psi.$ As in the previous example, we compute \n $$(\\nu\\lrcorner\\Omega^{\\CC \\PP^2})\\bullet\\phi = i (\\nu\\cdot \\Ric^{\\CC\n\\PP^2}(\\nu)\\cdot\\psi)_{\\mid_M} = -3i \\phi.$$ \nFinally, $\\Omega\\bullet\\phi = 3i\\phi.$ Using Equation (\\ref{diracgauss}), we have that $-\\frac 32 H$ is an eigenvalue of $D$. Since the norm of $\\phi$ is constant, equality holds in Theorem \\ref{thm4} and hence\n$$\\frac 94 H^2 + \\frac 32 = \\frac{S}{4}+\\frac{1}{4}\\vert II\\vert^2.$$\n\\end{example}\n\n\\section{Characterization of surfaces in $\\mathbb{S}^2\\times \\mathbb{R}$}\\label{charac} \n\\setcounter{equation}{0}\nIn this section, we characterize the surfaces in $\\mathbb{S}^2\\times \\mathbb{R}$ by solutions of the generalized Killing spinors equation which are restrictions of parallel spinors of the canonical $\\Spinc$-structure on $\\mathbb{S}^2\\times \\mathbb{R}$ (see also \\cite{r} for a different proof). First recall the compatibility equations for characterization of surfaces in $\\mathbb{S}^2\\times \\mathbb{R}$ established by B. Daniel \\cite[Thm 3.3]{daniel}: \n\\begin{thm} \\label{daniel}Let $(M,g)$ be a simply connected Riemannian manifold of dimension $2$, $A:TM\\rightarrow TM$ a field of symmetric operator and $T$ a vector field on $TM$. We denote by $f$ a real valued function such that $f^2+||T||^2=1$. Assume that $M$ satisfies the Gauss-Codazzi equations, i.e. $G={\\rm det} A+f^2$ and\n$$ d^\\nabla A(X,Y):=(\\nabla_XA)Y-(\\nabla_YA)X=f(g(Y,T)X-g(X,T)Y),$$ \nwhere $G$ is the gaussian curvature, and the following equations \n$$\\nabla_X T=fA(X), \\,\\,\\, X(f)=-g(AX,T).$$ Then there exists an isometric immersion of $M$ into $\\mathbb{S}^2\\times \\mathbb{R}$ such that the Weingarten operator is $A$ and $\\partial t=T+f\\nu,$ where $\\nu$ is the normal vector field to the surface $M$. \n\\end{thm}\nNow using this characterization theorem, we state our result:\n\\begin{thm} \\label{carac}\nLet $M$ be an oriented simply connected Riemannian manifold of dimension $2$. Let $T$ be a vector field and denote by $f$ a real valued function such that $f^2+||T||^2=1$. Denote by $A$ a symmetric endomorphism field of $TM$. The following statements are equivalent: \n\\begin{enumerate}\n\\item There exists an isometric immersion of $M$ into $\\mathbb{S}^2\\times \\mathbb{R}$ of Weingarten operator $A$ such that $\\partial t=T+f\\nu,$ where $\\nu$ is the unit normal vector field of the surface.\n\\item There exists a $\\Spinc$ structure on $M$ whose line bundle has a connection of curvature given by $i\\Omega=-if \\ltimes,$ such that it carries a non-trivial solution $\\phi$ of the generalized Killing spinor equation $\\nabla_X\\phi=-\\frac{1}{2}AX\\bullet\\phi$, with $T\\bullet\\phi=-f\\phi+\\bar\\phi.$\n\\end{enumerate}\n\\end{thm}\n{\\bf Proof.} We begin with $1\\Rightarrow 2$. The existence of such a $\\Spinc$ structure is assured by the restriction of the canonical one on $\\cercle^2\\times \\mathbb{R}$. Moreover, using the spinorial Gauss formula \\eqref{spingauss}, any parallel spinor $\\psi$ on $\\cercle^2\\times \\mathbb{R}$ induces a generalized Killing spinor $\\phi=\\psi|_M$ with $A$ the Weingarten map of the surface $M$. Hence it remains to show the relation $T\\bullet\\phi=-f\\phi+\\bar\\phi$. In fact, using that $\\Omega^{\\cercle^2\\times \\mathbb{R}}\\cdot\\psi=i\\psi,$ we write in the frame $\\{e_1,e_2,\\nu\\}$ \n\\begin{equation}\\label{eq:3}\n\\Omega^{\\cercle^2\\times \\mathbb{R}}(e_1,e_2)e_1\\cdot e_2\\cdot\\psi+\\Omega^{\\cercle^2\\times \\mathbb{R}}(e_1,\\nu)e_1\\cdot\\nu\\cdot\\psi+\\Omega^{\\cercle^2\\times \\mathbb{R}}(e_2,\\nu)e_2\\cdot\\nu\\cdot\\psi=i\\psi.\n\\end{equation}\nBy the previous example in Section \\ref{sect:3}, we know that $\\Omega^{\\cercle^2\\times \\mathbb{R}}(e_1,e_2)=-f$. For the other terms, we compute \n$$\n\\Omega^{\\cercle^2\\times \\mathbb{R}}(e_1,\\nu)=\\Omega^{\\cercle^2\\times \\mathbb{R}}(e_1,\\frac{1}{f}\\partial t-\\frac{1}{f}T)=-\\frac{1}{f}g(T,e_2)\\Omega^{\\cercle^2\\times \\mathbb{R}}(e_1,e_2)=g(T,e_2),\n$$\nwhere the term $\\Omega^{\\cercle^2\\times \\mathbb{R}}(e_1,\\partial t)$ vanishes since we can split $e_1$ into a sum of vectors on the sphere and on $\\mathbb{R}.$ Similarly, we find that $\\Omega^{\\cercle^2\\times \\mathbb{R}}(e_2,\\nu)=-g(T,e_1).$ By substituting these values into \\eqref{eq:3} and taking Clifford multiplication with $e_1\\cdot e_2$, we get the desired property. For $2\\Rightarrow 1$, a straightforward computation for the spinorial curvature of the generalized Killing spinor $\\phi$ yields on a local frame $\\{e_1,e_2\\}$ of $TM$ that \n\\begin{equation}\n(-G+{\\rm det}\\,A)e_1\\bullet e_2\\bullet\\phi=-(d^\\nabla A)(e_1,e_2)\\bullet \\phi + if\\phi.\n\\label{eq:6}\n\\end{equation}\nIn the following, we will prove that the spinor field $\\theta:=i\\phi-if\\bar{\\phi}+JT\\bullet\\phi$ is zero. For this, it is sufficient to prove that its norm vanishes. Indeed, we compute \n\\begin{equation}\n|\\theta|^2=|\\phi|^2+f^2|\\phi|^2+||T||^2|\\phi|^2-2\\Re(i\\phi,if\\bar{\\phi})+2\\Re(i\\phi,JT\\bullet\\phi)\n\\label{eq:5}\n\\end{equation}\nFrom the relation $T\\bullet\\phi=-f\\phi+\\bar{\\phi}$ we deduce that $\\Re(\\phi,\\bar\\phi)=f|\\phi|^2$ and the equalities\n$$g(T,e_1)|\\phi|^2=\\Re(ie_2\\bullet\\phi,\\phi) \\quad\\text{and}\\quad g(T,e_2)|\\phi|^2=-\\Re(ie_1\\bullet\\phi,\\phi).$$\nTherefore, Equation \\eqref{eq:5} becomes \n\\begin{eqnarray*}\n|\\theta|^2&=&2|\\phi|^2-2f^2|\\phi|^2+2\\Re(i\\phi,JT\\bullet\\phi)\\\\\n&=&2|\\phi|^2-2f^2|\\phi|^2+2 g(JT,e_1)\\Re(i\\phi,e_1\\bullet\\phi)+2g(JT,e_2) \\Re(i\\phi,e_2\\bullet\\phi)\\\\\n&=&2|\\phi|^2-2f^2|\\phi|^2+2 g(JT,e_1) g(T,e_2)|\\phi|^2-2g(JT,e_2) g(T,e_1)|\\phi|^2\\\\\n&=&2|\\phi|^2-2f^2|\\phi|^2-2g(JT,e_1)^2|\\phi|^2-2g(T,e_1)^2|\\phi|^2\\\\\n&=&2|\\phi|^2-2f^2|\\phi|^2-2||T||^2|\\phi|^2=0.\n\\end{eqnarray*}\nThus, we deduce $if\\varphi=-f^2e_1\\cdot e_2\\cdot\\varphi-fJT\\cdot\\varphi$, where we use the fact that $\\bar\\phi=i e_1\\bullet e_2\\bullet\\phi$. In this case, Equation \\eqref{eq:6} can be written as \n$$(-G+{\\rm det}\\,A+f^2)e_1\\bullet e_2\\bullet\\phi=-((d^\\nabla A)(e_1,e_2)+fJT)\\bullet \\phi.$$\nThis is equivalent to say that both terms $R_{1212}+{\\rm det}\\,A+f^2$ and $(d^\\nabla A)(e_1,e_2)+fJT$ are equal to zero. In fact, these are the Gauss-Codazzi equations in Theorem \\ref{daniel}. In order to obtain the two other equations, we simply compute the derivative of $T\\cdot\\varphi=-f\\varphi+\\bar\\varphi$ in the direction of $X$ to get \n\\begin{eqnarray*}\n\\nabla_X T \\bullet \\phi+T\\bullet\\nabla_X\\varphi&=&\\nabla_X T \\bullet \\phi-\\frac{1}{2}T\\bullet A(X)\\bullet\\varphi\\\\\n&=&-X(f)\\varphi-f\\nabla_X\\varphi+\\nabla_X\\bar\\varphi\\\\\n&=&-X(f)\\varphi+\\frac{1}{2}f AX\\bullet\\varphi+\\frac{1}{2} AX\\bullet\\bar{\\varphi}\\\\\n&=&-X(f)\\varphi+\\frac{1}{2}f AX\\bullet\\varphi+\\frac{1}{2} AX\\bullet(T\\bullet\\varphi+f\\varphi).\n\\end{eqnarray*}\nThis reduces to $\\nabla_X T \\bullet \\phi+g(T,A(X))\\varphi=-X(f)\\varphi+fA(X)\\bullet\\varphi.$ Hence we obtain $X(f)=-g(A(X),T)$ and $\\nabla_X T=fA(X)$ which finishes the proof.\n\\hfill$\\square$\n\\begin{remark} The second condition in Theorem \\ref{carac} is equivalent to the existence of a $\\Spinc$ structure whose line bundle $L$ verifies \n$c_1(L)=[\\frac{i}{2\\pi}f\\ltimes]$ and $f \\ltimes$ is a closed 2-form. This $\\Spinc$ structure carries a non-trivial solution $\\phi$ of the generalized Killing spinor equation $\\nabla_X\\phi=-\\frac{1}{2}AX\\bullet\\phi$, with $T\\bullet\\phi=-f\\phi+\\bar\\phi.$\n\\end{remark}\n{\\bf Acknowledgment}\\\\\\\\\nBoth authors are grateful to Oussama Hijazi for his encouragements and relevant remarks.\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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