# Superconducting Phases of the Square-Lattice Extended Hubbard Model

Wei-Chih Chen  Department of Physics, University of Alabama at Birmingham,
Birmingham, Alabama 35294, USA  Department of Physics and Astronomy, Clemson
University, Clemson, South Carolina 29631, USA  Yao Wang<EMAIL_ADDRESS>Department of Physics and Astronomy, Clemson University, Clemson, South
Carolina 29631, USA  Department of Chemistry, Emory University, Atlanta, GA,
30322, United States  Cheng-Chien Chen<EMAIL_ADDRESS>Department of Physics,
University of Alabama at Birmingham, Birmingham, Alabama 35294, USA

###### Abstract

We study the square-lattice extended Hubbard model with on-site $U$ and
nearest-neighbor $V$ interactions by exact diagonalization. We show that non-
equilibrium quench dynamics can help determine the equilibrium phase
transition boundaries, which agree with the calculations of fidelity metric,
dynamical structure factor, and correlation function. At half filling, the
phase diagrams in the strong-coupling regime include spin density wave and
$d_{x^{2}-y^{2}}$-wave superconductivity at large positive $U$, charge density
wave (extended $s^{*}$-wave superconductivity) at large positive (negative)
$V$, and $s$-wave superconductivity at large negative $U$ with vanishing $V$.
The energies of different particle sectors also help determine the phase
separation region. With carrier doping, charge fluctuation result in strong
competition between different orders, making it more difficult to identify the
leading instability on finite-size cluster. Nevertheless, the more exotic
$p$-wave superconducting pairing is found to be enhanced when the system is
heavily overdoped by $37.5\%-50\%$ holes, especially in interaction parameter
range relevant to the cuprate superconductors.

##  I Introduction

One of the most intriguing but challenging topics in condensed matter concerns
studying correlation effects in many interacting particles. Strongly
correlated systems exhibit various symmetry-breaking states, such as
superconductivity, which can arise from the interplay of spin, charge, and
lattice degrees of freedom [1, 2]. Already in 1950s, the
Bardeen–Cooper–Schrieffer theory successfully explained the mechanism of
conventional superconductors through phonon-mediated electron pairs [3]. In
late 1980s, the discovery of copper-based high-temperature superconductors –
later identified with a $d$-wave pairing symmetry [4] – reformed our
understanding of correlated materials [5]. Although the ultimate theory for
high-temperature superconductivity remains controversial, the strong electron
repulsive interactions in copper $3d$ orbitals are believed to play crucial
roles [6, 7, 8]. Unconventional superconductivity also has been explored in
other transition-metal oxides. For example, the infinite-layer nickelates
share similar electronic structures with the cuprates and become
superconducting at $\sim$20% hole doping [9]. Moreover, superconductivity in
the ruthenates was identified with a triplet $p$-wave pairing [10, 11, 12, 13,
14], although the claim was challenged by recent experiments [15, 16]. In any
case, these unconventional superconducting phases are believed to originate
from strong electron correlation effects.

The simplest toy model to describe electronic correlation is the single-band
Hubbard model with a local on-site interaction $U$. This simple model already
can explain the behaviors of Mott insulator, stripe order [17, 18, 19, 20],
strange metal [21, 22, 23], and to some extent $d$-wave superconductivity [24,
25, 26, 27]. As a natural extension, more recent studies have considered a
non-local Coulomb interaction (denoted as $V$) in the so-called extended
Hubbard model (EHM). A repulsive $V$ can induce a charge density wave (CDW),
and an attractive $V$ was argued to favor spin-triplet superconductivity [28,
29]. Already in 1D systems, a $p$-wave superconducting phase was predicted for
repulsive $U$ and attractive $V$ in EHM [30, 31, 32, 33, 34]. Similar studies
have been explored in 2D systems [35, 36, 37, 38, 39, 40], using weak-coupling
theory or with approximation in treating interaction effect. To understand and
design exotic phases like $p$-wave superconductivity, it is necessary to study
EHM in the strong-coupling regime.

Moreover, recent ARPES experiments on 1D cuprate chains have revealed a
sizeable attractive Coulomb interaction between nearest-neighbor (NN)
electrons [41]. Although not as strong as the on-site Coulomb repulsion, this
attractive interaction is comparable to the electron hopping integral, and
thereby should not be ignored [42]. The structural similarities among the
cuprates also suggest that attractive NN interaction may exist in quasi-2D
cuprate materials in general. Since electron-phonon coupling may be its
origin, this attractive interaction should be even stronger in a 2D copper-
oxide plane, because of the richer phonon modes and stronger impact of
ligands. Therefore, studying the 2D EHM could be important for understanding
high-$T_{c}$ pairing mechanism.

Figure 1:  Schematic phase diagram of the square-lattice extended Hubbard
model. Depending on the onsite $U$ and nearest-neighbor $V$ interactions, as
well as hole doping levels, the model can support various symmetry breaking
phases, such as spin density wave (SDW), charge density wave (CDW), $s$-wave
superconductivity, $d_{x^{2}-y^{2}}$-wave superconductivity, and the more
exotic $p$-wave superconductivity. A phase separation (PS) also can occur near
the $d$-wave phase especially with negative $V$.

In this paper, we systematically study the phase diagrams of the square-
lattice EHM as functions of interaction strengths and doping levels. Studying
strong correlation effect is challenging, and most theoretical and
computational methods are based on various levels of approximation in handling
the interaction. Here we choose to use exact diagonalization (ED), which has
the advantage that electron interaction effect can be treated exactly. We
employ ED to compute quench dynamics, fidelity metrics, and various
correlation functions. While the ED study is restricted to finite-size
clusters, we will show that the phase boundaries determined by ED agree with
the functional renormalization group (fRG) results in the weak-coupling limit
[38], and our approach remains valid in the strong-coupling limit. Figure 1
illustrates the main results of our study. At half filling, spin density wave
(SDW) is dominant at positive $U$, and $s$-wave superconductivity is dominant
at negative $U$ and small $|V|$. On the other hand, a positive $V$ induces a
charge density wave (CDW), and a negative $V$ can cause phase separation. When
the system is heavily overdoped by hole carriers, the more exotic $p$-wave
superconducting correlation can be enhanced at large positive $U$ and small
negative $V$. We also determine the phase separated region by computing the
energies of different particle sectors. These results will be relevant to
various physical systems such as the cuprates and cold-atoms experiments [43,
44].

##  II Model and Methods

The extended Hubbard model (EHM) reads

$\displaystyle H=-t_{h}\sum_{\langle
ij\rangle,\sigma}(c^{\dagger}_{i\sigma}c_{j\sigma}+h.c.)+U\sum_{i}n_{i\uparrow}n_{i\downarrow}+V\sum_{\begin{subarray}{c}\langle
ij\rangle,\\\
\sigma\sigma^{\prime}\end{subarray}}n_{i\sigma}n_{j\sigma^{\prime}}.$

Here, $c^{\dagger}_{i\sigma}$ creates an electron with spin $\sigma$ (=
$\uparrow$ or $\downarrow$) at lattice site $i$, and
$n_{i\sigma}=c^{\dagger}_{i\sigma}c_{i\sigma}$ is the corresponding fermion
number operator. $\langle...\rangle$ represents a pair of near-neighbor (NN)
sites. $t_{h}$ is the hopping amplitude. $U$ and $V$ are respectively the on-
site and NN interaction strengths; a positive (negative) value corresponds to
repulsive (attractive) force. Here, $t_{h}\equiv 1$ and we explore both
positive and negative interactions in the range $|U|\leq 10$ and $|V|\leq 10$.
We note that a small, attractive NN Coulomb interaction $V$ has been
identified in ARPES experiments in doped cuprate chains [41]. While longer-
range interactions beyond NN might also be present, their strengths are
expected to be weaker and remain to be established experimentally. We solve
Eq. (II) using the ED technique, which can treat correlation effects exactly
for arbitrary interaction strength. The main challenge of ED is that the
Hilbert space size grows exponentially with the lattice size, so the
calculations are limited to finite-size clusters. Here, we focus on a $4\times
4$ square cluster (under periodic boundary condition), which exhibits the
proper $D_{4}$ symmetry for studying the competition between $d-$wave and
$p$-wave superconductivity. On this $N=16$ site cluster, we study hole doping
levels at 0% (half filling with 1 electron per site on average), 12.5%, 25%,
37.5%, and 50% (quarter filling with 0.5 electron per site on average).

The ED algorithm proceeds as follows. We first construct the Hamiltonian
matrix using an eigenbasis of the fermion number operator. We then perform
matrix diagonalization to obtain the ground state $|G_{\lambda}\rangle$ with
interaction parameters $\lambda\equiv(U,V)$. Depnding on $\lambda$ and the
doping level, ground state degeneracy can occur, and these degenerate states
have to be considered when computing the expectation values of physical
observables. The actual diagonalization is performed using iterative Krylov
sub-space methods as implemented in the PETSc [45, 46] and SLEPc [47, 48]
libraries. The Krylov-Schur technique is utilized to resolve degenerate
eigenstates.

After obtaining the ground state(s) $|G_{\lambda}\rangle$ for a Hamiltonian
$H_{0}$ with interactions $\lambda=(U,V)$, we next perform an interaction
quench, where the Hamiltonian undergoes a sudden change to $H_{1}$ with
interactions $\lambda^{*}=(U^{*},V^{*})$ at time $t=0^{+}$. The state at time
$t+\delta t$ is obtained by acting the time evolution operator on the state at
time $t$:

$|\psi(t+\delta t)\rangle=e^{-iH_{1}\delta t}|\psi(t)\rangle,\,\textrm{for
}t\geq 0.$ (2)

Here, $|\psi(t=0)\rangle\equiv|G_{\lambda}\rangle$ is the ground state of
$H_{0}$. The time evolution calculation is again performed using a Krylov
method in PETSc, which requires only repeated matrix-vector multiplications to
construct the Krylov subspace
$\\{|\psi(t)\rangle,H^{1}_{1}|\psi(t)\rangle,H^{2}_{1}|\psi(t)\rangle,H^{3}_{1}|\psi(t)\rangle,...\\}$,
without the need to explicitly construct the matrix exponential operator
$e^{-iH_{1}\delta t}$. In our calculations, we typically evolve the system to
a final time $t=50{t_{h}}^{-1}$, with a time step $\delta
t=0.01-0.05{t_{h}}^{-1}$. For selected sets of interaction parameters, we have
performed additional calculations with varying time steps to ensure that the
numerical error due to a finite time discretization is negligible.

With $|\psi(t)\rangle$ available, we compute the equal-time correlation
functions $\langle\psi(t)|\hat{O}^{\dagger}\hat{O}|\psi(t)\rangle$ for
different order parameters. In particular, the correlations with the following
operators $\hat{O}$ are predominant at half filling:

$\displaystyle\rho_{\mathbf{q}}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{N}}\sum_{i}e^{i\mathbf{q}\cdot\mathbf{r}_{i}}(c^{\dagger}_{i\uparrow}c_{i\uparrow}+c^{\dagger}_{i\downarrow}c_{i\downarrow}),$
(3) $\displaystyle\rho^{s}_{\mathbf{q}}$ $\displaystyle=$
$\displaystyle\frac{1}{2\sqrt{N}}\sum_{i}e^{i\mathbf{q}\cdot\mathbf{r}_{i}}(c^{\dagger}_{i\uparrow}c_{i\uparrow}-c^{\dagger}_{i\downarrow}c_{i\downarrow}),$
(4) $\displaystyle\Delta_{s}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{N}}\sum_{i}c_{i\uparrow}c_{i\downarrow},$ (5)
$\displaystyle\Delta_{d_{x^{2}-y^{2}}}$ $\displaystyle=$
$\displaystyle\frac{1}{2\sqrt{N}}\sum_{i}(c_{i\uparrow}c_{i+\hat{x}\downarrow}+c_{i\uparrow}c_{i-\hat{x}\downarrow}$
$\displaystyle-$ $\displaystyle
c_{i\uparrow}c_{i+\hat{y}\downarrow}-c_{i\uparrow}c_{i-\hat{y}\downarrow}).$

Here, $\rho_{\mathbf{q}}$ ($\rho_{\mathbf{q}}^{s}$) is the charge (spin)
density operator in momentum space, relevant to a CDW (SDW) phase at ordering
vector $\mathbf{q}$. $\Delta_{s}$ and $\Delta_{d_{x^{2}-y^{2}}}$ are real-
space $s$-wave and $d_{x^{2}-y^{2}}$-wave pairing operators, respectively.
Upon doping, other superconducting instabilities can occur, and we also study
the pairing operators for extended $s$-wave ($s^{*}$-wave), $d_{xy}$-wave, and
$p_{x}$-wave superconductivity:

$\displaystyle\Delta_{s^{*}}$ $\displaystyle=$
$\displaystyle\frac{1}{2\sqrt{N}}\sum_{i}(c_{i\uparrow}c_{i+\hat{x}\downarrow}+c_{i\uparrow}c_{i-\hat{x}\downarrow}$
$\displaystyle+$ $\displaystyle
c_{i\uparrow}c_{i+\hat{y}\downarrow}+c_{i\uparrow}c_{i-\hat{y}\downarrow}),$
$\displaystyle\Delta_{d_{xy}}$ $\displaystyle=$
$\displaystyle\frac{1}{2\sqrt{N}}\sum_{i}(c_{i\uparrow}c_{i+\hat{x}+\hat{y}\downarrow}+c_{i\uparrow}c_{i-\hat{x}-\hat{y}\downarrow}$
$\displaystyle-$ $\displaystyle
c_{i\uparrow}c_{i-\hat{x}+\hat{y}\downarrow}-c_{i\uparrow}c_{i+\hat{x}-\hat{y}\downarrow}),$
$\displaystyle\Delta_{p_{x}}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2N}}\sum_{i}(c_{i\uparrow}c_{i+\hat{x}\downarrow}-c_{i\uparrow}c_{i-\hat{x}\downarrow}).$
(9)

By symmetry, the value of $p_{y}$-wave correlation is identical to that of
$p_{x}$-wave in our ED calculations. After obtaining
$\langle\psi(t)|\hat{O}^{\dagger}\hat{O}|\psi(t)\rangle$ for the above order
parameters, we Fourier transform the time-domain data to extract the spectral
features in frequency space. As shown later, the Fourier spectra are closely
related to the equilibrium system’s charge and spin excitation energies, which
exhibit distinct behaviors in different broken-symmetry phases.

To extract the charge and spin gaps of the equilibrium system, we also compute
the charge $N(\mathbf{q},\omega)$ and spin $S(\mathbf{q},\omega)$ dynamical
structure factors:

$\displaystyle N(\mathbf{q},\omega)\\!$ $\displaystyle=$
$\displaystyle\\!\frac{1}{\pi}\mathrm{Im}\left\langle
G_{\lambda*}\\!\left|\rho_{-\mathbf{q}}\frac{1}{H_{1}\\!-\\!E_{G^{*}}\\!-\\!\omega\\!-\\!i\Gamma}\rho_{\mathbf{q}}\right|\\!G_{\lambda*}\\!\right\rangle,$
(10) $\displaystyle S(\mathbf{q},\omega)$ $\displaystyle=$
$\displaystyle\frac{1}{\pi}\mathrm{Im}\left\langle\\!G_{\lambda*}\\!\left|\rho^{s}_{-\mathbf{q}}\frac{1}{H_{1}\\!-\\!E_{G^{*}}\\!-\\!\omega\\!-\\!i\Gamma}\rho^{s}_{\mathbf{q}}\right|\\!G_{\lambda*}\\!\right\rangle.$
(11)

Here, $|G_{\lambda^{*}}\rangle$ is the equilibrium ground state of $H_{1}$
with energy $E_{G^{*}}$, and $\Gamma$ is a finite spectral broadening chosen
to be $0.1t_{h}$ in this study.

Finally, we also compute the fidelity of quantum state overlap
$f(\lambda^{*},\lambda)\equiv|\langle G_{\lambda^{*}}|G_{\lambda}\rangle|$,
where $|G_{\lambda}\rangle$ is the ground state with interactions
$\lambda\equiv(U,V)$, and $\lambda^{*}\equiv\lambda+\delta\lambda$ represents
a small deviation from $\lambda$. If ground state degeneracy occurs, the
fidelity value can be obtained by a singular value decomposition of the
overlap matrix with matrix elements $M_{k\ell}\equiv\langle
G_{\lambda^{*},k}|G_{\lambda,\ell}\rangle$. Here, $|G_{\lambda,\ell}\rangle$
($|G_{\lambda^{*},k}\rangle$) is the $\ell$-th ($k$-th) degenerate ground
state for interaction $\lambda$ ($\lambda^{*}$), and the largest singular
value is chosen to represent the fidelity. The fidelity approach is powerful
for determining phase transitions in quantum many-body systems [49, 50]. In
particular, if $|G_{\lambda}\rangle$ and $|G_{\lambda^{*}}\rangle$ belong to
the same broken-symmetry phase, $f$ is expected to be equal or close to unity.
On the other hand, if $|G_{\lambda}\rangle$ and $|G_{\lambda^{*}}\rangle$ have
a distinct nature, $f$ will be reduced from unity, which thereby signals a
quantum phase transition with a small change $\delta\lambda$ in the
interaction parameters. Using quench dynamics, spectral gaps, and fidelity
metrics, we are able to probe quantum phase transition boundaries using ED
with results in agreement with fRG in the weak-coupling limit, and to propose
phase diagrams in the strong-coupling regime.

##  III Results and Discussion

Figure 2:  Correlation functions for different order parameters computed by
exact diagonalization (ED) for half-filled extended Hubbard model in the weak-
coupling regime: $|U|\leq 1$ and $|V|\leq 1$. Panel (h) depicts the
corresponding phase diagram with phase boundaries determined by ED
calculations of fidelity metric, correlation function, and total energy. Gray
dashed lines indicate regions where phase separation (PS) would occur.

###  III.1 Weak-Coupling Limit

We first focus on a small interaction range $|U|\leq 1$ and $|V|\leq 1$, where
we can benchmark ED against fRG [38, 51]. Figure 2 shows equilibrium
correlation functions $\langle
G_{\lambda}|\hat{O}^{\dagger}\hat{O}|G_{\lambda}\rangle$ for different order
parameters in Eqs. (3)-(9) as functions of interaction strengths
$\lambda=(U,V)$. The phase separation (PS) region is determined from the
energy difference between different particle sectors (see Appendix). Overall,
a positive $V>0$ will stabilize a $(\pi,\pi)$ charge density wave (CDW), as
manifested in Fig. 2(a). Moreover, $V>0$ with $U<0$ will further enhance the
tendency towards the CDW state. In contrast, a predominant positive $U>0$ is
expected to support a $(\pi,\pi)$ spin density wave (SDW), as seen in Fig.
2(c).

The ED results show several salient features for the evolution of
superconducting correlations with interactions. In particular, $s$-wave
correlation is largely enhanced when $-0.5\leq V\leq 0$ and $U<0$ [Fig. 2(b)].
The $d_{x^{2}-y^{2}}$-wave correlation can increase when $V<0$ and $U>V$ [Fig.
2(d)]. The $p$-wave correlation also can increase when $V<0$ and $U>0$ [Fig.
2(f)]. Other correlations with different pairing symmetries, such as $d_{xy}$
or $s^{*}$, also can be suppressed or enhanced depending on the interactions
[Fig. 2(e) and (g)]. As discussed later, these more exotic superconductivity
pairings may be enhanced at larger interaction strengths or at higher doping
levels.

In principle, correlation functions computed on a finite-size cluster cannot
determine directly the symmetry breaking phases, as we cannot observe true
phase transitions but only cross-over phenomena. Other approaches such as
finite-size extrapolation for all orders are needed to determine the leading
instability in the thermodynamic limit [52, 53, 27, 54, 55], which for ED is
not practical beyond 16-site calculations. In our results, however, it is
clear that the regions where correlation functions show apparent enhancement
or suppression are already reminiscent of the actual phase boundaries obtained
by previous fRG studies [38]. Our weak-coupling phase diagram at half filling
is summarized in Fig. 2(h). Compared to fRG [38], our result includes
additionally a PS region, and a region with strong $p$-wave superconducting
instability, which is also suggested by mean-field theory [39].

To help determine phase boundaries, we first consider equilibrium calculations
of the quantum fidelity $f(\lambda^{*},\lambda)$. Figure 3(a) shows the
fidelity calculations along the path $V:-1\rightarrow 1$ at a fixed $U=-1$,
with an interaction step $\delta\lambda=\Delta V=0.1$. This path is depicted
by the arrow line in the figure inset. The fidelity (blue line) exhibits two
dips around $V=-0.5$ and 0, and the number of ground state degeneracy (green
line) also changes at $V=0$. We emphasize that our correlation and fidelity
calculations are always performed on a “homogeneous state”, instead of a
phase-separated one. Our 16-site cluster at a fixed particle sector is too
small to show the spatially inhomogeneous PS with both hole-rich and hole-
deficient regions. Instead, the PS information is obtained by computing the
ground-state energies in different particle sectors (see Appendix). Therefore,
although in Fig. 3(a) the fidelity path shows a transition between $d$-wave
and $s$-wave superconductivity in the homogeneous state, it is understood that
the actual ground state of the system should be phase separated when there is
a strong negative $V$. Gray dashed lines are utilized to indicate the PS
region in Fig. 3. The same understanding and labeling scheme apply to all the
following discussion and figures.

Figure 3:  Quantum fidelity (blue line) and the number of ground state
degeneracy (green line) for the extended Hubbard model at half filling. The
calculations are performed along different paths shown by arrow lines on the
inset weak-coupling phase diagram: (a) $V:-1.0\rightarrow 1.0$ at fixed
$U=-1.0$, with an interaction step $\Delta V=0.1$. (b) $U:-1.0\rightarrow 1.0$
at fixed $V=-0.2$, with an interaction step $\Delta U=0.1$. Dips in the
fidelity indicate phase transition boundaries. The calculations are performed
on a homogeneous state; gray dashed lines indicate regions where phase
separation would occur.

With the above caveat in mind, Fig. 3(a) together with the correlation
functions in Fig. 2 suggest that the system starts from PS at $(U=-1,V=-1)$,
transits to $s$-wave superconductivity around $(U=-1,V=-0.5)$, and enters the
$(\pi,\pi)$ CDW state at $(U=-1,V>0)$. Since $s$-wave superconductivity and
the $(\pi,\pi)$ CDW phase are expected to be degenerate when $V=0$ and $U<0$,
this explains why our calculations appear to show a broader boundary near
$V=0$.

Figure 3(b) shows the fidelity calculations along the path $U:-1\rightarrow 1$
at a fixed $V=-0.2$, with an interaction step $\Delta U=0.1$. The fidelity
exhibits three dips around $U=-0.2$, 0.1, and 0.3, while the ground state
remains non-degenerate throughout the whole path (indicated by the arrow line
in the figure inset). In accord with our phase diagram in Fig. 2(h), the
system starts from the $s$-wave phase at $(U=-1,V=-0.2)$, transits to
$d_{x^{2}-y^{2}}$-wave superconductivity around $(U=-0.2,V=-0.2)$, and enters
the $(\pi,\pi)$ SDW state at $(U=0.3,V=-0.2)$. We note that the fidelity
calculation and correlation functions in Fig. 2 appear to show additional
presence of a $p$-wave state. Since our ground state is a true quantum many-
body wavefunction, both leading and sub-leading instabilities may be picked up
in the ED calculations. In fact, since $d_{x^{2}-y^{2}}$ and $p$-wave
superconductivities have similar mean-field energies when $V<0$ and $U>0$
[39], these different orders may coexist in certain regions of the phase
diagram.

Figure 4:  (a) Time evolution of the equal-time correlation function for
$(\pi,\pi)$ charge order for the extended Hubbard model at half filling. The
initial parameters $(U,V)=(-1,1)$ at time $t<0$ are quenched to
$(U^{*},V^{*})=(-1,0.9)$ at $t=0^{+}$. (b)-(e) Fourier spectra of the equal-
time measurements for $(\pi,\pi)$ CDW, $(\pi,\pi)$ SDW, $s$-wave, $d$-wave
superconducting correlations, respectively. The horizontal axis represents
$V^{*}$ after the quench. Vertical dashed red lines indicate phase boundaries
near $V=0$ and $V=-0.6$, which agree well with the fidelity calculations. The
calculations are performed on a homogeneous state; gray dashed lines indicate
regions where phase separation would occur.  Figure 5:  The charge
$N(\mathbf{q},\omega)$ and spin $S(\mathbf{q},\omega)$ dynamical structure
factors at $\mathbf{q}=(\pi,\pi)$ for the extended Hubbard model at half
filling. The false color intensities are plotted in a log scale. The strong
elastic peak in $N(\mathbf{q},\omega)$ for $V\gtrapprox 0$ is caused by doubly
degenerate ground states in a $(\pi,\pi)$ CDW phase. The calculations are
performed on a homogeneous state; gray dashed lines indicate regions where
phase separation would occur.

We next discuss using non-equilibrium quench dynamics to determine equilibrium
phase boundaries. Unlike the fidelity metric, which is mainly a theoretical
tool, quantum quench of interaction can be prepared for example in optical-
lattice or ultrafast experiments [56, 57, 58, 59], and the subsequent dynamics
can be obtained by measuring correlation functions in time domain. Figure 4(a)
shows an example equal-time measurement for charge correlation with
$(\pi,\pi)$ ordering $N(\pi,\pi,t)$ [corresponding to the operator in Eq.
(3)]. The original equilibrium system at time $t<0$ is the ground state of
Hamiltonian $H_{0}$ with interactions $(U=-1,V=1)$, which supports a
$(\pi,\pi)$ CDW. At time $t=0^{+}$, the Hamiltonian is quenched to $H_{1}$
with interactions $(U=-1,V=0.9)$. $N(\pi,\pi,t)$ then oscillates in time at
$t>0$. For each quenched calculation, we perform Fourier transformations for
different equal-time correlations for the operators in Eqs. (3)-(9), in order
to extract the characteristic oscillation frequencies associated with quenched
dynamics.

Figures 4(b)-(e) show the resulting Fourier spectra of quenched dynamics for
CDW, SDW [for $\mathbf{q}=(\pi,\pi)$], $s$-wave, and $d_{x^{2}-y^{2}}$-wave
superconductivity [for $\mathbf{q}=(0,0)$], respectively. The horizontal axis
represents the post-quench value of $V$. We follow the same path as depicted
in Fig. 3(a) inset: $V:-1\rightarrow 1$ at a fixed $U=-1$, with a quench step
of $\Delta V=0.1$. The Fourier spectra exhibit distinct behaviors depending on
the quenched Hamiltonian. In Fig. 4(b), the Fourier spectra of charge
correlation $N(\pi,\pi)$ show a clear gap for $V\gtrapprox 0$. The gap size
rises with increasing $V$, changing from $\sim 3.7$ at $V=0.0$ to $\sim 7.7$
at $V=1$. In Fig. 4(c), the Fourier spectra of spin correlation $S(\pi,\pi)$
also show a similar gap for $V\gtrapprox 0$. The spectral intensity of this
gap is largely suppressed between $-0.5\lessapprox V\lessapprox 0$.
Noticeably, new low-energy modes emerge below $V\lessapprox-0.5$. In general,
the Fourier spectra of quenched $s$-wave [Fig. 4(d)] and
$d_{x^{2}-y^{2}}$-wave [Fig. 4(e)] superconducting correlations resemble those
of charge and spin correlations. We note again that the calculations are
performed on a homogeneous state; gray dashed lines in the figure indicate
regions where phase separation would occur. Based on the quench behaviors,
Figs. 4(b)-(e) can be separated into three regions [indicated by the vertical
dashed red lines]: a phase separated state for $V\lessapprox-0.5$, $s$-wave
superconductivity for $-0.5\lessapprox V\lessapprox 0$, and $(\pi,\pi)$ CDW
for $V\gtrapprox 0$. The quench calculations agree well with the fidelity
results, demonstrating the applicability of using a novel non-equilibrium
approach to probe equilibrium phase transitions.

To understand the spectral features of quenched measurements, we compute the
charge $N(\mathbf{q},\omega)$ and spin $S(\mathbf{q},\omega)$ dynamical
structure factors at the ordering vector $\mathbf{q}=(\pi,\pi)$. Here,
$|G_{\lambda^{*}}\rangle$ in Eqs. (10)-(11) is the equilibrium ground state of
$H_{1}$ (the Hamiltonian with the after-quench interactions). As seen in Fig.
5(a), $N(\mathbf{q},\omega)$ shows a clear charge gap for $V\gtrapprox 0$, and
the gap behaves similarly as that in Fig. 4(b): The gap starts as $\sim 3.7$
at $V=0$ and increases with increasing $V$, and it reaches $\sim 7.7$ at
$V=1$. For $V\gtrapprox 0$, the system develops $(\pi,\pi)$ CDW order by
spontaneously breaking the discrete charge symmetry, so it lacks a low-energy
excitation (Goldstone mode). On the other hand, $N(\mathbf{q},\omega)$
exhibits a strong elastic peak for $V\gtrapprox 0$. This elastic peak in the
CDW phase is due to doubly degenerate ground states: one with zero momentum
and the other with momentum $(\pi,\pi)$. As seen in Fig. 5(a),
$N(\mathbf{q},\omega)$ changes its behavior for $V\lessapprox 0$, where the
elastic peak disappears and the charge gap increases with decreasing $V$. The
result implies a phase transition near $V=0$. Another phase boundary near
$V=-0.5$ can be inferred from $S(\mathbf{q},\omega)$ in Fig. 5(b), where a low
energy mode emerges below $V\lessapprox-0.5$. The above results show that the
Fourier spectra of equal-time correlations after quantum quench can track the
excitation energies in $N(\mathbf{q},\omega)$ and $S(\mathbf{q},\omega)$.
However, while dynamical correlations provide information on the underlying
broken-symmetry phase (if there is one), they cannot inform directly how close
the ground state is to a phase transition, which instead can potentially be
inferred by experiments with varying strengths of quenched interaction to
examine whether the system resides in proximity to a phase boundary or in a
deeply ordered state.

Figure 6:  Correlation functions for different order parameters computed by
exact diagonalization (ED) for half-filled extended Hubbard model in the
strong-coupling regime: $|U|\leq 10$ and $|V|\leq 10$. Panel (h) depicts the
corresponding phase diagram with phase boundaries determined by ED
calculations of fidelity metric, correlation function, and total energy. Gray
dashed lines indicate regions where phase separation (PS) would occur.  Figure
7:  (a) Evolution of correlation strengths for different order parameters as a
function of $V$ for $U=8$ at half filling. The gray area near $V\sim-4$
indicates the phase separation regime (same as the gray dashed regime in Fig.
6). (b) Same as panel (a) but for the evolution along the $U$ direction with
$V=-0.2$, which is above the phase separation boundary. Figure 8:  Quantum
fidelity (blue line) and the number of ground state degeneracy (green line)
for the extended Hubbard model at half filling. The calculations are performed
along different paths shown by arrow lines on the inset _s_ trong-coupling
phase diagram: (a) $V:-1.4\rightarrow 1.4$ at fixed $U=6$, with an interaction
step $\Delta V=0.2$. (b) $V:-5\rightarrow 2.8$ at fixed $U=8$, with an
interaction step $\Delta V=0.1$. Dips in the fidelity indicate phase
transition boundaries. The calculations are performed on a homogeneous state;
gray dashed lines indicate regions where phase separation would occur.  Figure
9:  (a) Time evolution of the equal-time correlation function for $(\pi,\pi)$
charge ordering computed for the extended Hubbard model at half filling. The
initial parameters $(U,V)=(8,2.6)$ at time $t<0$ are quenched to
$(U^{*},V^{*})=(8,2.5)$ at $t=0^{+}$. (b)-(e) Fourier spectra of the equal-
time measurements for $(\pi,\pi)$ CDW, $(\pi,\pi)$ SDW, $s$-wave, $d$-wave
superconducting correlations, respectively. The horizontal axis represents
$V^{*}$ after the quench. Vertical dashed red lines indicate phase boundaries
near $V=-4.6$ and $V=2$, which agree with the fidelity calculations. The
calculations are performed on a homogeneous state; gray dashed lines indicate
regions where phase separation would occur.  Figure 10:  The charge
$N(\mathbf{q},\omega)$ and spin $S(\mathbf{q},\omega)$ dynamical structure
factors at $\mathbf{q}=(\pi,\pi)$ for the extended Hubbard model at half
filling for strong coupling. The false color intensities are plotted in a log
scale. The strong elastic peak in $N(\mathbf{q},\omega)$ for $V\gtrapprox 0$
is caused by doubly degenerate ground states in a $(\pi,\pi)$ CDW phase. The
vertical lines indicate phase boundaries near $V=0$ and $V=-0.6$. The
calculations are performed on a homogeneous state; gray dashed lines indicate
regions where phase separation would occur.

###  III.2 Strong-Coupling Regime

After establishing the abilities of using quantum fidelity and quench dynamics
to probe phase boundaries in the weak-coupling limit, we now focus on the
strong-coupling regime: $|U|\leq 10$ and $|V|\leq 10$. Figure 6 shows
equilibrium correlation functions $\langle
G_{\lambda}|\hat{O}^{\dagger}\hat{O}|G_{\lambda}\rangle$, which exhibit
overall four distinct regions: A positive $V>0$ will favor a $(\pi,\pi)$ CDW
[Fig. 6(a)], and together with $U<0$ it will further enhance the CDW state. In
contrast, a predominant positive $U>0$ will stabilize a $(\pi,\pi)$ SDW [Fig.
6(c)], and the SDW phase boundary along the $V$-axis is enlarged with
increasing $|V|$. In the atomic limit, on a half-filled $N$-site square
lattice, the energy of the CDW state is $U\times N/2$ (since there are $N/2$
doubly occupied sites), while that of the SDW phase is $V\times 2N$ (since
there are $2N$ distinct interaction bonds without double counting). Therefore,
the CDW to SDW transition boundary should occur at $U\geq 4V$, which is
consistent with our calculations and previous studies [60, 36, 37].

Near $V\sim 0$ and $U<0$, the system exhibits strong $s$-wave pairing
correlations, as indicated by the dark red strip in Fig. 6(b). When $V<0$, the
phase diagram has a substantial region dominated by $d_{x^{2}-y^{2}}$-wave
pairing [Fig. 6(d)]. However, we note that with a predominant negative $V$,
the system will tend to be phase separated (into hole-rich and hole-deficient
regions) [60], which again can be understood qualitatively using energy
consideration in the atomic limit. The actual phase boundary between a
homogeneous ground state and a phase-separated one can be quantitatively
determined from the total energy calculations in different particle sectors
(see Appendix). Based on the above results, we depict the strong-coupling
half-filled EHM phase diagram in Fig. 6(h).

Following the strategy used in the weak-coupling system, we first resort to
fidelity calculations to help determine the phases boundaries. For $U<0$, the
ED correlation functions in Fig. 6 suggest that the phase diagram consists of
three regions, separated by phase boundaries around $V=0$. The evolutions of
correlation strengths for different order parameters as functions of $V$ (with
$U=8$) and of $U$ (with $V=-0.2$) are further displayed in Fig. 7. Figure 8(a)
shows the fidelity (blue line) and the number of degenerate ground states
(green line) along the path $V:-1.4\rightarrow 1.4$ at $U=-6$, as depicted in
the figure inset. The fidelity deviates from unity between $-0.6\lesssim
V\lesssim 0.4$, and the number of ground state degeneracy also changes
accordingly. Therefore, based on the correlation function and fidelity
results, the system begins as PS at $(U=-6,V\lesssim-0.6)$, then transits to
the $s$-wave phase at $(U=-6,-0.6\lesssim V\lesssim 0.4)$, and reaches the CDW
state at $(U=-6,V\gtrsim 0.4)$. Compared to the $U=-1$ case in Fig. 3(a), a
more attractive $U$ will slightly enlarge the $s$-wave phase boundary along
the $V$-axis.

For $U>0$, the $s$-wave pairing is largely suppressed, and the phase diagram
shows three regions consisting of CDW, SDW, and PS. Figure 8(b) shows the
fidelity and the number of ground state degeneracy along the path
$V:-4.8\rightarrow 2.4$ at $U=8$, as depicted in the figure inset. The
fidelity basically remains unity except at the critical values $V=-4.3$ and
$2.2$, where the fidelity drops to almost zero, signaling two phase transition
boundaries. Therefore, the system begins as PS at $(U=8,V\lesssim-4.3)$,
transits to the $(\pi,\pi)$ SDW phase at $(U=8,-4.3\lesssim V\lesssim 2.2)$,
and reaches the CDW state at $(U=8,V\gtrsim 2.2)$. Overall, the phase
boundaries suggested by fidelity match the regions where correlation functions
change more abruptly.

We next discuss the quench dynamics. Figure 9(a) shows the equal-time
measurement for charge correlation at $(\pi,\pi)$. The original system at time
$t<0$ is the ground state of Hamiltonian $H_{0}$ with interactions
$(U=8,V=2.6)$, which supports a $(\pi,\pi)$ CDW. At time $t=0^{+}$, the
Hamiltonian is quenched to $H_{1}$ with interactions $(U=8,V=2.5)$.
$N(\pi,\pi,t)$ then oscillates in time at $t>0$. Figures 9(b)-(e) show the
resulting Fourier spectra of quenched dynamics for $(\pi,\pi)$ CDW,
$(\pi,\pi)$ SDW, $s$-wave, and $d_{x^{2}-y^{2}}$-wave superconductivity,
respectively. The horizontal axis represents the post-quench value of $V$.
Here we follow the same path as depicted in the inset of Fig. 8(b),
$V:-4.8\rightarrow 2.4$ at a fixed $U=8$, with a quench step of $\Delta
V=0.1$. As shown in Fig. 9(b), the “charge gap” behaves quite differently
below $V\lesssim-4.6$ and above $V\gtrsim 2.0$. Above $V\gtrsim 2$, the gap is
enhanced by $V$, showing an increasingly more robust CDW phase. Between
$-4.6\lesssim V\lesssim 2$, the spectra for the $S(\pi,\pi)$ correlation in
Fig. 9(c) exhibits low-energy excitation (Goldstone mode) of the spontaneous
spin symmetry breaking $(\pi,\pi)$ SDW state. We note again that the
calculations are performed on a homogeneous state; gray dashed lines in the
figure indicate regions where phase separation would occur. Overall, the
quench dynamics spectra can track the charge and spin gaps in the dynamical
structure factors $N(\mathbf{q},\omega)$ and $S(\mathbf{q},\omega)$ shown in
Fig. 10.

###  III.3 Doped Systems

As shown above, the phase boundaries determined by fidelity and quench
dynamics agree well with the correlation function calculations. In hole-doped
systems, we thereby use mainly the correlation function results with
additional fidelity calculations. We note that in doped systems, since the
momentum dependence and the competition with electron itinerary become more
important, it can become challenging to determine the leading instability on a
finite-size cluster. Therefore, instead of specifying the exact phases, we
will focus on the trends of different order parameters evolving with the
interaction strength. In particular, we concentrate on the heavily overdoped
regime with 50% hole doping. In this case, the system exhibits strong $p$-wave
superconductivity correlation in the interaction parameter space
$(U,V)=(8,-1)$ relevant to the cuprate superconductors [41]. In general, the
boundaries of phase transition (or cross-over phenomenon on a finite-size
cluster) evolve smoothly from the half-filled phase diagram in Fig. 6 to the
50% hole doped system in Fig. 11. The results for other hole fillings are
shown in the Appendix.

Figure 11 shows the correlation functions for different order parameters
computed at 50% hole doping in the strong-coupling regime: $|U|\leq 10$ and
$|V|\leq 10$. When $V>0$, the correlations are dominated by CDW for $U>0$ and
$s$-wave superconductivity for $U<0$. $d_{xy}$-pairing correlation is also
found to be enhanced when $V>0$ and $U>0$, and it may be the leading
instability for a small, positive $V$ [36]. When $V<0$, most of the phase
diagram resides in the PS region, especially for $U<0$. The PS boundary for
each doping is determined from the total energy calculations in different
particle sectors and is shown in the Appendix. Even if $U$ is positive but
with $U<|V|$ (where $V$ is negative), the system remains phase-separated.
Interestingly, when $U$ is positive and dominates over $|V|$, both $p$-wave
and $d$-wave superconducting correlations can be strongly enhanced.

The evolution of the correlation strengths for different order parameters as a
function of $V$ (at fixed $U=8$) is shown in Fig. 11(h). For the parameter set
$U=8$ with a negative $V$ ($\sim-1$) relevant to the cuprates, $p$-wave and
$d$-wave correlations have comparable strengths. In general, a repulsive $U$
will favor an inter-site phase-changing superconducting order parameter (like
$p$-wave or $d$-wave pairing), and an attractive $V$ can enhance
superconducting correlations while suppressing other competing states like CDW
[61]. This enhancement of superconductivity by $V$ is beyond the impact of a
next-nearest-neighbor hopping [27, 61]. An attractive $V$ can further favor
$p$-wave superconductivity due to enhanced NN triplet states especially near
50% hole doping [34]. In mean-field calculations [39], the two phases could
possibly coexist. This is consistent with the dominance of the $p$-wave
instability in 1D systems [28, 29, 34], where the $d$-wave competitor is
geometrically forbidden. Therefore, it would be interesting to investigate
more extensively near this parameter regime both theoretically and
experimentally, which may hold the promise of realizing the more exotic
$p$-wave superconductivity.

Figure 11:  Correlation functions for different order parameters computed by
exact diagonalization (ED) for 50% hole-doped extended Hubbard model in the
strong-coupling regime: $|U|\leq 10$ and $|V|\leq 10$. Panel (h) depicts the
evolution of correlation strengths for different order parameters as a
function of $V$ (at fixed $U=8$). The path is also specified by the dashed
arrow in panel (a). Gray shade indicates the phase separation region.

##  IV Conclusion

We have performed extensive exact diagonalization (ED) calculations to study
the extended Hubbard model (EHM) for both attractive and repulsive $U$ and $V$
values, ranging from weak to strong coupling regimes. We have utilized the
correlation functions and fidelity metrics to explore how different
superconducting phases and their boundaries evolve with the interaction
parameters and doping levels. We also have shown that a more novel,
experimentally accessible non-equilibrium quench approach can help determine
the phase boundary of the equilibrium system. While our ED studies are limited
to finite-size clusters, the resulting phase boundaries agree well with other
numerical techniques like functional Renormalization Group [38] in the weak-
coupling limit. The ED approach remains valid in the strong-coupling limit,
and the results should be reliable at a semi-quantitative level. The quenched
dynamics in general cannot be easily and accurately computed by other
numerical methods, either. Therefore, our exact treatment of the model in
finite-size systems also provide systematic benchmark results for other
computational techniques [62, 63] capable of addressing the thermodynamic
limit, and an experimental roadmap for exploring different superconducting
order parameters.

Our results indicate that the EHM is a potential platform to realize the more
exotic $p$-wave superconductivity in the repulsive-$U$ and (slightly)
attractive-$V$ parameter regime, especially when the system is heavily
overdoped by hole carriers. This study is timely, since an attractive $V\sim-
t_{h}$ has recently been identified experimentally in doped 1d cuprate chains
[41]. The parameter regime $(U,V)=(8t_{h},-t_{h})$ is also where $p$-wave
superconductivity shows strong instability in our calculations. In the actual
cuprate materials, the $U$ and $V$ values can behave differently in response
to external stimuli, such as strain, pressure, and laser field, so it is
likely to induce transitions near a phase boundary to favor certain
instability via heterostructure, high pressure, or ultrafast techniques.
Studying a possible $p$-wave state in EHM using other computational approaches
or experiments can be intriguing and important area for future research.

## ACKNOWLEDGMENTS

The authors thank Zecheng Shen for the help in preparing the manuscript.
W.C.C. and Y.W. acknowledge support from the National Science Foundation (NSF)
award DMR-2132338. C.C.C. acknowledges support from NSF Awards OIA-1738698 and
DMR-2142801. The calculations were performed on the Frontera computing system
at the Texas Advanced Computing Center. Frontera is made possible by NSF Award
OAC-1818253.

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  * Jiang  _et al._ [2023] S. Jiang, D. J. Scalapino, and S. R. White, A Single-band Model with Enhanced Pairing from DMRG-based Downfolding of the Three-band Hubbard Model, arXiv preprint arXiv:2303.00756 (2023).

## APPENDIX

##  Appendix A Phase Separation Region

Our exact diagonalization calculations are performed on a square-lattice
$N=16$-site cluster, with a fixed particle sector at a given filling. The
cluster size is too small to observe directly the inhomogeneous sparation of
hole-rich and hole-deficient regions. Instead, the PS information for a given
set of interaction parameters $(U,V)$ can be obtained by computing the total
energies in different particle sectors. For example, at the filling with
$N_{e}$ electrons on an $N$-site cluster, the energy difference $\Delta E$ is
computed:

$\Delta E=E(N_{e})-\frac{N_{e}}{2N}\times[E(2N)+E(0)].$ (12)

Here, $E(N_{e})$ is the energy of the homogeneous ground state with electron
occupation number $N_{e}$; $E(0)$ and $E(2N)$ are respectively the energies of
the hole-rich and hole-deficient states on an $N$-site cluster. Figure 12
shows the $\Delta E$ false-color intensity plots as functions of $(U,V)$ for
different hole-doped systems. A sign change (from negative to positive) in
$\Delta E$ would signal transition to a phase-separated state, which helps
determine the phase boundary of the PS region.

Figure 12:  False-color intensity plot of the energy difference $\Delta E$ for
determining the phase separation boundary as discussed in the Appendix text. A
sign change (from negative to positive) in $\Delta E$ would signal transition
to a phase-separated state.

##  Appendix B Correlation Functions at Other Doping Levels

The correlation functions for different order parameters computed in the
strong-coupling regime ($|U|\leq 10$ and $|V|\leq 10$) for 12.5%, 25%, and
37.5% hole-doped systems are shown in Figs. 13, 14, and 15, respectively. As
discussed above, the PS region for each doping is determined from the energy
difference calculation in different particle sectors and is indicated by gray
shade in panel (h) of the figure.

Figure 13:  Correlation functions for different order parameters computed by
exact diagonalization (ED) for 12.5% hole-doped extended Hubbard model in the
strong-coupling regime: $|U|\leq 10$ and $|V|\leq 10$. Panel (h) depicts the
evolution of correlation strengths for different order parameters as a
function of $V$ (at fixed $U=8$). The path is also specified by the dashed
arrow in panel (a). Gray shade indicates the phase separation region.  Figure
14:  Correlation functions for different order parameters computed by exact
diagonalization (ED) for 25% hole-doped extended Hubbard model in the strong-
coupling regime: $|U|\leq 10$ and $|V|\leq 10$. Panel (h) depicts the
evolution of correlation strengths for different order parameters as a
function of $V$ (at fixed $U=8$). The path is also specified by the dashed
arrow in panel (a). Gray shade indicates the phase separation region.  Figure
15:  Correlation functions for different order parameters computed by exact
diagonalization (ED) for 37.5% hole-doped extended Hubbard model in the
strong-coupling regime: $|U|\leq 10$ and $|V|\leq 10$. Panel (h) depicts the
evolution of correlation strengths for different order parameters as a
function of $V$ (at fixed $U=8$). The path is also specified by the dashed
arrow in panel (a). Gray shade indicates the phase separation region.


# Mapping class groups of simply connected
high-dimensional manifolds need not be arithmetic

Manuel Krannich<EMAIL_ADDRESS>and  Oscar Randal-Williams
<EMAIL_ADDRESS>Centre for Mathematical Sciences,
Wilberforce Road, Cambridge CB3 0WB, UK

###### Abstract.

It is well known that Sullivan showed that the mapping class group of a simply
connected high-dimensional manifold is commensurable with an arithmetic group,
but the meaning of “commensurable” in this statement seems to be less well
known. We explain why this result fails with the now standard definition of
commensurability by exhibiting a manifold whose mapping class group is not
residually finite. We do not suggest any problem with Sullivan’s result:
rather we provide a gloss for it.

###### 2010 Mathematics Subject Classification:

57R50, 11F06, 20E26

In the landmark paper [Sul77] Sullivan proves a structural result on the
groups $\pi_{0}\mathrm{Diff}(M)$ of isotopy classes of diffeomorphisms of a
simply connected compact manifold $M$ of dimension at least $5$, which has as
consequence that these groups are “commensurable” with arithmetic groups
(Theorem 13.3 of loc. cit.).

Sullivan defines “commensurability” of groups (at the bottom of p. 307 of loc.
cit.) as the equivalence relation generated by passing to subgroups of finite
index and taking quotients by finite normal subgroups. This differs from the
current common usage of this term as the equivalence relation generated only
by passing to subgroups of finite index; from now on let us reserve
_commensurable_ for the latter term, and refer to Sullivan’s notion as
_commensurable up to finite kernel_.

These two notions differ. Recall that a group is _residually finite_ if the
intersection of all its finite index subgroups is trivial, or equivalently, if
each nontrivial element remains nontrivial in some finite quotient. As
arithmetic groups are residually finite [Ser79, p. 108], and residual
finiteness is clearly preserved by passing to finite index sub- and
supergroups, a group commensurable to an arithmetic group is in particular
residually finite. However, work of Deligne [Del78] shows that there are
central extensions

$0\longrightarrow\mathbf{Z}/n\longrightarrow\Gamma\longrightarrow\operatorname{Sp}_{2g}(\mathbf{Z})\longrightarrow
0$

for which $\Gamma$ is not residually finite. Such a $\Gamma$ is clearly
commensurable up to finite kernel with the arithmetic group
$\operatorname{Sp}_{2g}(\mathbf{Z})$, but cannot be commensurable with an
arithmetic group.

The aim of this note is to explain how Deligne’s example may be imported into
manifold theory to provide a mapping class group of a simply connected high-
dimensional manifold that is not residually finite, and hence not
commensurable with an arithmetic group.

######  Theorem.

For $n\equiv 5\ (\mathrm{mod}\ 8)$ and $g\geq 5$, the mapping class group
$\pi_{0}\operatorname{Diff}(\sharp^{g}S^{n}\times S^{n})$ is not residually
finite.

The best known family of groups commensurable up to finite kernel with an
arithmetic group but not residually finite is Deligne’s, and the best known
(to the authors, as we have studied them elsewhere [GRW16, Kra19]) family of
mapping class groups of simply connected high-dimensional manifolds are those
of $\sharp^{g}S^{n}\times S^{n}$. Fortunately these examples are often close
to each other; the proof of the Theorem will be to make this precise.

######  Remark.

Sullivan also proves that the groups $\pi_{0}\mathrm{hAut}(X)$ of homotopy
classes of homotopy equivalences of a simply connected finite CW complex $X$
are commensurable up to finite kernel with an arithmetic group [Sul77, Theorem
10.3 (i)]. However, as explained by Serre [Ser79, p. 108], in this case
further results of Sullivan may be used to show that such groups are
commensurable to arithmetic groups in our sense. Indeed, the groups
$\pi_{0}\mathrm{hAut}(X)$ are residually finite as a consequence of [Sul74,
Theorem 3.2], and among residually finite groups being commensurable up to
finite kernel and being commensurable agree: using the second description of
residual finiteness mentioned above, one can deduce that for any finite normal
subgroup $K\leq G$ of a residually finite group $G$, there is a finite group
$C$ and a morphism $G\rightarrow C$ such that the induced morphism
$G\rightarrow(G/K)\times C$ is injective and hence exhibits $G$ as a finite
index subgroup of $(G/K)\times C$, so $G$ and $G/K$ are commensurable.

As a point of terminology, the _finite residual_ of a group is the
intersection of all its finite index subgroups; a group is residually finite
precisely if its finite residual is trivial.

## Deligne’s extensions

We begin by explaining the special case of Deligne’s work [Del78] mentioned
above in a bit more detail (see also [Mor09]). The fundamental group of the
Lie group $\operatorname{Sp}_{2g}(\mathbf{R})$ can be identified with
$\mathbf{Z}$, so the pullback of its universal cover to
$\operatorname{Sp}_{2g}(\mathbf{Z})$ yields a central extension of the form

(1)
$0\longrightarrow\mathbf{Z}\longrightarrow\widetilde{\operatorname{Sp}}_{2g}(\mathbf{Z})\longrightarrow\operatorname{Sp}_{2g}(\mathbf{Z})\longrightarrow
0.$

As long as $\operatorname{Sp}_{2g}(\mathbf{Z})$ has the congruence subgroup
property, i.e. for $g\geq 2$, Deligne’s work implies that the finite residual
of $\widetilde{\operatorname{Sp}}_{2g}(\mathbf{Z})$ agrees with the subgroup
$2\cdot\mathbf{Z}\subset\widetilde{\operatorname{Sp}}_{2g}(\mathbf{Z})$, so
$\widetilde{\operatorname{Sp}}_{2g}(\mathbf{Z})$ is in particular not
residually finite. Moreover, this shows that for $n\geq 3$ the quotients
$\widetilde{\operatorname{Sp}}_{2g}(\mathbf{Z})/(n\cdot\mathbf{Z})$ are not
residually finite either, and hence give rise to non-residually finite central
extensions of $\operatorname{Sp}_{2g}(\mathbf{Z})$ as asserted in in the
introduction.

There is an alternative description of the extension (1), more convenient for
our purposes, which we now explain.

## The signature class

Let us describe Meyer’s _signature class_
$\operatorname{sgn}\in\operatorname{H}^{2}(\operatorname{BSp}_{2g}(\mathbf{Z}))$,
following [Mey72]. For $g\geq 3$ the group
$\operatorname{Sp}_{2g}(\mathbf{Z})$ is perfect, i.e. the first homology
$\operatorname{H}_{1}(\operatorname{BSp}_{2g}(\mathbf{Z}))$ vanishes, so by
the universal coefficient theorem it suffices to specify a morphism

(2)
$\operatorname{sgn}\colon\operatorname{H}_{2}(\operatorname{BSp}_{2g}(\mathbf{Z}))\longrightarrow\mathbf{Z}.$

To do so, recall that, as a consequence of classical work of Thom [Tho54], the
second integral homology group of a space $X$ is canonically isomorphic to the
group of cobordism classes of continuous maps $f\colon S\rightarrow X$ for
oriented closed surfaces $S$. As a result, we may represent a homology class
in $\operatorname{H}_{2}(\operatorname{BSp}_{2g}(\mathbf{Z}))$ by a map
$f\colon S\rightarrow X$ of this kind and associate to it a symmetric bilinear
form given by the composition

$\operatorname{H}^{1}(S;f^{*}\mathcal{H})\otimes\operatorname{H}^{1}(S;f^{*}\mathcal{H})\overset{\smile}{\longrightarrow}\operatorname{H}^{2}(S;f^{*}(\mathcal{H}\otimes\mathcal{H}))\xrightarrow{\lambda_{*}}\operatorname{H}^{2}(S;\mathbf{Z})\overset{\langle[S],-\rangle}{\longrightarrow}\mathbf{Z}.$

Here $\mathcal{H}$ denotes the local system
$\mathcal{H}\rightarrow\operatorname{BSp}_{2g}(\mathbf{Z})$ induced by the
standard module $\mathbf{Z}^{2g}$ of $\operatorname{Sp}_{2g}(\mathbf{Z})$,
which comes equipped with a canonical fibrewise symplectic form
$\lambda\colon\mathcal{H}\otimes\mathcal{H}\rightarrow\mathbf{Z}$. In [Mey72,
§3] Meyer notes that this form is nondegenerate after passing to the torsion
free quotient of $\operatorname{H}^{1}(S;f^{*}\mathcal{H})$ (as a consequence
of Poincaré duality). Moreover, he shows that the signature of the induced
non-degenerate form on the torsion free quotient depends only on the bordism
class of $f$ (a a consequence of Lefschetz duality), so taking signatures
yields a morphism of the form (2).

The signature class is closely related to the extension (1): by construction,
this extension is pulled back from the central extension of
$\operatorname{BSp}_{2g}(\mathbf{R})\simeq\operatorname{BU}(g)$ that, under
the classification of central extensions in terms of second cohomology,
corresponds to a generator in
$\operatorname{H}^{2}(\operatorname{BSp}_{2g}(\mathbf{R}))\cong\operatorname{Hom}(\pi_{1}\operatorname{Sp}_{2g}(\mathbf{R}),\mathbf{Z})\cong\mathbf{Z}$.
This group is also generated by the first Chern class
$c_{1}\in\operatorname{H}^{2}(\operatorname{BSp}_{2g}(\mathbf{R}))$, so up to
possibly rechoosing the identification
$\pi_{1}\operatorname{Sp}_{2g}(\mathbf{R})\cong\mathbf{Z}$, the extension (1)
is classified by the pullback
$c_{1}\in\operatorname{H}^{2}(\operatorname{BSp}_{2g}(\mathbf{Z}))$ along the
inclusion
$\operatorname{Sp}_{2g}(\mathbf{Z})\subset\operatorname{Sp}_{2g}(\mathbf{R})$.
It is a consequence of Meyer’s signature formula [Mey72, §4] (see also [Mey73,
p. 246]) that

(3) $\operatorname{sgn}=-4\cdot
c_{1}\in\operatorname{H}^{2}(\operatorname{BSp}_{2g}(\mathbf{Z})).$

## The theta subgroup

The mapping class groups we shall consider are closely connected to a certain
finite index subgroup
$\operatorname{Sp}_{2g}^{q}(\mathbf{Z})\leq\operatorname{Sp}_{2g}(\mathbf{Z})$,
sometimes called the _theta group_ , which consists of those symplectic
matrices that preserve the standard quadratic refinement, i.e. the function
$q\colon\mathbf{Z}^{2g}\rightarrow\mathbf{Z}/2$ mapping $\sum_{i}(a_{i}\cdot
e_{i}+b_{i}\cdot f_{i})$ to $\sum_{i}a_{i}b_{i}$, where $(e_{i},f_{i})_{1\leq
i\leq g}$ is the usual symplectic basis. By [GRW16, Lemma 7.5], the signature
morphism (2) becomes divisible by $8$ when restricted to
$\operatorname{H}_{2}(\operatorname{BSp}^{q}_{2g}(\mathbf{Z}))$, but the
resulting morphism
$\operatorname{sgn}/8\colon\operatorname{H}_{2}(\operatorname{BSp}^{q}_{2g}(\mathbf{Z}))\rightarrow\mathbf{Z}$
does not uniquely determine a class in
$\operatorname{H}^{2}(\operatorname{BSp}^{q}_{2g}(\mathbf{Z}))$, because in
contrast to $\operatorname{Sp}_{2g}(\mathbf{Z})$, the subgroup
$\operatorname{Sp}_{2g}^{q}(\mathbf{Z})$ is not perfect. To specify such a
class, we use that the abelianisation of
$\operatorname{Sp}_{2g}^{q}(\mathbf{Z})$ is by [JM90, p. 147] cyclic of order
$4$ and generated by

$X_{g}=\operatorname{diag}\left(\left(\begin{smallmatrix}0&-1\\\
1&0\end{smallmatrix}\right),\left(\begin{smallmatrix}1&0\\\
0&1\end{smallmatrix}\right),\ldots,\left(\begin{smallmatrix}1&0\\\
0&1\end{smallmatrix}\right)\right)$

as long as $g\geq 3$. This implies that the morphism
$\mathbf{Z}/4\rightarrow\operatorname{Sp}_{2g}^{q}(\mathbf{Z})$ defined by
$X_{g}$ induces an isomorphism on torsion subgroups of second cohomology, so
induces a splitting

(4)
$\operatorname{H}^{2}(\operatorname{BSp}_{2g}(\mathbf{Z}))\overset{\cong}{\longrightarrow}\operatorname{Hom}(\operatorname{H}_{2}(\operatorname{BSp}_{2g}(\mathbf{Z})),\mathbf{Z})\oplus\operatorname{H}^{2}(\mathrm{B}\mathbf{Z}/4),$

which we can use to define a class

$\mu\in\operatorname{H}^{2}(\operatorname{BSp}^{q}_{2g}(\mathbf{Z}))$

by declaring it to agree with the divided signature $\operatorname{sgn}/8$ on
the first summand and to be trivial on the second. This class is related to
the pullback of
$c_{1}\in\operatorname{H}^{2}(\operatorname{BSp}_{2g}(\mathbf{Z}))$ by

(5) $8\cdot\mu=-4\cdot
c_{1}\in\operatorname{H}^{2}(\operatorname{BSp}_{2g}^{q}(\mathbf{Z})),$

since both sides agree in the first summand of (4) as a result of (3) and in
the second because $\operatorname{H}^{2}(\mathrm{B}\mathbf{Z}/4)$ is
annihilated by $4$. By Deligne’s result the finite residual of the central
extension (1) classified by $c_{1}$ is $2\cdot\mathbf{Z}$, which implies that
the analogous central extension of $\operatorname{Sp}_{2g}(\mathbf{Z})$
classified by $-4\cdot c_{1}$ has finite residual $8\cdot\mathbf{Z}$, using
that

  1. (i)

for a central extension $0\rightarrow A\rightarrow E\rightarrow G\rightarrow
0$ classified by a class
$\lambda\in\operatorname{H}^{2}(\operatorname{BG};A)$, the multiple
$k\cdot\lambda\in\operatorname{H}^{2}(\operatorname{BG};A)$ for
$k\in\mathbf{Z}$ classifies the pushout of this extension along the
multiplication map $k\cdot(-)\colon A\rightarrow A$ (see [Bro82, p. 94]), and
that

  2. (ii)

taking finite residuals is functorial with respect to all group homomorphisms,
and maps inclusions of finite index to isomorphisms.

Since
$\operatorname{Sp}_{2g}^{q}(\mathbf{Z})\leq\operatorname{Sp}_{2g}(\mathbf{Z})$
has finite index and we just argued that the central extension of
$\operatorname{Sp}_{2g}(\mathbf{Z})$ classified by $-4\cdot
c_{1}\in\operatorname{H}^{2}(\operatorname{BSp}_{2g}(\mathbf{Z}))$ has finite
residual $8\cdot\mathbf{Z}$, we conclude that that the central extension of
$\operatorname{Sp}_{2g}^{q}(\mathbf{Z})$ classified by $-4\cdot
c_{1}\in\operatorname{H}^{2}(\operatorname{BSp}^{q}(\mathbf{Z}))$ also has
finite residual $8\cdot\mathbf{Z}$. Using (i) and (ii) once more, we derive
from (5) that the central extension

(6) $0\longrightarrow\mathbf{Z}\longrightarrow
E_{g}\longrightarrow\operatorname{Sp}_{2g}^{q}(\mathbf{Z})\longrightarrow 0$

classified by $\mu$ has finite residual $\mathbf{Z}$.

######  Remark.

It does not matter for our argument, but the relation (5) cannot be divided by
4. By definition $\mu$ vanishes when restricted to $\mathrm{B}\mathbf{Z}/4$,
but $c_{1}$ restricts to a generator of
$\operatorname{H}^{2}(\mathrm{B}\mathbf{Z}/4)\cong\mathbf{Z}/4$ because the
composition
$\mathrm{B}\mathbf{Z}/4\to\operatorname{BSp}_{2}^{q}(\mathbf{Z})\to\operatorname{BSp}_{2}(\mathbf{R})\simeq\operatorname{BU}(1)$
classifies the tautological line bundle.

## Proof of the Theorem

Using work of Kreck [Kre79], it is shown in [GRW16, Equation 7.3 and Theorem
7.7] that for $g\geq 5$ and $n\equiv 5\ (\mathrm{mod}\ 8)$ the mapping class
group of orientation preserving diffeomorphisms of
$W_{g}=\sharp^{g}S^{n}\times S^{n}$ admits a splitting

$\pi_{0}\operatorname{Diff}^{\scaleobj{0.8}{+}}(W_{g})\cong\widetilde{E}_{g}\times\operatorname{coker}(J)_{2n+1}$

into the (finite) cokernel $\operatorname{coker}(J)_{2n+1}$ of the stable
$J$-homomorphism in dimension $2n+1$ and the central extension

$0\longrightarrow\mathbf{Z}/\lvert\operatorname{bP}_{2n+2}\rvert\longrightarrow\widetilde{E}_{g}\longrightarrow\operatorname{Sp}_{2g}^{q}(\mathbf{Z})\longrightarrow
0,$

obtained from (6) by taking quotients by the subgroup
$\lvert\operatorname{bP}_{2n+2}\rvert\cdot\mathbf{Z}\leq\mathbf{Z}$ generated
by the order of Kervaire–Milnor’s finite cyclic group
$\operatorname{bP}_{2n+2}$ of homotopy $(2n+1)$-spheres bounding a
parallelisable manifold [KM63]. Since the finite residual of $E_{g}$ is
$\mathbf{Z}$ by the discussion above, that of $\widetilde{E}_{g}$ is
$\mathbf{Z}/\lvert\operatorname{bP}_{2n+2}\rvert$, which is nontrivial, since
under our assumption that $n\equiv 5\ (\mathrm{mod}\ 8)$, the group
$\operatorname{bP}_{2n+2}$ is known not to vanish (see Section 7 loc. cit.).
This implies that $\widetilde{E}_{g}$ is not residually finite, and hence
neither is $\pi_{0}\operatorname{Diff}^{\scaleobj{0.8}{+}}(W_{g})$ nor its
non-orientation preserving variant $\pi_{0}\operatorname{Diff}(W_{g})$.

## Acknowledgements

The authors were supported by the ERC under the European Union’s Horizon 2020
research and innovation programme (grant agreement No. 756444), and by a
Philip Leverhulme Prize from the Leverhulme Trust.

## References

  * [Bro82] K. S. Brown, _Cohomology of groups_ , Graduate Texts in Mathematics, vol. 87, Springer-Verlag, New York-Berlin, 1982.
  * [Del78] P. Deligne, _Extensions centrales non résiduellement finies de groupes arithmétiques_ , C. R. Acad. Sci. Paris Sér. A-B 287 (1978), no. 4, A203–A208.
  * [GRW16] S. Galatius and O. Randal-Williams, _Abelian quotients of mapping class groups of highly connected manifolds_ , Math. Ann. 365 (2016), no. 1-2, 857–879.
  * [JM90] D. Johnson and J. J. Millson, _Modular Lagrangians and the theta multiplier_ , Invent. Math. 100 (1990), no. 1, 143–165.
  * [KM63] M. A. Kervaire and J. W. Milnor, _Groups of homotopy spheres. I_ , Ann. of Math. (2) 77 (1963), 504–537.
  * [Kra19] M. Krannich, _Mapping class groups of highly connected $(4k+2)$-manifolds_, 2019, arXiv:1902.10097.
  * [Kre79] M. Kreck, _Isotopy classes of diffeomorphisms of $(k-1)$-connected almost-parallelizable $2k$-manifolds_, Algebraic topology, Aarhus 1978, Lecture Notes in Math., vol. 763, Springer, Berlin, 1979, pp. 643–663.
  * [Mey72] W. Meyer, _Die Signatur von lokalen Koeffizientensystemen und Faserbündeln_ , Bonn. Math. Schr. (1972), no. 53, viii+59.
  * [Mey73] by same author, _Die Signatur von Flächenbündeln_ , Math. Ann. 201 (1973), 239–264.
  * [Mor09] D. W. Morris, _A lattice with no torsion-free subgroup of finite index (after P. Deligne)_ , Available at http://people.uleth.ca/~dave.morris/talks/deligne-torsion.pdf.
  * [Ser79] J.-P. Serre, _Arithmetic groups_ , Homological group theory (Proc. Sympos., Durham, 1977), London Math. Soc. Lecture Note Ser., vol. 36, Cambridge Univ. Press, Cambridge-New York, 1979, pp. 105–136.
  * [Sul74] D. Sullivan, _Genetics of homotopy theory and the Adams conjecture_ , Ann. of Math. (2) 100 (1974), 1–79.
  * [Sul77] by same author, _Infinitesimal computations in topology_ , Inst. Hautes Études Sci. Publ. Math. (1977), no. 47, 269–331 (1978).
  * [Tho54] R. Thom, _Quelques propriétés globales des variétés différentiables_ , Comment. Math. Helv. 28 (1954), 17–86.


note that $y\,\longmapsto\,s(y)+\omega(y)$ is a holomorphic section, over $U$,
of the projection $\rho$ in (8.5).

######  Corollary 8.2.

  1. (1)

The moduli space ${\mathcal{H}}_{g}(r)$ in (5.7) has a canonical algebraic
symplectic structure $\Omega_{{\mathcal{H}}_{g}(r)}$.

  2. (2)

The symplectic form $\Omega_{{\mathcal{H}}_{g}(r)}$ on ${\mathcal{H}}_{g}(r)$
is compatible with the $T^{*}{\mathcal{B}}_{g}(r)$–torsor structure of
${\mathcal{H}}_{g}(r)$ obtained in Theorem 5.3.

  3. (3)

There is a holomorphic line bundle $\mathbf{L}$ on ${\mathcal{H}}_{g}(r)$ and
a holomorphic connection $\nabla^{\mathbf{L}}$ on $\mathbf{L}$ such that the
curvature of $\nabla^{\mathbf{L}}$ is the symplectic form
$\Omega_{{\mathcal{H}}_{g}(r)}$.

###### Proof.

Using the isomorphism ${\mathcal{H}}$ in Theorem 8.1, the above algebraic
symplectic form $\Omega_{\mathcal{L}}$ on ${\rm Conn}({\mathcal{L}})$ (see
(8.30)) produces an algebraic symplectic form

$\Omega_{{\mathcal{H}}_{g}(r)}\,:=\,{\mathcal{H}}^{*}\Omega_{\mathcal{L}}$

on ${\mathcal{H}}_{g}(r)$.

The compatibility condition in the statement (2) says that for any locally
defined holomorphic section of the projection $\gamma$ in (5.7)

${\mathcal{B}}_{g}(r)\,\supset\,U\,\stackrel{{\scriptstyle
s}}{{\longrightarrow}}\,{\mathcal{H}}_{g}(r)$

and any holomorphic $1$–form $\omega\,\in\,H^{0}(U,\,T^{*}U)$, the equality

$s^{*}\Omega_{{\mathcal{H}}_{g}(r)}+d\omega\,=\,(s+\omega)^{*}\Omega_{{\mathcal{H}}_{g}(r)}$
(8.31)

holds; note that $y\,\longmapsto\,s(y)+\omega(y)$ is a holomorphic section,
over $U$, of the projection $\gamma$ in (5.7). Now, (8.31) follows immediately
from (8.30).

We recall that $\Omega_{\mathcal{L}}$ is the curvature of the holomorphic
connection $f_{0}\circ h_{0}$ (see (8.29)) on the holomorphic line bundle
$\rho^{*}{\mathcal{L}}$. Therefore, $\Omega_{{\mathcal{H}}_{g}(r)}$ is the
curvature of the holomorphic connection

$\nabla^{\mathbf{L}}\,:=\,{\mathcal{H}}^{*}(f_{0}\circ h_{0})$

on the holomorphic line bundle
$\mathbf{L}\,:=\,{\mathcal{H}}^{*}\rho^{*}{\mathcal{L}}$. ∎

##  9\. Isomonodromy and symplectic form

###  9.1. Projective structure on a Riemann surface

Let $Y$ be a compact connected Riemann surface. A holomorphic coordinate chart
on $Y$ is a pair of the form $(U,\,\varphi)$, where $U\,\subset\,Y$ is an open
subset and $\varphi\,:\,U\,\longrightarrow\,{\mathbb{C}}{\mathbb{P}}^{1}$ is a
holomorphic embedding. A holomorphic coordinate atlas on $Y$ is a collection
of holomorphic coordinate charts $\\{(U_{i},\,\varphi_{i})\\}_{i\in J}$ such
that $Y\,=\,\bigcup_{i\in J}U_{i}$. A projective structure on $Y$ is given by
a holomorphic coordinate atlas $\\{(U_{i},\,\varphi_{i})\\}_{i\in J}$
satisfying the following condition: For every $i,\,j\,\in\,J\times J$ with
$U_{i}\cap U_{j}\,\not=\,\emptyset$, and every connected component
$V_{c}\,\subset\,U_{i}\bigcap U_{j}$, there is a
$\tau^{c}_{j,i}\,\in\,\text{PSL}(2,{\mathbb{C}})$ such that the map
$(\varphi_{j}\circ\varphi^{-1}_{i})\big{|}_{\varphi_{i}(V_{c})}$ is the
restriction, to $\varphi_{i}(V_{c})$, of the automorphism of
${\mathbb{C}}{\mathbb{P}}^{1}$ given by $\tau_{j,i}$. Recall that
$\text{PSL}(2,{\mathbb{C}})\,=\,\text{Aut}({\mathbb{C}}{\mathbb{P}}^{1})$.

Two holomorphic coordinate atlases $\\{(U_{i},\,\varphi_{i})\\}_{i\in J}$ and
$\\{(U_{i},\,\varphi_{i})\\}_{i\in J^{\prime}}$ satisfying the above condition
are called equivalent if their union $\\{(U_{i},\,\varphi_{i})\\}_{i\in J\cup
J^{\prime}}$ also satisfies the above condition. A projective structure on $Y$
is an equivalence class of holomorphic coordinate atlases satisfying the above
condition.

Giving a projective structure on $Y$ is equivalent to giving a holomorphic
Cartan geometry on $Y$ for the pair of groups
$(\text{PSL}(2,{\mathbb{C}}),\,B)$, where
$B\,\subset\,\text{PSL}(2,{\mathbb{C}})$ is the Borel subgroup

$B\,:=\,\Big{\\{}\begin{pmatrix}a&b\\\
c&d\end{pmatrix}\,\in\,\text{PSL}(2,{\mathbb{C}})\,\mid\,c\,=\,0\Big{\\}}\,;$

see [Sh] for Cartan geometry. Let
$\widetilde{B}\,\subset\,\text{SL}(2,{\mathbb{C}})$ be the Borel subgroup that
projects to $B$ defined above. Giving a holomorphic Cartan geometry on $Y$ for
the pair of groups $(\text{SL}(2,{\mathbb{C}}),\,\widetilde{B})$ is equivalent
to giving a projective structure on $Y$ together with a theta characteristic
on $Y$ [Gu].

For $g\,\geq\,2$, let

$\delta\,\,:\,\,{\mathcal{P}}_{g}\,\longrightarrow\,{\mathcal{M}}^{\theta}_{g}$
(9.1)

be the moduli space of triples of the form $(X,\,K^{1/2}_{X},\,P)$, where

  * •

$X$ is a compact connected Riemann surface of genus $g$,

  * •

$K^{1/2}_{X}$ is a theta characteristic on $X$, and

  * •

$P$ is a projective structure on $X$.

The map $\delta$ in (9.1) sends any $(X,\,K^{1/2}_{X},\,P)$ to
$(X,\,K^{1/2}_{X})$.

######  Proposition 9.1.

The moduli space ${\mathcal{H}}_{g}(r)$ in (5.7) admits a natural projection

$f\,\,:\,\,{\mathcal{H}}_{g}(r)\,\longrightarrow\,{\mathcal{P}}_{g}\,,$

where ${\mathcal{P}}_{g}$ is defined in (9.1).

###### Proof.

Take any $(X,\,K^{1/2}_{X},\,F,\,D)\,\in\,{\mathcal{H}}_{g}(r)$. Let

$s\,\,\in\,\,\frac{H^{0}\left(3\mathbf{\Delta},\,\left((F\otimes
K^{3/2}_{X})\boxtimes(F^{*}\otimes K^{3/2}_{X})\otimes{\mathcal{O}}_{X\times
X}(3{\mathbf{\Delta}})\right)\big{|}_{3\mathbf{\Delta}}\right)}{H^{0}(X,\,{\rm
ad}(F)\otimes K^{\otimes 2}_{X})}$ (9.2)

be the element corresponding to it given by Corollary 6.5. Restricting $s$ to
$2\mathbf{\Delta}\,\subset\,3\mathbf{\Delta}$, we get a section

$\displaystyle s_{1}\,\in\,H^{0}\left(2\mathbf{\Delta},\,\left((F\otimes
K^{3/2}_{X})\boxtimes(F^{*}\otimes K^{3/2}_{X})\otimes{\mathcal{O}}_{X\times
X}(3{\mathbf{\Delta}})\right)\big{|}_{2\mathbf{\Delta}}\right)$
$\displaystyle\,=\,H^{0}\left(2\mathbf{\Delta},\,\left((F\boxtimes
F^{*})\otimes((K^{3/2}_{X}\boxtimes K^{3/2}_{X})\otimes{\mathcal{O}}_{X\times
X}(3{\mathbf{\Delta}}))\right)\big{|}_{2\mathbf{\Delta}}\right)\,.$

Since $((K^{3/2}_{X}\boxtimes K^{3/2}_{X})\otimes{\mathcal{O}}_{X\times
X}(3{\mathbf{\Delta}}))\big{|}_{2\mathbf{\Delta}}$ is canonically trivialized
[BR, p. 688, Theorem 2.2] (this was noted in (7.3)), there is a unique section

$s^{\prime}_{1}\,\,\in\,\,H^{0}(2\mathbf{\Delta},\,(F\boxtimes
F^{*})\big{|}_{2\mathbf{\Delta}})$ (9.3)

such that $s_{1}\,=\,s^{\prime}_{1}\otimes\textbf{t}$, where t is the section
of $((K^{3/2}_{X}\boxtimes K^{3/2}_{X})\otimes{\mathcal{O}}_{X\times
X}(3{\mathbf{\Delta}}))\big{|}_{2\mathbf{\Delta}}$ that trivializes it. On the
other hand, $s_{1}\big{|}_{\mathbf{\Delta}}\,=\,\text{Id}_{F}$ (see Corollary
6.5), which implies that
$s^{\prime}_{1}\big{|}_{\mathbf{\Delta}}\,=\,\text{Id}_{F}$, because the
restriction of t to $\mathbf{\Delta}$ is the constant function $1$ on $X$
using the identification of $\mathbf{\Delta}$ with $X$ given by
$x\,\longmapsto\,(x,\,x)$. Consequently, $s^{\prime}_{1}$ defines a
holomorphic connection on $F$. This holomorphic connection in turn provides an
extension of the section $s^{\prime}_{1}$ to a section

$s_{2}\,\in\,H^{0}\left(3\mathbf{\Delta},\,(F\boxtimes
F^{*})\big{|}_{3\mathbf{\Delta}}\right)\,;$

we note that using parallel translations, for the integrable connection on $F$
defined by $s^{\prime}_{1}$, we get a holomorphic isomorphism between
$q^{*}_{1}F$ and $q^{*}_{2}F$ over an analytic neighborhood of
$\mathbf{\Delta}\,\subset\,X\times X$ (recall that $q_{i}$ is the projection
of $X\times X$ to the $i$-th factor). Now, invoking the isomorphism

$s_{2}\,:\,(q^{*}_{1}F)\big{|}_{3\mathbf{\Delta}}\,\longrightarrow\,(q^{*}_{2}F)\big{|}_{3\mathbf{\Delta}}\,,$

the element $s$ in (9.2) becomes

$s\,\,\in\,\,\frac{H^{0}\left(3\mathbf{\Delta},\,\left((F\otimes F^{*}\otimes
K^{3/2}_{X})\boxtimes(K^{3/2}_{X})\otimes{\mathcal{O}}_{X\times
X}(3{\mathbf{\Delta}})\right)\big{|}_{3\mathbf{\Delta}}\right)}{H^{0}(X,\,{\rm
ad}(F)\otimes K^{\otimes 2}_{X})}\,.$

Composing with the trace homomorphism $F\otimes
F^{*}\,=\,\text{End}(F)\,\longrightarrow\,{\mathcal{O}}_{X}$ defined by
$A\,\longmapsto\,\frac{1}{r}\text{trace}(A)$, the above element $s$ gives a
section

$\phi_{2}\,\,\in\,\,H^{0}\left(3\mathbf{\Delta},\,\left((K^{3/2}_{X})\boxtimes
K^{3/2}_{X})\otimes{\mathcal{O}}_{X\times
X}(3{\mathbf{\Delta}})\right)\big{|}_{3\mathbf{\Delta}}\right).$

This section $\phi_{2}$ defines a projective structure on $X$ [BR, p. 688,
Theorem 2.2]. Therefore, we have a map

$f\,:\,{\mathcal{H}}_{g}(r)\,\longrightarrow\,{\mathcal{P}}_{g}$

that sends any $(X,\,K^{1/2}_{X},\,F,\,D)\,\in\,{\mathcal{H}}_{g}(r)$ to
$(X,\,K^{1/2}_{X},\,\phi_{2})\,\in\,{\mathcal{P}}_{g}$, where $\phi_{2}$ is
constructed above from $D$. ∎

###  9.2. Isomonodromy

For any compact Riemann surface $X$ of genus $g$, with $g\,\geq\,2$, let
${\mathcal{D}}_{X}(r)$ denote the moduli space of pairs of the form $(F,\,D)$,
where $F$ is a stable vector bundle on $X$ of rank $r$ and degree zero, and
$D$ is a holomorphic connection on $F$.

######  Proposition 9.2.

For the map $f$ in Proposition 9.1, the fiber over any
$(X,\,K^{1/2}_{X},\,P)\,\in\,{\mathcal{P}}_{g}$ is canonically identified with
the moduli space ${\mathcal{D}}_{X}(r)$.

###### Proof.

Take any $(X,\,K^{1/2}_{X},\,F,\,D)\,\in\,{\mathcal{H}}_{g}(r)$ in the fiber
of $f$ over $(X,\,K^{1/2}_{X},\,P)\,\in\,{\mathcal{P}}_{g}$. As seen before,
$s^{\prime}_{1}$ in (9.3) defines a holomorphic connection on $F$. So we get a
map from the fiber of $f$ over $(X,\,K^{1/2}_{X},\,P)$ to
${\mathcal{D}}_{X}(r)$ that sends any $(X,\,K^{1/2}_{X},\,F,\,D)$ to
$(F,\,s^{\prime}_{1})$.

Conversely, take any $(F,\,D)\,\in\,{\mathcal{D}}_{X}(r)$. Then $D$ gives a
section

$\widehat{s}_{D}\,\,\in\,\,H^{0}(3\mathbf{\Delta},\,(F\boxtimes
F^{*})\big{|}_{3\mathbf{\Delta}})\,.$

On the other hand, the projective structure $P$ gives a section

$P^{\prime}\,\,\in\,\,H^{0}(3\mathbf{\Delta},\,((K^{3/2}_{X}\boxtimes
K^{3/2}_{X})\otimes{\mathcal{O}}_{X\times
X}(3{\mathbf{\Delta}}))\big{|}_{3\mathbf{\Delta}})$

[BR, p. 688, Theorem 2.2]. So we have

$\widehat{s}_{D}\otimes
P^{\prime}\,\,\in\,\,H^{0}\left(3\mathbf{\Delta},\,\left((F\otimes
K^{3/2}_{X})\boxtimes(F^{*}\otimes K^{3/2}_{X})\otimes{\mathcal{O}}_{X\times
X}(3{\mathbf{\Delta}})\right)\big{|}_{3\mathbf{\Delta}}\right)\,.$

Consider the image of $\widehat{s}_{D}\otimes P^{\prime}$

$\widetilde{\widehat{s}_{D}\otimes
P^{\prime}}\,\in\,\frac{H^{0}\left(3\mathbf{\Delta},\,\left((F\otimes
K^{3/2}_{X})\boxtimes(F^{*}\otimes K^{3/2}_{X})\otimes{\mathcal{O}}_{X\times
X}(3{\mathbf{\Delta}})\right)\big{|}_{3\mathbf{\Delta}}\right)}{H^{0}(X,\,{\rm
ad}(F)\otimes K^{\otimes 2}_{X})}$

in the quotient space. Now
$(X,\,K^{1/2}_{X},\,F,\,\widetilde{\widehat{s}_{D}\otimes P^{\prime}})$ gives
an element of ${\mathcal{H}}_{g}(r)$ in the fiber of $f$ over
$(X,\,K^{1/2}_{X},\,P)\,\in\,{\mathcal{P}}_{g}$ (see Corollary 6.5). ∎

In view of Proposition 9.2, the isomonodromy condition for integrable
holomorphic connections defines a holomorphic foliation

${\mathcal{F}}\,\,\subset\,\,T{\mathcal{H}}_{g}(r)$ (9.4)

which gives the following decomposition:

$T{\mathcal{H}}_{g}(r)\,=\,{\mathcal{F}}\oplus\text{kernel}(df)\,,$ (9.5)

where $df\,:\,T{\mathcal{H}}_{g}(r)\,\longrightarrow\,f^{*}T{\mathcal{P}}_{g}$
is the differential of the projection $f$ in Proposition 9.1. Consequently,
the differential $df$ identifies ${\mathcal{F}}$ with
$f^{*}T{\mathcal{P}}_{g}$.

We note that ${\mathcal{D}}_{X}(r)$ has a natural holomorphic symplectic form
[Go], [AB]. Also, ${\mathcal{P}}_{g}$ has a holomorphic symplectic form which
is constructed using the monodromy representation associated to any projective
structure [Go], [AB], [He]. Therefore, using the decomposition in (9.5) we
obtain two closed holomorphic $2$-forms on ${\mathcal{H}}_{g}(r)$: one is
given by the symplectic form on ${\mathcal{D}}_{X}(r)$ and the other is given
by the symplectic form on ${\mathcal{P}}_{g}$.

We end with the following conjecture:

######  Conjecture 9.3.

The holomorphic symplectic form on ${\mathcal{H}}_{g}(r)$ in Corollary 8.2(1)
is a constant linear combination of the above two holomorphic $2$-forms on
${\mathcal{H}}_{g}(r)$.

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