aainstitutetext: Kavli Institute for Theoretical Physics, University of
California, Santa Barbara, CA 93106-4030, USAbbinstitutetext: Department of
Physics, University of California, Santa Barbara, CA 93106-9530,
USAccinstitutetext: Department of Physics and Astronomy, University of Utah,
Salt Lake City, UT 84112-0830, USAddinstitutetext: Center for Theoretical
Physics and Department of Physics, University of California, Berkeley, CA
94720-7300, USAeeinstitutetext: Microsoft Station Q, Santa Barbara, CA
93106-6105, USAffinstitutetext: Department of Mathematics, University of
California, Santa Barbara, CA 93106-3080, USA

# Establishing strongly-coupled 3D AdS quantum gravity with Ising dual using
all-genus partition functions

Chao-Ming Jian b Andreas W. W. Ludwig a,c Zhu-Xi Luo a,d Hao-Yu Sun e,f and
Zhenghan Wang

###### Abstract

We study 3D pure Einstein quantum gravity with negative cosmological constant,
in the regime where the AdS radius $l$ is of the order of the Planck scale.
Specifically, when the Brown-Henneaux central charge $c=3l/2G_{N}$ ($G_{N}$ is
the 3D Newton constant) equals $c=1/2$, we establish duality between 3D
gravity and 2D Ising conformal field theory by matching gravity and conformal
field theory partition functions for AdS spacetimes with general asymptotic
boundaries. This duality was suggested by a genus-one calculation of Castro et
al. [Phys. Rev. D 85, 024032 (2012)]. Extension beyond genus-one requires new
mathematical results based on 3D Topological Quantum Field Theory; these turn
out to uniquely select the $c=1/2$ theory among all those with $c<1$,
extending the previous results of Castro et al..
Previous work suggests the reduction of the calculation of the gravity
partition function to a problem of summation over the orbits of the mapping
class group action on a “vacuum seed”. But whether or not the summation is
well-defined for the general case was unknown before this work. Amongst all
theories with Brown-Henneaux central charge $c<1$, the sum is finite and
unique only when $c=1/2$, corresponding to a dual Ising conformal field theory
on the asymptotic boundary.

##  1 Introduction and Summary of Results

A way of looking at 3D pure gravity with negative cosmological constant using
partition functions and modular properties was initiated in work by Dijkgraaf
et al. FareyTail , Witten Witten , and Maloney and Witten MaloneyWitten . In a
pioneering paper Castro , Castro et al. argued that two well-known Conformal
Field Theories (CFT) in two-dimensional space time, i.e., Ising and
tricritical Ising minimal models BPZ ; FQS , are dual to pure Einstein quantum
gravity in three-dimensional spacetime with negative cosmological constant,
i.e., in Anti-de Sitter spacetime (AdS3).111In the same paper similar
arguments are also presented for certain versions of theories of higher spin
quantum gravity. See also 3. These are theories of strongly-coupled gravity
where the AdS radius $l$ is of the order of the Planck scale. The arguments
provided by Castro et al. in support of these dualities at the corresponding
values of the Brown-Henneaux BrownHenneaux  central charges $c=3l/2G_{N}$
($G_{N}$ is the 3D Newton constant) consisted in demonstrating a match between
the gravity partition function of Euclidean AdS${}_{3}\,\,$spacetime when its
asymptotic boundary is a 2D torus, with the torus partition function of the
corresponding 2D minimal model CFT.

To be specific, one can think of the finite-temperature partition function of
pure Einstein gravity in Euclidean AdS${}_{3}\,\,$as being written as a path
integral. The latter is formally a sum over every smooth 3-manifold $X$ whose
asymptotic boundary is a torus $T^{2}$,

$Z_{\text{grav}}(\tau,\bar{\tau})=\int_{\partial
X=T^{2}}\mathcal{D}g_{\mu\nu}~{}e^{-c\ S_{E}[g_{\mu\nu}]},$ (1)

with $\tau$ the conformal structure parameter of the boundary torus, the
Brown-Henneaux central charge $c=3l/2G_{N}$ playing the role of the inverse
gravity coupling constant (large in the semi-classical regime).
$S_{E}[g_{\mu\nu}]$ is the Einstein-Hilbert action with $g_{\mu\nu}$ the
complete Riemannian metric tensor on $X$. One will need to both sum over all
different geometries of the bulk 3-manifold $X$ with the same equivalence
class of conformal structures (i.e., the same conformal class) on the boundary
torus, as well as integrate over all different boundary metrics connected by
small diffeomorphisms, i.e., those isotopic to the identity. The full
gravitational path integral can then be written as

$Z_{\text{grav}}=\sum_{X\ ({\rm where}\ \partial X=T^{2})}Z(X,\tau),$ (2)

where $Z(X,\tau)$ denotes the contribution from the sum over all metrics
related by small diffeomorphisms on a particular 3-manifold $X$ with a fixed
conformal structure parameter $\tau$ on the asymptotic torus boundary, while
the summation over $X$ means summing over different $\tau$’s in the same
conformal class.

In the semi-classical (large $c$) limit, the smooth 3-manifolds $X$
contributing to the path integral turn out to be only those which admit
classical solutions, i.e., which are saddle points222One main conclusion of
MaloneyWitten  is that in the weak-coupling/semiclassical regime, one has to
include geometries corresponding to complex saddle points of $S_{E}[g]$ in
order to have a Hilbert space interpretation of the gravity theory. However,
since here we are only concerned with the strongly coupled regime, we are not
bound by these considerations. of the Einstein-Hilbert action $S_{E}[g]$, and
only solid tori are commonly considered, see MaldacenaStrominger ; FareyTail ;
MaloneyWitten . Following the logic pursued in previous work Castro ;
FareyTail ; Witten  on this problem, the gravitational path integral can then
be thought of as being organized as a sum over classical solutions, along with
a full treatment of all quantum fluctuations around each saddle point. For the
case of solid tori $X$, different saddles correspond to inequivalent ways of
filling in the bulk $X$ of the boundary torus $T^{2}$, and are related to each
other by $SL(2,\mathbb{Z})$ modular transformations. The gravity partition
function (2) can then be obtained as the sum of inequivalent images of a
certain “vacuum seed” partition function in Euclidean AdS${}_{3}\,\,$under the
action of $SL(2,\mathbb{Z})$. Physically, this “vacuum seed” describes the
gravitational partition function of thermal AdS${}_{3}\,\,$where the spatial
cycle of the boundary torus $T^{2}$ is contractible in the bulk, whereas the
cycle of Euclidean time is not. This corresponds to a particular solid torus
$X$, for example see Figure 1. As argued in Castro , the gravitational “vacuum
seed” partition function can be obtained exactly by using the remarkable and
fundamental results of Brown and Henneaux BrownHenneaux ; this is reviewed in
Section 2.1 below, and the result summarized in the next paragraph. After
action on the “vacuum seed” gravitational partition function with a non-
trivial modular transformation, the spatial cycle may no longer be
contractible in the bulk while the temporal cycle may now be; in that case the
corresponding gravitational partition function describes physically that of a
BTZ black hole BTZ . The sum over modular transformations in
$SL(2,\mathbb{Z})$ appearing in (2) can also be seen to originate from general
coordinate invariance, independent of invoking semi-classical notions such as
saddle points, because non-trivial modular transformations correspond to large
diffeomorphisms, not continuously connected to the identity; in the
gravitational path integral for the “vacuum seed” partition function, on the
other hand, these large diffeomorphisms are thought to be excluded. (This
complementary point of view was also stressed in Castro .)

As was argued in Castro , with certain assumptions the gravitational “vacuum
seed” partition function turns out to be precisely equal to the vacuum
character of the dual CFT. Furthermore, owing to the fact that the vacuum
character of rational CFTs is invariant under a certain finite index subgroup
of the modular group $SL(2,\mathbb{Z})$ Bantay ; NgSch , the modular sum in
(2) over the infinite group $SL(2,\mathbb{Z})$ of modular transformations
reduces in fact to the sum over a finite number of right cosets of that finite
index subgroup in $SL(2,\mathbb{Z})$ when the Brown-Henneaux central charge
$c$ is equal to that of a unitary conformal minimal model CFT BPZ ; FQS . For
Brown-Henneaux central charge $c=1/2$, the resulting finite sum was shown in
Castro  to be proportional to the partition function of the 2D Ising CFT on
the torus $T^{2}$.

Based on the above analysis of solid tori $X$, Castro et al. argued in Castro
that amongst all BPZ ; FQS  the unitary Virasoro minimal models with central
charge $c<1$, only the Ising and tricritical Ising CFTs are dual to pure
Einstein gravity at the corresponding values of the Brown-Henneaux central
charge.333Some of the $\mathcal{W}_{N}$ minimal models, were also conjectured
in Castro  to be possibly dual to higher-spin gravity theories instead of
being dual to pure Einstein gravity. This is due to the existence of an
extended chiral conformal algebra, generated by conserved currents possessing
(conformal) spins with values ranging from $3$ up to $N$. These currents
generalize the spin-2 stress-energy tensor $T_{\mu\nu}$ which generates “pure
graviton” excitations in the pure Einstein gravity discussed in Section 2.1,
and lead to a “truncated version” of higher spin Vasiliev gravity, the latter
containing generalized graviton excitations of arbitrary integer spin (see
e.g. Vasiliev ; Gopakumar ). As it is well known, the presence of extended
chiral conformal algebras can lead to multiple modular invariants in 2D CFTs,
but by extending the “vacuum seed” to the vacuum representation of the
extended chiral algebra, a single modular invariant can be built, and
generalizations to higher spin gravity of the Virasoro arguments leading to
Ising and Tricritical Ising are possible as described in Castro  based on
genus-one considerations. A possible gravitational explanation of this
observation could be as follows: It turns out that amongst all unitary
Virasoro minimal CFTs with central charge $c<1$, only the Ising and
tricritical Ising CFTs satisfy the condition that the conformal weights $h$ of
all non-trivial primary states are larger than $c/24$. In CFTs with large
central charge $c$, this inequality describes a necessary condition that a
primary state of conformal weight $h$ can be interpreted as being dual to a
black hole Strominger . Assuming that this condition is still valid in the
strong-coupling regime where $c$ is not large, Ising and tricritical Ising
would be the only unitary minimal model CFTs with $c<1$ in which all primary
states can be interpreted as being dual to black holes. All other $c<1$
unitary minimal model CFTs would then contain, in addition to black holes,
other primary matter fields, and these CFTs could thus not be dual to pure
Einstein gravity. (We will come back in Appendix G to the interpretation of
primary states in the Ising CFT as states dual to black holes in strongly-
coupled Einstein gravity, by suggesting a possible expression for their
Bekenstein-Hawking entropy.)

The focus of the present paper is pure Einstein quantum gravity on Euclidean
3-manifolds $X$ whose asymptotic boundaries $\partial X$ are higher-genus
Riemann surfaces. This arises physically because 3-manifolds $X$ whose
boundaries are Riemann surfaces of higher genus $g\geq 2$, are known
Aminneborg ; Brill ; Krasnov1 ; Krasnov2 ; Yin1 ; Giombi ; Skenderis  to be
the Euclidean spacetimes corresponding to multi-boundary wormholes in
Lorentzian signature. $X$ is commonly restricted to handlebodies, and we will
follow this assumption here; more complicated saddles such as non-handlebodies
were studied in Yin2  in the semi-classical regime. We plan to come back to
this issue in future work. There is a variety of interesting and important
physical questions related to such multi-boundary wormholes (see, e.g.,
Multibdry  for a relatively recent discussion), and a complete description of
the duality between 3D quantum gravity and the associated 2D CFT at the
asymptotic boundary must include all those spacetimes. In other words, any
proposed duality must also be valid in any such multi-boundary wormhole
spacetime. For that reason, it is important to investigate the duality between
quantum gravity in AdS${}_{3}\,\,$and the CFT on the asymptotic boundary at
higher genus $g\geq 2$.

For the gravitational partition function at general genus $g$, there is again
formally a sum over geometries of the smooth handlebody $X$ and over its
boundary geometries

$Z_{\text{grav}}(\Omega,\bar{\Omega})=\int_{\partial X=\Sigma_{g}}{\cal
D}g_{\mu\nu}\ e^{-c~{}S_{E}[g_{\mu\nu}]},$ (3)

where the “period matrix” $\Omega$, a $g\times g$-dimensional symmetric
complex matrix, completely parametrizes the conformal structure of the genus
$g$ Riemann surface $\Sigma_{g}$ constituting the boundary of $X$. The
gravitational path integral can then again be written in the form

$Z_{\text{grav}}=\sum_{X\ ({\rm where}\ \partial X=\Sigma_{g})}\ Z(X,\Omega),$
(4)

where $Z(X,\Omega)$ stands for the contribution from the sum over all metrics
connected by small diffeomorphisms on a particular smooth handlebody $X$ with
a fixed period matrix $\Omega$ on its asymptotic boundary $\Sigma_{g}$, while
the summation over $X$ means summing over different period matrices $\Omega$
in the same conformal class.

Different Euclidean saddles can be constructed by specifying which cycles of
the Riemann surface $\Sigma_{g}$ are contractible in the interior of the
3-manifold $X$, and such cycles are mapped into each other under the action of
the mapping class group (MCG) $\Gamma_{g}$ of the Riemann surface
$\Sigma_{g}$. To compute the gravitational path integral in (4), our strategy
is analogous to the torus case: We again start with the contribution from a
certain gravitational “vacuum seed” partition function
$Z_{\text{vac}}(\Omega,\bar{\Omega})$ corresponding to the trivial saddle and
perform a modular sum to write the complete partition function in (4) in the
following more explicit form

$Z_{\text{grav}}(\Omega,\bar{\Omega})=\sum_{\gamma\in\Gamma_{c}\backslash\Gamma_{g}}Z_{\text{vac}}(\gamma\Omega,\bar{\gamma}\bar{\Omega}).$
(5)

Here, $\Gamma_{c}$ denotes the subgroup of the mapping class group
$\Gamma_{g}$ of the Riemann surface $\Sigma_{g}$ (the latter being an infinite
group) which leaves the “vacuum seed partition function”
$Z_{\text{vac}}(\Omega,\bar{\Omega})$ invariant, and
$\Gamma_{c}\backslash\Gamma_{g}$ is the right coset space; it is over this
coset space that the sum in (5) is performed. Whether this sum has an infinite
or a finite number of terms depends in general (a): on the value of the Brown-
Henneaux central charge, and (b): on the genus $g$. When the sum is infinite,
there is no natural procedure to associate a value to it.444A natural
regularization scheme would require a probability measure on the (infinite)
mapping class group that is also invariant under “translations” (i.e., under
group multiplications). A group with such a translation-invariant measure that
is further finitely additive (the measure of a finite disjoint union of sets
is the sum of the measures of these sets) is called amenable. All mapping
class groups are non-amenable as they contain non-abelian free groups as
subgroups. Subgroups of amenable groups are amenable and non-abelian free
groups are known to be non-amenable. It follows that there are no natural
regularization schemes to sum over mapping class groups in this sense.
However, this theorem does not apply to summations over cosets such as the
regularized sum considered for genus one in MaloneyWitten . It is not clear
how to generalize their treatment to higher genus at the current stage. In
this paper, we show that for all theories of pure Einstein gravity in
AdS${}_{3}\,\,$with Brown-Henneaux central charge $c<1$, this sum is finite
and unique only when $c=1/2$, corresponding to the dual CFT at the asymptotic
boundary to be the Ising CFT. Therefore we argue that in the strong-coupling
regime of Brown-Henneaux central charge $c=3l/2G_{N}<1$, pure Einstein gravity
is only dual to a 2D CFT if $c=1/2$.

We arrive at this conclusion by extending the results obtained for genus one
by Castro et al. Castro . Recall that, as mentioned above, Castro et al.
argued solely based on genus-one considerations that the only 2D CFTs with
central charge $c<1$ that can be dual to pure Einstein gravity in
AdS${}_{3}\,\,$at the corresponding Brown-Henneaux central charges are the
Ising and the Tricritical Ising CFTs of central charges $c=1/2$ and $c=7/10$,
respectively. The results we obtain in the present paper, based on
consideration of arbitrary genus $g$, are two-fold:

(i) For Brown-Henneaux central charge $c=1/2$. After first identifying the
gravitational genus-$g$ “vacuum seed” partition function, we observe that the
orbit of the vacuum seed under the MCG action is dictated by a projective
representation $\rho_{g}$ of the MCG $\Gamma_{g}$ that is identical to the
projective representation induced by the holomorphic conformal blocks of the
2D Ising CFT. We then show, using the properties of $\rho_{g}$, that the
action of the MCG $\Gamma_{g}$ on the vacuum seed generates an orbit that is
always a finite set for any genus $g$ and, hence, leads only to a finite sum
in (5). We further prove that this projective representation $\rho_{g}$ is
irreducible, which, by Schur’s Lemma, leads to the conclusion that the finite
sum in (5) for the gravitational partition function is unique, and is
precisely proportional to the partition function of the 2D Ising CFT.555The
physical significance of the factor of proportionality is not entirely clear
at this point. The key mathematical results that we prove in this paper and
that underlie our physics conclusions on the quantum gravity partition
function at $c=1/2$ are: (1) The representation $\rho_{g}$, when viewed as a
mapping from the MCG $\Gamma_{g}$ to a unitary group, has a finite image set
for any genus $g$, and (2) the projective representation $\rho_{g}$ of the MCG
$\Gamma_{g}$ is always irreducible for any genus $g$. These results are
obtained by exploiting the connection between the 2D Ising CFT and the 3D
Ising topological quantum field theory (TQFT). We first provide a simplified
discussion on these results in Section 3 for the genus-two case, and continue
with the discussion of the general genus $g$ case in Section 4.

(ii) For Brown-Henneaux central charge $c=7/10$. While the genus-one
considerations by Castro et al. Castro  would permit the conclusion that pure
Einstein gravity in AdS${}_{3}\,\,$at $c=7/10$ is dual to the 2D Tricritical
Ising CFT at the asymptotic boundary, their arguments do not carry over to
higher genus $g\geq 2$. (As discussed above, consideration of arbitrary genus
is necessary for a complete description of a duality.) We arrive at this
conclusion by considering the 3D TQFT related to the 2D Tricritical Ising CFT
at $c=7/10$. We show that, at Brown-Henneaux central charge $c=7/10$, the sum
occurring in the $g\geq 2$ gravitational partition function (5) has an
infinite number of terms and cannot be naturally regularized, as explained in
4. A detailed discussion will be provided in Section 4.5.

The remainder of the paper is organized as follows. In Section 2, we review
the torus case. Section 3 presents a discussion of the genus-two case, while
Section 4 presents a complete discussion and proof for general genus $g$,
which is independent of the previous section and is more mathematically
involved. The difficulty in extending to the Tricritical Ising case is
discussed in more detail at the end of Section 4. Several Appendices spell out
various details. In the last appendix G, the duality is used to compute the
gravitational entropy and we find a resemblance to the topological correction
to the entanglement entropy occurring in the context of topological phases of
matter.

##  2 Gravitational partition function with torus asymptotic boundary

The simplest Euclidean smooth 3-manifold $X$ that contributes to the sum in
(2) is that of thermal AdS3, topologically a solid torus. It is described in
the semi-classical limit ($c\gg 1$) by the following metric

$ds^{2}=l^{2}\left(d\rho^{2}+\cosh^{2}\rho~{}dt_{E}^{2}+\sinh^{2}\rho~{}d\phi^{2}\right),$
(6)

where $\phi\sim\phi+2\pi$ denotes a spatial cycle which is contractible in the
bulk of $X$, and the Euclidean time $t_{E}$ parametrizes a non-contractible
cycle. Defining $z=-t_{E}+i\phi$, the complex coordinate $z$ parametrizes
points on the asymptotic boundary ($\rho\to\infty$) of $X$, and it is
periodically identified according to

$z\sim z+2\pi in\sim z+2\pi im\tau,~{}m,n\in\mathbb{Z},$ (7)

where the first identification is automatic (due to the periodicity of
$\phi$), while the second is to construct the thermal AdS3 space-time, and
$\tau$ is the complex parameter specifying the conformal structure of the
boundary torus. Large diffeomorphisms, i.e., elements of the MCG, act on this
conformal structure parameter as

$\tau\rightarrow\gamma\cdot\tau=\frac{a\tau+b}{c\tau+d},\quad\gamma=\left(\begin{matrix}a&b\\\
c&d\end{matrix}\right)\in SL(2,\mathbb{Z}).$ (8)

Note that the large diffeomorphisms do not change the conformal structure on
the boundary torus. Therefore, all conformal structure parameters
$\gamma\cdot\tau$ with $\gamma\in SL(2,\mathbb{Z}))$ (in other words, all
$\tau$’s related to each other by the MCG) specify the same conformal
structure. Each of $\gamma\tau$ gives a classical Euclidean solution to the
Einstein’s equation MaldacenaStrominger , i.e., a valid saddle point of (1).
These may or may not be different saddle points depending on the choice of
$\gamma$. For example, the combination $a=0,$ $b=1,$ $c=-1,$ $d=0$ realizes a
modular $S$ transformation, which maps $\tau\mapsto-1/\tau$ and the resultant
saddle is the Euclidean BTZ black hole BTZ . It is related to thermal
AdS${}_{3}\,\,$by exchanging the spatial and temporal cycles, consistent with
the defining feature of a Euclidean BTZ black hole - the existence of a a
temporal cycle contractible in the bulk. It was shown in MaloneyWitten  that
the only smooth solutions to the equation of motion with torus boundary
conditions are the ones above, but not all these solutions labeled by $\gamma$
are inequivalent. Specifically, an overall sign flip of $a,b,c,d$ does not
change the saddle, neither does a constant integer shift
$(a,b)\rightarrow(a,b)+n(c,d)$ generated by the modular $T$ transformation.
Physically, the latter observation corresponds to the fact that adding a
contractible cycle to a non-contractible cycle leaves the non-contractible
cycle still non-contractible. We denote the subgroup of $SL(2,\mathbb{Z})$
generated by $T$ by $\Gamma_{\infty}$. So in the semi-classical regime,
different saddles are labeled by different right cosets of $\Gamma_{\infty}$
in $SL(2,\mathbb{Z})$, or equivalently by integers $(c,d)$ corresponding to
solid tori $M_{c,d}$. Notice that all solid tori $M_{c,d}$ share the same
hyperbolic metric (6), because by a famous theorem of Sullivan Sullivan ;
McMullen ; Finiteness , for a fixed conformal class of the asymptotic
boundary, the bulk is a unique smooth and infinite-volume hyperbolic
3-manifold, with a rigid complete metric.

These saddle-point Euclidean spacetimes $M_{c,d}$ can be obtained from the
corresponding Lorentzian ones via analytical continuation, which amounts to
taking the Schottky double of its Lorentzian $t=0$ constant time slice
Krasnov1 ; Krasnov2 . The Schottky double of a surface is essentially two
copies of the surface glued along their boundaries, i.e., a closed surface.
(For a surface without a boundary, the Schottky double is two disconnected
copies of the surface, with all moduli replaced by their complex conjugates in
the second copy.) In Figure 1, we depict the examples of Euclidean thermal
AdS${}_{3}\,\,$with $(c,d)=(0,1)$ and the Euclidean BTZ black hole
$(c,d)=(1,0)$, as well as their constant time slices. Both are non-
rotating666For a definition see Appendix E. and possess an equal time $t=0$
surface with a $\mathbb{Z}_{2}$ time-reversal symmetry.

Figure 1:  From left to right: (a) Geometry for Euclidean thermal AdS3, with
time going along the longitudinal direction; (b) The constant time slice of
(a) is a disk; (c) Geometry for Euclidean BTZ black hole, with the event
horizon being the dashed line in the core of the solid torus; (d) Constant
time slice of (c) is an annulus,where the inner boundary is the event horizon.

It turns out that in the strongly coupled regime, $\Gamma_{\infty}$ is
enhanced to a larger group $\Gamma_{c}$ (a “new gauge symmetry”) Castro ,
which is a finite index subgroup of $SL(2,\mathbb{Z})$. Hence the inequivalent
manifolds $X$ are then labeled by right cosets $\gamma\in\Gamma_{c}\backslash
SL(2,\mathbb{Z})\equiv\Gamma$ and one can write

$Z_{\text{grav}}(\tau,\bar{\tau})=\sum_{\gamma\in\Gamma}Z_{\text{vac}}(\gamma\tau,\gamma\bar{\tau}),$
(9)

where $Z_{\text{vac}}$ is the partition function of the “vacuum seed”, by
which we here mean here that of thermal AdS3 spacetime.

###  2.1 Vacuum seed

To compute $Z_{\text{vac}}(\tau,\bar{\tau})$, one needs to evaluate in the
path integral (1) the contribution from metrics that are continuously
connected to thermal AdS3. In this subsection (and only here), we temporarily
resort to Lorentzian signature for convenience. These metrics differ from that
of empty AdS3, the Lorentzian counterpart of the Euclidean thermal AdS3, by
small diffeomorphisms that preserve the Brown-Henneaux boundary conditions

$ds^{2}\sim
l^{2}\left[d\rho^{2}+\frac{1}{4}e^{2\rho}(-dt^{2}+d\phi^{2})+O(\rho^{0})\right]$
(10)

at large $\rho$.

The classical phase space of the theory is the same as the configuration space
of all classical excitations that are continuously connected to the global
AdS${}_{3}\,\,$ground state metric (6). Brown and Henneaux BrownHenneaux
observed777See also, e.g., the compact review in Strominger . that the phase
space charges $H(\zeta_{n})$ corresponding to such diffeomorphisms $\zeta_{n}$
satisfy the Virasoro algebra with central charge $c=3l/2G_{N}$. Acting on the
ground state with these charge operators, one obtains the boundary graviton
states, whose norms must be positive. Upon performing canonical quantization
as proposed in Castro ; BrownHenneaux , these charge operators are promoted to
operators representing the generators of the Virasoro algebra,

$L_{n}\equiv H[\zeta_{n}],\quad\bar{L}_{n}\equiv H[\bar{\zeta}_{n}].$ (11)

Then the vacuum state is annihilated by $L_{0}$ and $\bar{L}_{0}$, as well as
other Virasoro lowering operators. This state corresponds semi-classically to
empty Lorentzian AdS3. The conformal symmetry constrains the theory strongly,
and the boundary gravitons are described by the states obtained by acting with
chains of Virasoro raising operators on the vacuum, i.e., these are the
descendant states $L_{-n_{1}}\cdots L_{-n_{k}}|0\rangle$, with $n_{i}>1$. Our
desired partition function $Z_{\text{vac}}$ is then the generating function
that counts these states.

In the strongly coupled regime of Brown-Henneaux central charge $c<1$, the
requirement of unitarity constrains the central charge to the values
$c=1-6/m(m+1)$ corresponding to the ‘Virasoro minimal model’ CFTs, where $m$
is an integer larger than two. Furthermore, eliminating the null states gives
the vacuum (identity) character of an irreducible highest-weight
representation of the Virasoro algebra (see for example BigYellowBook ),

$Z_{\text{vac},\,g=1}=\text{Tr}_{\text{vac}}\
q^{L_{0}}\bar{q}^{\bar{L}_{0}}=\absolutevalue{\chi_{1,1}(\tau)}^{2},\quad{\rm
where}\ q=e^{2\pi i\tau}.$ (12)

Here the subscript $r,s$ of a general character $\chi_{r,s}$ denotes indices
of the Kac table that label all possible irreducible representations of the
Virasoro algebra at this central charge (where $r=1,2,...,m-1$ and
$s=1,2,...,m$), and888The character is known RochaCaridi ; Cappelli  to take
the form $\eta(\tau)q^{-(1-c)/24}\chi_{r,s}=$
$\sum_{k\in\mathbb{Z}}\left(q^{h_{r+2km,s}}-q^{h_{r+2km,-s}}\right)\equiv S$.
Using the identity $h_{r+lm,s+l(m+1)}=h_{r,s}$ for all $l\in\mathbb{Z}$, which
follows from (14) by inspection, one can bring the expression above into the
form $S=q^{h_{r,s}}$
$+\sum_{k=1}^{\infty}\left[q^{h_{r+2km,s}}+q^{h_{r,s+2k(m=1)}}\right]$
$-\left[\left(\sum_{k=0}^{\infty}q^{h_{r+(2k+1)m,-s+(m+1)}}\right)+\left(\sum_{k=1}^{\infty}q^{h_{r,-s+2k(m+1)}}\right)\right]$.
After expressing this as a single sum over $l$ from $l=1$ to $\infty$, where
$l=2k$ for even $l$, and $l=(2k+1)$ for odd $l$, the sum $S$ is easily seen to
yield the result presented in the following equation.

$\displaystyle\chi_{r,s}=$ (13)
$\displaystyle=\frac{q^{(1-c)/24}}{\eta(\tau)}\left[q^{h_{r,s}}+\sum\limits_{l=1}^{\infty}(-1)^{l}\left(q^{h_{r+lm,s(-1)^{l}+(m+1)[1-(-1)^{l}]/2}}+q^{h_{r,s(-1)^{l}+l(m+1)+(m+1)[1-(-1)^{l}]/2}}\right)\right],\qquad\quad$

with $\eta(\tau)=q^{1/24}\prod_{n=1}^{\infty}(1-q^{n})$ the Dedekind eta
function and the highest weight $h_{r,s}$ given by

$h_{r,s}=\frac{[(m+1)r-ms]^{2}-1}{4m(m+1)}.$ (14)

###  2.2 Modular sum and duality to the Ising CFT

The simplest minimal model is the Ising CFT with $c=1/2$. There are three
irreducible representations of the Virasoro algebra satisfying $h_{1,1}=0,$
$h_{2,1}=1/2$ and $h_{1,2}=1/16$. The partition function of the theory is
simply the diagonal modular invariant

$Z_{\text{Ising}}(\tau,\bar{\tau})=|\chi_{1,1}(\tau)|^{2}+|\chi_{1,2}(\tau)|^{2}+|\chi_{2,1}(\tau)|^{2},$
(15)

where the three summands are conformal characters of the identity, energy and
spin operators, respectively BigYellowBook . These characters can also (for
Ising) be expressed in terms of the Riemann or Jacobi theta function, as
reviewed in Appendix C, equation (LABEL:eq:TorusChiTheta).

On the other hand, the gravitational partition function is obtained by summing
all images of the vacuum character under $\Gamma\equiv\Gamma_{c}\backslash
SL(2,\mathbb{Z})$, the right coset space of $\Gamma_{c}$ in
$SL(2,\mathbb{Z})$, as

$Z_{\text{grav}}(\tau,\bar{\tau})=\sum_{\gamma\in\Gamma}|\chi_{1,1}(\gamma\tau)|^{2},$
(16)

where $\Gamma_{c}$ is the set of all “pure gauge transformations” of the
vacuum, which are defined to be those elements of $SL(2,\mathbb{Z})$ that act
trivially on the modulus of the vacuum character, $|\chi_{1,1}|$:

$\Gamma_{c}=\\{\gamma\in SL(2,\mathbb{Z})\large{\mid}\
\absolutevalue{\chi_{1,1}(\gamma\tau)}=|\chi_{1,1}(\tau)|\\}.$ (17)

This is a finite index subgroup as proven in Bantay , so the summation in (16)
has a finite number of terms, unlike the $c>1$ Farey-tail cases that were
discussed in Refs FareyTail ; MaloneyWitten . We will see in later sections
that the finiteness property, seen here at genus one, extends (for Ising) to
the case of higher genus. Starting from the “vacuum seed” $|\chi_{1,1}|^{2}$,
(12), and repeatedly acting on it with the generators

$S=\left(\begin{matrix}0&-1\\\ 1&0\end{matrix}\right),\quad
T=\left(\begin{matrix}1&1\\\ 0&1\end{matrix}\right)$ (18)

of $SL(2,\mathbb{Z})$, one finds 24 inequivalent contributions, which sum up
to

$Z_{\text{grav}}=8Z_{\text{Ising}}.$ (19)

The physical meaning of this constant factor of 8 is at present unclear, while
its mathematical meaning, along with extra new results on $\Gamma_{c}$ that go
beyond those presented in Castro , are collected in Appendix A. Therefore we
see the equality of the partition functions of pure Einstein gravity in
AdS${}_{3}\,\,$at Brown-Henneaux central charge $c=3l/2G_{N}=1/2$ and that of
the Ising CFT, at genus one.

##  3 Gravitational partition functions with genus-2 asymptotic boundaries

Now we generalize the discussion of the duality between Euclidean
AdS${}_{3}\,\,$and Virasoro minimal model CFTs to genus two. The current
section is more “physical” or intuitive, compared to Section 4 which discusses
the case for arbitrary genus and will be more mathematically involved. We will
focus on the $c=3l/2G_{N}=1/2$ theory and present its gravitational partition
function as well as its relation to the Ising CFT in Section 3.1, followed by
a review of the relevant mathematical concepts in Section 3.2.

###  3.1 Gravitational partition function

Similar to the genus-one case, the key assumption in the computation of the
gravitational partition function is that the path integral is equal to the
contribution from classical saddle points and the full set of quantum
fluctuations around them, irrespective of the fact that the Brown-Henneaux
central charge is now of order one.

As briefly reviewed in the last section, the analytical continuation from
Lorentzian to Euclidean signature basically amounts to taking a Schottky
double. When the Lorentzian geometry contains three asymptotic regions, its
constant time slice is a pair of pants. The boundary of the corresponding
Euclidean spacetime is thus obtained (following the notion of the Schottky
double, mentioned above) by gluing two pairs of pants together, thereby
obtaining a genus-two Riemann surface. Different ways of gluing give distinct
saddles and correspond to different choices of contractible cycles in the
bulk. In Figure 2, we sketch three bulk geometries that possess a
$\mathbb{Z}_{2}$ time-reflection symmetry Multibdry . The left one depicts the
case which corresponds to three disconnected thermal AdS${}_{3}\,\,$spacetimes
in Lorentzian signature. The green circles label the interfaces between the
two pairs of pants. The middle panel describes the Euclidean version of the
three-sided wormhole. The right figure is the case with one copy of thermal
AdS${}_{3}\,\,$and a BTZ black hole. Different bulk saddles can be transformed
into each other by the action of the mapping class group (whose definition
will be reviewed in Section 3.2).

Figure 2: Different Euclidean saddles with $\mathbb{Z}_{2}$ time-reflection
symmetry. They analytically continue to three copies of thermal
AdS${}_{3}\,\,$(left), three-sided wormhole (middle), and one thermal
AdS${}_{3}\,\,$plus a BTZ black hole (right). The areas encircled by the green
lines are the sets of fixed points of the action of the $\mathbb{Z}_{2}$
symmetry.

The full partition function can thus be written as the modular sum of one of
the saddles, namely as that of the vacuum saddle without black holes,

$Z_{\text{grav}}(\Omega,\bar{\Omega})=\sum_{\gamma\in\Gamma}Z_{\text{vac}}(\gamma\Omega,\bar{\gamma}\bar{\Omega}),$
(20)

where $\Gamma=\Gamma_{c}\backslash\Gamma_{g}$ is the right coset space of the
mapping class group (MCG) $\Gamma_{g}$ of the Riemann surface $\Sigma_{g}$
with respect to $\Gamma_{c}$, the symmetry group that leaves $Z_{\text{vac}}$
invariant. - Here $g=2$. The $2\times 2$-dimensional complex, symmetric period
matrix $\Omega$ is a higher-genus generalization of the modular parameter
$\tau$ in genus one, whose definition is presented in Section 3.2 below. The
conformal structure on the asymptotic boundary is specified by the period
matrix $\Omega$. All period matrices $\Omega$ related to each other by the MCG
correspond to the same conformal structure.

####  3.1.1 Vacuum seed

We are interested in the case where the bulk gravity is a genus-two
handlebody, which can be viewed as three solid cylinders that meet at a cup
and a cap (each being a “3-ball” - the interior of a 2-dimensional sphere),
compare e.g., Figure 4. We will choose the notation for the elementary cycles
depicted in Figure 3 below.

Figure 3: The canonical homology basis for $\Sigma_{g}$.

The vacuum sector $Z_{\text{vac}}$ dominates the full partition function in
the low-temperature limit, which we define to be the limit where the three
solid cylinders are long and thin, like in Figure 4. (This is analogous to the
genus-one case, where in the low-temperature limit, the dominant geometry is
the one whose boundary torus has a longitude much larger than its meridian.)
In this limit, a natural local coordinate system can be chosen, such that a
constant time slice is a disjoint union of three disks, i.e., the cross
sections of the three solid cylinders (see Figure 4), while the time direction
is along the longitudinal direction of the cylinders.999From a TQFT point of
view, this corresponds to the case where only the trivial anyons propagate in
the long cylinders. The relationship with TQFT is discussed briefly in
Appendix C and will be generally described in Section 4. Such a topology
analytically continues to three copies of thermal AdS3. Namely, all the
$\alpha$-cycles in Figure 3 need to be contractible in the bulk.

Figure 4: The low-temperature or long-cylinder limit of the genus-two
geometry.

For a bulk geometry with a higher-genus asymptotic boundary, we believe that
the association with the Brown-Henneaux central charge $c=3l/2G_{N}$ is still
valid.101010In their original paper BrownHenneaux , given the global AdS3
metric with $\rho$ being the radial direction,
$ds^{2}=-\left(\frac{\rho^{2}}{l^{2}}\right)dt^{2}+\left(\frac{l^{2}}{\rho^{2}}\right)d\rho^{2}+\rho^{2}d\phi^{2},$
after quotienting it by some discrete subgroup of the isometry group of global
AdS3, off-diagonal entries of the new metric need to satisfy the asymptotic
conditions  $g_{t\rho}\sim\mathcal{O}(1/\rho^{3}),\quad
g_{t\phi}\sim\mathcal{O}(1),\quad g_{\rho\phi}\sim\mathcal{O}(1/\rho^{3}),$ in
order to produce two copies of Virasoro algebras with central charge $c$ on
the boundary. In principle, these conditions can be checked here using the
Fefferman-Graham metric for asymptotic AdSd+1, constructed basically by
shooting geodesics inwards from the boundary Fefferman1 ; Fefferman2 :
$ds^{2}=g_{\rho\rho}d\rho^{2}+g_{ij}dx^{i}dx^{j}$
$=\displaystyle{\frac{l^{2}d\rho^{2}}{4\rho^{2}}+\frac{1}{\rho}\tilde{g}_{ij}(\rho,x)dx^{i}dx^{j}}$,
where the $d$-dimensional metric $\tilde{g}_{ij}(\rho,x)$ is the
$\rho$-dependent Euclidean boundary metric. Recall in the genus-one case, the
boundary torus describes the time evolution of graviton states living on the
boundary of a disk. When the Brown-Henneaux central charge is $c=1/2$, these
states correspond to the quantum states of the 2D Ising CFT in the vacuum
sector $\chi_{1,1}$ (and $\bar{\chi}_{1,1}$). For genus two and in the local
coordinate system where a constant time slice consists of three disjoint disks
(see Figure 4), the boundary graviton states live on the boundary of each
disk. Hence locally, the boundary graviton states correspond to three copies
of $\chi_{1,1}$ states (and $\bar{\chi}_{1,1}$ states). Globally, the former
should correspond to states in the vacuum conformal block of the Ising CFT at
genus two (the analogue of the $\chi_{1,1}$ sector for genus one), which we
denote by $\chi_{\text{vac}}$. Therefore, we assume $Z_{\text{vac}}$ to be of
the same form as the partition function of the Ising vacuum conformal block.
This assumption is a natural extension of results in Yin1 ; Bootstrap . In the
large-$c$ and the pinching limit of the genus-2 asymptotic boundary, Ref. Yin1
calculated the vacuum seed of AdS${}_{3}\,\,$to order$1/c^{2}$. This was then
shown to match exactly with the partition function of the vacuum conformal
block of a 2D large-$c$ CFT Bootstrap . Naturally, we expect this match to
hold to all orders of $1/c$, thereby justifying the assumption.

The full partition function of the 2D Ising CFT theory on a Riemann surface of
arbitrary genus was worked out in Ref. DVV  using a $\mathbb{Z}_{2}$ orbifold
of the free compactified boson theory, and in Ref. AGMV  using a single, non-
interacting Majorana fermion. In the formulation using the Majorana fermion, a
choice of boundary conditions (spin structure) has to be imposed. The
contribution from each choice of boundary conditions or spin structure can be
written as the norm of the regularized determinant of the corresponding chiral
Dirac operator. The determinant can further be separated into two factors, one
being the Riemann theta function of the corresponding spin structure (whose
definition will be reviewed in Section 3.2), while the other is independent of
spin structures and only a function of the metric. In what follows, the former
will be denoted as the classical contribution to the partition function, and
the latter will be called the quantum contribution. (Note this has a different
meaning from the “quantum” used to describe gravitational theories which are
beyond semiclassical regime. The word “quantum” here stems from the fact that
this universal factor accounts for the quantum fluctuations of the boson
fields in the $\mathbb{Z}_{2}$ orbifold.) For more details about the quantum
contribution, we refer to Appendix C. In fact, not only the full partition of
the 2D Ising CFT, but each of the conformal blocks also factorizes into a
classical and a quantum piece. Given the identification of the gravitational
vacuum seed and the vacuum conformal block of the 2D Ising CFT, discussed
above, we can write $Z_{{\rm vac}}=Z_{{\rm vac}}^{\rm cl}Z_{{\rm vac}}^{\rm
qu}$ (where “cl” stands for classical and “qu” stands for quantum). In the
following discussion, we will be interested in how the different sectors or
conformal blocks in the theory transform into each other under the mapping
class group. For this purpose, it is enough to temporarily ignore the overall
quantum factor that is the same for all conformal blocks and focus on the
classical contribution of the gravitational vacuum seed

$Z_{\text{vac}}^{\text{cl}}(\Omega,\bar{\Omega})=|\chi_{\text{vac}}^{\text{cl}}(\Omega)|^{2}=\frac{1}{16}\left|\sum_{b_{1},b_{2}\in\\{0,1/2\\}}\vartheta^{1/2}\left[\begin{matrix}a_{1}=0&a_{2}=0\\\
b_{1}&b_{2}\end{matrix}\right](0|\Omega)\right|^{2},$ (21)

where $|\chi_{\text{vac}}^{\text{cl}}(\Omega)|^{2}$ is the classical
contribution to the vacuum conformal block of the 2D Ising CFT, and where
$\vartheta$ denotes the conventional Riemann theta function (see (30) of
Section 3.2 for a review of relevant notations). Here, $a_{1,2}$ should be
viewed as the two components of the characteristic vector ${\bf
a}=(a_{1},a_{2})$ appearing in the theta function for the genus-2 case.
Similarly, $b_{1,2}$ are the two components of the characteristic vector ${\bf
b}=(b_{1},b_{2})$. The number of components of these characteristic vectors is
given by the genus $g$ in general. We explain in the following the specific
choice of theta functions $\vartheta$ appearing in the above expression.

We know that along a contractible cycle, the boundary condition for a fermion
has to be anti-periodic.111111This is the natural boundary condition for
fermions since they anti-commute. See also for example AGMV ; BigYellowBook ;
Multibdry . Periodic boundary conditions for fermions would imply a
singularity inside the cycle, often called a $\mathbb{Z}_{2}$-vortex, or
Majorana fermion zero mode. Since, as discussed above, in the gravitational
vacuum seed all the $\alpha$-cycles in Figure 3 need to be contractible, all
the corresponding boundary conditions on the (Majorana) fermion along those
cycles need to be anti-periodic. Consequently, the top characteristic vector
of the theta functions that are relevant for the vacuum sector is zero, i.e.,
$a_{1}=a_{2}=0.$ Furthermore, the vacuum sector must be a equal-weight
summation over both even and odd fermion number parities along every
$\beta$-cycle. This means $Z_{\text{vac}}^{\text{cl}}$ has to be the modulus
square of a equal-weight linear combination of the square root of Riemann
theta functions that appear in equation (21), as displayed in that same
equation.

The above form (21) for $Z_{\text{vac}}^{\text{cl}}$ is also analogous to the
classical contribution to the vacuum seed on the torus. The latter, as
reviewed in (LABEL:eq:TorusChiTheta) of Appendix C, is the equal-weight sum of
the square roots of all theta functions whose characteristic ‘vector’ ${\bf
a}=(a_{1})$ is zero. On the torus, this has a natural Hamiltonian
interpretation that exists due to a global notion of time (leading to a clean
separation of 1D space and 1D time), which is absent at higher
genus.121212Namely, at genus one there are four (one of them vanishing)
holomorphic partition functions, $\chi_{\sigma_{x},\sigma_{t_{E}}}\equiv
Tr_{{\cal H}_{\sigma_{x}}}(\sigma_{t_{E}})^{F}q^{L_{0}}$, where $F$ denotes
the fermion parity operator, $\sigma_{x},\sigma_{t_{E}}=\pm 1$. Here
$\mp\sigma_{x}$ denotes spatial (anti-)periodicity whereas $\mp\sigma_{t_{E}}$
denotes Euclidean temporal (anti-)periodicity. These holomorphic partition
functions are proportional to $\vartheta^{1/2}\left[\begin{matrix}a\\\
b\end{matrix}\right]$ with $a=(1-\sigma_{x})/2$ and $b=(1-\sigma_{t_{E}})/2$
One then sees from (LABEL:eq:TorusChiTheta) of Appendix C that the torus
vacuum character $\chi_{1,1}$ is proportional to the sum of the square-roots
of theta functions with $a=0$, summed over $b=0$ and $b=1/2$. The sum
appearing in (21) is the natural generalization of this genus-one expression
to genus two.

In the pinching limit where the bulk ‘cylinder’ connecting the two tori
pinches off DVV , i.e., where $\Omega_{12},\Omega_{21}\to 0$, the (classical)
vacuum seed partition function (21) reduces to
$|\chi_{1,1}^{\text{cl}}(\tau_{1})\ \chi_{1,1}^{\text{cl}}(\tau_{2})|^{2}$,
which is the product of the classical parts of the two torus vacuum seeds
$|\chi_{1,1}(\tau_{1})|^{2}$ and $|\chi_{1,1}(\tau_{2})|^{2}$ of two tori with
modular parameters $\tau_{1}$ and $\tau_{2}$.

One can check that (21) is invariant under a genus-two generalization of
$\Gamma_{\infty}$, see Appendix B. This is a subgroup of the genus-two mapping
class group $\Gamma_{g}$,131313Basic facts about this genus-two generalization
of $\Gamma_{\infty}$ will be discussed in Appendix B where this group is
referred to as $\Gamma_{\infty}^{[2]}$. generated Yin1  by integer shifts of
matrix elements of the period matrix $\Omega$, as well as the
$SL(2,\mathbb{Z})$ transformation that acts on $\Omega$ by conjugation
$\Omega\mapsto A\Omega A^{T}$. The genus-$2$ generalization of the group
$\Gamma_{\infty}$ is the classical symmetry of the vacuum seed at large $c$,
and it is enhanced in the case of strong coupling ($c<1$) to the previously
mentioned group $\Gamma_{c}$, a subgroup of $\Gamma_{g=2}$ which is larger
than $\Gamma_{\infty}$. This new “gauge symmetry” will be relevant in the
modular sum as it turns out to be a finite-index subgroup.

As a consistency check of (21), in the low-temperature or long-cylinder limit
depicted in Figure 4, the leading contribution to $Z_{\text{vac}}^{\text{cl}}$
needs to be equal to that of the total classical contribution to the full
Ising partition function at genus two in the same limit, as explained in
Appendix D. The long-cylinder limit can be taken in the following way: The
genus-two Riemann surface can be described as a hyperelliptic curve, which is
the set of solutions to the following equation (see for example
CardyMaloneyMaxfield  for a recent discussion of this)

$y(z)^{2}=\prod_{k=1}^{3}\frac{z-u_{k}}{z-v_{k}}.$ (22)

Such a surface is a two-sheeted branched cover of the Riemann sphere, the
points on which are parametrized by $z$, and the two sheets are labeled by the
choice of the root $y$ which solves (22). There is a $\mathbb{Z}_{2}$ “replica
symmetry” generated by $y\rightarrow-y$, which physically corresponds to the
time reversal symmetry discussed above in Figure 2, and the corresponding
text. The covering map has $2\times 3=6$ branch points $(u_{k},v_{k})$.
Monodromy of $z$ around one of the six branch points shifts $y\rightarrow-y$
and moves from one sheet to the other. The locations of the branch points span
the moduli space141414This is a $g=2$ coincidence, for general genus $g$ the
moduli space $\mathcal{M}_{g}$ of a Riemann surface $\Sigma_{g}$ has real
dimension $6g-6$, while the number of real branches is $2g+2$. of the Riemann
surface. Consequently, the period matrix can be expressed in terms of the
branch points CoserTagliacozzoTonni , and the long-cylinder limit corresponds
to taking $u_{k}-v_{k}$ to be small for $k=1,2,3,$. To obtain the vacuum seed
partition function, the resulting period matrix $\Omega$ is inserted into
(21). For the case of the long-cylinder limit at general genus $g$, one simply
replaces the number $3$ appearing in (22) by $g+1$, and proceeds in an
analogous fashion.

A final remark is that, for our gravitational vacuum seed partition function
$Z_{\text{vac}}$ to be identical with that of the vacuum conformal block of
the boundary CFT (up to some constant factor), we further need to discuss the
cup and cap regions, where three cylinders join. We argue that the three-point
correlation functions that describe the graviton scattering processes in the
gravity theory match those in the boundary conformal theory151515At genus one,
a related but somewhat different two-to-one scattering process in AdS3 between
the bulk duals of the light primaries $O$ and $\chi$, whose conformal weights
less than $c/12$, is described in Kraus . Their CFT three-point function is
found to have the same form as the gravitational scattering amplitude between
their gravitational duals in the BTZ background, with a proportionality factor
only dependent on the saddle geometry. Although this process is not
necessarily in pure gravity, this result is in support of our argument about
the form of $Z_{\text{vac}}$..

####  3.1.2 Genus two modular sum

With the above expression for the vacuum seed, we now perform the sum over the
images of the action with the MCG (“modular sum”) as in (20) at
$g=2$.161616Instead of $Z_{\text{vac}}^{\text{cl}}$, we use the full quantum
conformal block $Z_{\text{vac}}=|\chi_{\text{vac}}|^{2}$ in the modular sum.
The quantum contribution and issues related to it, are discussed in Appendices
B and C. . We will first provide the numerical results, and then give a
mathematical argument for the finiteness of the modular sum. Independently, we
will present later in Section 4.2 another simple proof from a TQFT perspective
for arbitrary genus.

As reviewed in Section 3.2, the subgroup of the mapping class group which acts
non-trivially on the period matrix is $Sp(4,\mathbb{Z})$. The generators of
$Sp(2g,\mathbb{Z})$ are reviewed in Appendix B. By acting repeatedly with the
two generators of $Sp(4,\mathbb{Z})$ on the vacuum seed partition function, we
find 3840 inequivalent contributions with the aid of Mathematica.171717This
set is invariant under the action of Torelli group introduced in Section 3.2
below. The Torelli group acts by multiplying
$\left\\{\vartheta[1/2,1/2;1/2,1/2](\Omega)\cdot\vartheta^{*}[1/2,1/2;1/2,1/2](\Omega)\right\\}^{1/2}$
by a minus sign, which can be explicitly verified in the pinching limit using
the formalism in DVV  and straightforwardly carries over to the general case
away from that limit. Only this specific theta function product is affected by
the Torelli group action, because it is related to the sector (conformal
block) $\chi_{\sigma\psi\sigma}$ (in the language of Appendix C), where there
is a fermion $\psi$ in the middle of the genus-two handlebody (denoted by $b$
in Figure 12, i.e., $b\to\psi$), which acquires a negative sign upon the Dehn
twist along the separating curve. These modular images sum up to 384 times the
partition function $Z_{\rm Ising}$ of the 2D Ising CFT at genus two of
Appendix C):

$Z_{\text{grav}}=384Z_{\text{Ising}}.$ (23)

The factor $10=3840/384$ is simply the dimension of the conformal block basis,
or simply the number of linearly independent Riemann theta functions. The
physical meaning of the constant factor $384$ in (23) is unclear at this
point.

We emphasize that all the above arguments are gravitational ones that solely
come from the three-dimensional bulk. In the remainder of this section, we
support the above computation by a mathematical explanation for the finiteness
of the summation in the partition function (as in (20)).

In the Ising case, there exists181818This is a generalization of the
mathematical result in Wright  which is explained in Appendix F. a short exact
sequence for any genus $g$,

$1\rightarrow\rho_{g}(D_{g})\rightarrow\rho_{g}(\Gamma_{g})\rightarrow
Sp(2g,\mathbb{Z}_{2})\rightarrow 1,$ (24)

where $\rho_{g}(\Gamma_{g})$ is the image group of the mapping class group
$\Gamma_{g}$ represented as matrices in the basis of Riemann theta functions,
$D_{g}$ is the subgroup of the mapping class group that acts trivially on
$H_{1}(\Sigma_{g},\mathbb{Z}_{2})$, and $\rho_{g}(D_{g})$ is the corresponding
image group. The latter turns out to always be a subgroup of
$\mathbb{Z}_{8}^{N}$, where $N$ is a finite positive integer.

Since $\rho_{g}(D_{g})$ is abelian (see Appendix F), (24) gives a central
extension of $Sp(2g,\mathbb{Z}_{2})$. Such central extensions are classified
by the second cohomology group
$H^{2}(Sp(2g,\mathbb{Z}_{2}),~{}\rho_{g}(D_{g}))$: For every group element
$h\in Sp(2g,\mathbb{Z}_{2})$ and $n\in\rho_{g}(D_{g})$, there is an element
$(n,h)$ in $\rho_{g}(\Gamma_{g})$, satisfying the group multiplication
$(n_{1},h_{1})\cdot(n_{2},h_{2})=\left(\omega(h_{1},h_{2})n_{1}n_{2},h_{1}h_{2}\right),$
where $\omega(h_{1},h_{2})$ is a 2-cocycle with $\rho_{g}(D_{g})$
coefficients. Alternatively, one can interpret the above short exact sequence
in terms of projective representations. Irreducible representations of the
mapping class group $\Gamma_{g}$ correspond to the irreducible projective
representations of $Sp(2g,\mathbb{Z}_{2})$, where the projective phases are
given by $\rho_{g}(D_{g})$.

Since $Z_{\text{vac}}$ involves taking the modulus square of the vacuum
character, the overall phases of $\rho_{g}(D_{g})$ will not matter. We can
simply focus on the summation over elements of $Sp(2g,\mathbb{Z}_{2})$ that
act non-trivially on the absolute values of the theta functions. At genus
$g=2$, $Sp(4,\mathbb{Z}_{2})$ turns out to be equal to the permutation group
$S_{6}$ and contains $6!=720$ elements. Due to the short exact sequence (24),
the image group of $\Gamma_{g}$ is clearly finite.

In Section 4, we will present an alternative simple proof for the finiteness
of $\rho_{g}(\Gamma_{g})$ that works for arbitrary genus, from a topological
field theory perspective.

###  3.2 Review of the relevant concepts

We first describe the homology of orientable, finite-type two-dimensional
surfaces $\Sigma_{g}$ of genus $g$. When $\Sigma_{g}$ is compact, its homology
groups are free, with dim$H_{0}(\Sigma_{g})=1$, dim$H_{1}(\Sigma_{g})=2g$,
dim$H_{2}(\Sigma_{g})=1$. One can choose a canonical homology basis
$\alpha_{i},\beta_{i}$ with $1\leq i\leq g$ for $H_{1}(\Sigma_{g})$ as in
Figure 3. Any closed curve on $\Sigma_{g}$ generates a homology class, which
can be uniquely decomposed into the classes generated by
$\alpha_{i},\beta_{i}$. They are normalized with respect to the algebraic
intersection number $J(C_{1},C_{2})$ between two simple closed curves $C_{1}$
and $C_{2}$, by

$J(\alpha_{i},\alpha_{j})=J(\beta_{i},\beta_{j})=0,\quad
J(\alpha_{i},\beta_{j})=-J(\beta_{i},\alpha_{j})=\delta_{ij}.$ (25)

There are $g$ pairs of holomorphic and anti-holomorphic one-forms on
$\Sigma_{g}$, denoted by $\\{\omega_{i},\bar{\omega}_{i}\\}$ $(i=1,\cdots,g)$,
which satisfy the normalization condition

$\oint_{\alpha_{i}}\omega_{j}=\delta_{ij}.$ (26)

The period matrix defined by

$\oint_{\beta_{i}}\omega_{j}=\Omega_{ij}$ (27)

is then a $g\times g$ complex symmetric matrix, with a positive-definite
imaginary part.191919An alternative normalization for $\Omega$ more suitable
for computation is considered in Appendix D. Analogous equations as above hold
for the anti-holomorphic counterparts $\bar{\omega}_{i}$ and
$\bar{\Omega}_{ij}$. The period matrix $\Omega$ generalizes the modular
parameter $\tau$ for the torus, completely parametrizing the conformal
structure of $\Sigma_{g}$. Note that a conformal structure of $\Sigma_{g}$ can
be specified by different period matrices that are related to each other by
the mapping class group.202020The moduli space $\mathcal{M}_{g}$, the space of
conformal structures of $\Sigma_{g}$, has real dimension $6g-6$. The Torelli
map from $\mathcal{M}_{g}$ to the space of $\Omega$’s quotiented by the
mapping class group $\Gamma_{g}$ is injective, intuitively because the latter
has real dimension $g(g+1)$, so the parametrization is complete.

The mapping class group (MCG) $\Gamma_{g}$ of a genus-$g$ Riemann surface
$\Sigma_{g}$ is the group of all isotopy classes of orientation preserving
diffeomorphisms of $\Sigma_{g}$. It is generated by Dehn twists around the
cycles $C$ of $\Sigma_{g}$. A Dehn twist acts by excising a tubular
neighborhood of $C$ inside $\Sigma_{g}$, twisting the latter by $2\pi$, and
then gluing it back to the rest of the surface. There are two generators for
each handle, and one for each closed curve linking the holes of two
neighboring handles.

$\Gamma_{g}$ leaves the intersections (25) invariant, thus acting on the
canonical homology basis by $Sp(2g,\mathbb{Z})$ transformations. The
$Sp(2g,\mathbb{Z})$ transformations act on the period matrix by

$\gamma=\left(\begin{matrix}A&B\\\ C&D\\\ \end{matrix}\right)\in
Sp(2g,\mathbb{Z}),\quad\gamma:\Omega\rightarrow(A\Omega+B)(C\Omega+D)^{-1},$
(28)

where $A,B,C,D$ are $g$ by $g$ matrices. At genus $g=2$, the minimal number of
generators of $Sp(4,\mathbb{Z})$ is two Bender ; these are reviewed in
Appendix B. For $Sp(2g,\mathbb{Z})$ with $g\geq 3$ the minimal number of
generators is three Lu .

Some elements of $\Gamma_{g}$ act trivially on the canonical homology basis,
leaving it invariant. These elements are diffeomorphisms homotopic to the
identity and they form a normal subgroup of $\Gamma_{g}$, known as the Torelli
group $\mathcal{I}_{g}$ Hatcher ; Primer  . For genus two, $\mathcal{I}_{g}$
is infinitely generated by Dehn twists around the separating curve, i.e., the
curve that separates the genus two surface into two tori. For $g\geq 3$,
besides the ones that twist around the separating curves, there exists another
type of generator, called the “bounding pair map”. A bounding pair map is the
composition of a twist along a non-separating curve $C_{1}$ and an inverse
twist along another non-separating curve $C_{2}$ which is disjoint from
$C_{1}$ but represents the same homology class as $C_{1}$. So $C_{1}\cup
C_{2}$ separates $\Sigma_{g}$ into two subsurfaces having $C_{1}\cup C_{2}$ as
their common boundary. These two kinds of generators are shown in Figure 5.

Figure 5: Generators of the Torelli group $\mathcal{I}_{g}$. Left: Dehn twist
along a separating curve. Right: the bounding pair map.

In summary, we have the following non-splitting short exact sequence,

$1\rightarrow\mathcal{I}_{g}\rightarrow\Gamma_{g}\rightarrow
Sp(2g,\mathbb{Z})\rightarrow 1.$ (29)

Riemann or Siegel theta functions, which depend on two $g$-dimensional row
vectors $\mathbf{a},\mathbf{b}\in\mathbb{R}^{g}$ called characteristics, are
defined by the following infinite sum Mumford ; Fay ; Igusa ,

$\vartheta\left[\begin{matrix}\mathbf{a}\\\
\mathbf{b}\end{matrix}\right](\mathbf{z}|\Omega)\equiv\sum_{\mathbf{n}\in\mathbb{Z}^{g}}\exp\left(i\pi(\mathbf{n}+\mathbf{a})\cdot\Omega\cdot(\mathbf{n}+\mathbf{a})+2\pi
i(\mathbf{n}+\mathbf{a})\cdot(\mathbf{z}+\mathbf{b})\right),$ (30)

where $\mathbf{z}\in\mathbb{R}^{g}$ is a $g$-dimensional vector.

In this paper, we will be interested in and limit our discussion to the Ising
case described by a single Majorana fermion species, where the characteristic
vectors are $\mathbf{a},\mathbf{b}\in(\frac{1}{2}\mathbb{Z})^{g}$. In this
case there is, associated with each theta function, the notion of a spin-
structure of characteristics $\left[\begin{matrix}\mathbf{a}\\\
\mathbf{b}\end{matrix}\right]$, denoting a $2\times g$-matrix. The spin
structure is called to be even or odd depending on whether
$4\mathbf{a}\cdot\mathbf{b}$ is even or odd, respectively. This can be seen
from the following identity

$\vartheta\left[\begin{matrix}\mathbf{a}\\\
\mathbf{b}\end{matrix}\right](-\mathbf{z}|\Omega)=(-1)^{4\mathbf{a}\cdot\mathbf{b}}\vartheta\left[\begin{matrix}\mathbf{a}\\\
\mathbf{b}\end{matrix}\right](\mathbf{z}|\Omega).$ (31)

Additionally, due to the identity

$\vartheta\left[\begin{matrix}\mathbf{a}+\mathbf{n}\\\
\mathbf{b}+\mathbf{m}\end{matrix}\right](\mathbf{z}|\Omega)=e^{2\pi
i\mathbf{a}\cdot\mathbf{m}}\vartheta\left[\begin{matrix}\mathbf{a}\\\
\mathbf{b}\end{matrix}\right](\mathbf{z}|\Omega),$ (32)

where $\mathbf{m},$ $\mathbf{n}\in\mathbb{Z}^{g}$, it is enough to only
consider $\mathbf{a},\mathbf{b}\in(\frac{1}{2}\mathbb{Z}_{2})^{g}$. At genus
$g$, there are $2^{g-1}(2^{g}-1)$ odd spin structures and $2^{g-1}(2^{g}+1)$
even ones. The theta functions $\vartheta(\Omega)$ always vanish for odd spin
structures, which is obvious from (31).

Riemann theta functions are also weight-1/2 modular forms. From now on we will
denote $\vartheta(0|\Omega)$ by $\vartheta(\Omega)$ for convenience. When
their argument $\Omega$ is acted on by $\gamma=\left(\begin{matrix}A&B\\\
C&D\end{matrix}\right)\in Sp(2g,\mathbb{Z})$, they transform as Igusa ; AGMV :

$\vartheta\left[\begin{matrix}\mathbf{a^{\prime}}\\\
\mathbf{b^{\prime}}\end{matrix}\right]\left(\gamma\Omega\right)=\epsilon(\gamma)e^{-i\pi\phi(\mathbf{a},\mathbf{b})}\det\left(C\Omega+D\right)^{1/2}\vartheta\left[\begin{matrix}\mathbf{a}\\\
\mathbf{b}\end{matrix}\right](\Omega),$ (33)

where

$\left\\{\begin{matrix}\mathbf{a^{\prime}}\\\
\mathbf{b^{\prime}}\end{matrix}\right\\}=\left(\begin{matrix}D&-C\\\
-B&A\end{matrix}\right)\left\\{\begin{matrix}\mathbf{a}\\\
\mathbf{b}\end{matrix}\right\\}+\frac{1}{2}\left\\{\begin{matrix}{\left(CD^{T}\right)_{d}}\\\
{\left(AB^{T}\right)_{d}}\end{matrix}\right\\},$ (34)

and

$\phi\left(\mathbf{a},\mathbf{b}\right)=\left(\mathbf{a}D^{T}B\mathbf{a}+\mathbf{b}C^{T}A\mathbf{b}\right)-\left[2\mathbf{a}B^{T}C\mathbf{b}+\left(\mathbf{a}D^{T}-\mathbf{b}C^{T}\right)\left(AB^{T}\right)_{d}\right],$
(35)

which is $\Omega$-independent.

In (34), $\left\\{\begin{matrix}\cdot\\\ \cdot\end{matrix}\right\\}$ means
concatenating two $g$-dimensional row vectors into a single $2g$-dimensional
column vector, i.e., $\left\\{\begin{matrix}\mathbf{a}\\\
\mathbf{b}\end{matrix}\right\\}$ $\equiv\left[\begin{matrix}\mathbf{a}^{T}\\\
\mathbf{b}^{T}\end{matrix}\right]$ where $\cdot^{T}$ denotes the matrix
transpose, whereas $(\cdot)_{d}$ denotes the $g$-dimensional row vector whose
entries are the diagonal elements of the $g\times g$ matrix appearing inside
the parentheses $(\,)$. The subtle phase $\epsilon(\gamma)$ is always an
eighth root of unity independent of $\mathbf{a}$ and $\mathbf{b}$, and
incidentally, if $\gamma=I_{2g}\,\,\text{mod}\,\,2$, then
$\epsilon^{2}(\gamma)=e^{\pi i\text{Tr}(D-1)/2}$.

We note that the action of the group $Sp(4,\mathbb{Z})$ on the Riemann theta
functions at genus $g=2$ defines a 10-dimensional projective, not a linear
representation. The explicit forms of the matrix representations of the (two)
generators of the group are displayed in Appendix B.

##  4 Gravitational partition functions with boundaries of arbitrary genus

In this section, we discuss the full gravitational partition function at
Brown-Henneaux central charge $c=1/2$ with an asymptotic boundary being a
Riemann surface of arbitrary genus following the same strategy as in the
genus-2 case. The full gravitational partition function $Z_{\text{grav}}$ at
Brown-Henneaux central charge $c=1/2$ with a genus-$g$ asymptotic boundary
$\Sigma_{g}$ is again formulated as a sum over the contributions from
different saddle points which are all related to the “vacuum seed”
contribution $Z_{\text{vac}}$ by the action of the mapping class group
$\Gamma_{g}$ of the asymptotic boundary $\Sigma_{g}$. Given the period matrix
$\Omega$ that specifies the conformal structure on the asymptotic boundary
$\Sigma_{g}$, we should write the full gravitational partition function as

$\displaystyle
Z_{\text{grav}}(\Omega,\bar{\Omega})=\sum_{\gamma\in\Gamma}Z_{\text{vac}}(\gamma\Omega,\bar{\gamma}\bar{\Omega}),$
(36)

where $\Gamma=\Gamma_{c}\backslash\Gamma_{g}$ is the right coset space of the
mapping class group $\Gamma_{g}$ by its subgroup $\Gamma_{c}$ that leaves the
vacuum seed invariant. In this sum, the term with trivial $\gamma$ represents
the contribution from the vacuum sector (as known as the “vacuum seed") while
other terms present the contributions from other saddle points.

In the following, we will first argue in Section 4.1 that the vacuum seed
$Z_{\text{vac}}(\Omega,\bar{\Omega})$ at Brown-Henneaux central charge $c=1/2$
can be identified with the vacuum conformal block of the 2D Ising CFT on the
asymptotic boundary $\Sigma_{g}$ with the same period matrix $\Omega$. Then,
we will show that the $\Gamma_{g}$-orbit
$\\{Z_{\text{vac}}(\gamma\Omega,\bar{\gamma}\bar{\Omega})|\gamma\in\Gamma_{g}\\}$
of the vacuum seed, which appears in (36), is dictated by the projective
representation $\rho_{g}$ of the MCG $\Gamma_{g}$ induced by the holomorphic
conformal blocks of the 2D Ising CFT on $\Sigma_{g}$. Subsequently, we will
prove in Section 4.2 a mathematical result stating that $\rho_{g}$, viewed as
a mapping from $\Gamma_{g}$ to a unitary group, has a finite image set ${\rm
im}(\rho_{g})$, which has the immediate consequence that the sum
$\sum_{\gamma\in\Gamma_{g}}$ in (36) is finite. Furthermore, in Section 4.3,
we will prove another mathematical result stating that the MCG representation
$\rho_{g}$ is irreducible. Using the irreducibility of $\rho_{g}$, we can show
that the finite sum in (36) for the full gravitational partition function is
precisely proportional to the partition function of the 2D Ising CFT on the
asymptotic boundary $\Sigma_{g}$. In Section 4.4, we establish duality between
3D AdS quantum gravity at Brown-Henneaux central charge $c=1/2$ and 2D Ising
CFT. There, we will also further comment on our arguments for the
gravitational vacuum seed $Z_{\text{vac}}(\Omega,\bar{\Omega})$. In Section
4.5, we will discuss, from the perspective of the higher-genus partition
function, the fundamental difficulty in extending the duality to the case with
Brown-Henneaux central charge $c=7/10$.

###  4.1 Vacuum seed

Similar to the discussion of the genus-2 asymptotic boundary, to identify the
vacuum seed, namely the gravitational partition function contributed by the
vacuum sector, we start with a handlebody $X$ with a genus-$g$ asymptotic
boundary $\partial X=\Sigma_{g}$. The classical saddle point geometry on such
a handlebody $X$ is asymptotically AdS3 Aminneborg ; Brill ; Skenderis ;
Krasnov1 ; Krasnov2 ; Yin1 ; Giombi . As stated in Section 3.1.1, we believe
that the asymptotic behavior of the geometry ensures that the Brown-Henneaux
central charge $c=3l/2G_{N}$ is still applicable even if the boundary genus
$g$ is larger than 1. In the following, we will always focus on the case with
Brown-Henneaux central charge $c=1/2$.

As far as topology goes, the genus-$g$ handlebody $X$ can be viewed as two
3-balls (the interiors of two 2-dimensional spheres) connected by $g+1$ solid
cylinders. A genus-$3$ example is shown in Figure 6.212121In this paper, we
only study handlebodies in 3 dimensions. A genus-$g$ (3-dimensional)
handlebody means a handlebody with a genus-$g$ 2-dimensional boundary. Similar
to the genus-2 discussion, we believe that the vacuum seed $Z_{\text{vac}}$
should dominate the (full) gravitational partition function on the 3-manifold
$X$ in the limit where the boundary period matrix $\Omega$ is chosen such
that, for each of the solid cylinder regions, the boundary circumference is
much shorter than the length of the cylinder. In such a limit, it is natural
to consider a (local) coordinate system such that the Euclidean time direction
is along the longitudinal direction of each solid cylinder region. The Hilbert
space of quantum gravity states should then be associated to a constant-time
slice, which is a disjoint union of the cross sections of each of the solid
cylinders, namely the disjoint union of $g+1$ disks. For example, for $g=3$,
the Hilbert space of quantum gravity states should be associated with a
disjoint union of 4 disks as shown in Figure 6.

Recall that in the discussion of the case with a genus-1 asymptotic boundary,
the quantum gravity states defined on a single disk are the boundary graviton
states that form the irreducible (identity) representation of the Virasoro
algebra with the corresponding Brown-Henneaux central charge $c$. For $c=1/2$
in particular, the boundary graviton states on a single disk are in one-to-one
correspondence with quantum states of the 2D Ising CFT within the
$|\chi_{1,1}|^{2}$ sector.

Coming back to the genus-$g$ handlebody, we now need to assign a Hilbert space
to the disjoint union of $g+1$ disks. We naturally expect the Hilbert space to
be identified as the tensor product of $g+1$ copies of boundary graviton
states obtained in the genus-1 discussion. In this picture, each solid
cylinder region physically describes the time evolution of the boundary
graviton states.

So far, we have have been discussing the solid cylinder regions of the
handlebody. Each of the 3-ball regions in the handlebody glues together all of
the solid cylinders. Physically, each of them should describe the scattering
process of $g+1$ boundary graviton states. Since the boundary graviton states
are in one-to-one correspondence with the quantum states of the 2D Ising CFT,
we further make the proposal that the vacuum seed,
$Z_{\text{vac}}(\Omega,\bar{\Omega})$, is identical to the vacuum conformal
block of the 2D Ising CFT on the asymptotic boundary $\Sigma_{g}$ with period
matrix $\Omega$222222In the vacuum conformal block of the 2D Ising CFT, the
states propagating along the boundary of the solid cylinder regions all belong
to the irreducible representation of the Virasoro algebra (and its anti-
holomorphic copy) associated with $|\chi_{1,1}|^{2}$., which we naturally
expect to factorize into holomorphic and the anti-holomorphic pieces, i.e.,

$\displaystyle
Z_{\text{vac}}(\Omega,\bar{\Omega})=\chi_{\text{vac}}(\Omega)\bar{\chi}_{\text{vac}}(\bar{\Omega}),$
(37)

where $\chi_{\text{vac}}(\Omega)$ and $\bar{\chi}_{\text{vac}}(\bar{\Omega})$
are the respective holomorphic and anti-homolorphic vacuum conformal blocks of
the 2D Ising CFT on the genus-$g$ surface $\Sigma_{g}$ with period matrix
$\Omega$. In fact, our proposed form of the vacuum seed is simply a natural
extension of results in Yin1  and Bootstrap . To be more specific, Yin1
calculates the vacuum seed of the pure 3D AdS gravity with a genus-2
asymptotic boundary in the large-$c$ limit and also in the degeneration limit
of the boundary. The result is obtained to the order $1/c^{2}$. Bootstrap
shows that the vacuum conformal block of a 2D large-$c$ CFT matches exactly
with the result of Yin1  to all the orders calculated. Naturally, such a
matching is expected to hold to all orders of $1/c$. Hence, (37) is a
reasonable assumption when we take $c=1/2$. In addition, we will also see in
the following subsections that a vacuum seed of the form of ((37)) does yield
a sensible expression for the full gravitational partition function through
the modular sum (36).

Figure 6: A handlebody with a genus-$3$ asymptotic boundary. Each shaded disk
shown should be associated with a Hilbert space of the boundary graviton
states which form the representation of the Virasoro algebra with $c=1/2$.

###  4.2 Finiteness of the Modular Sum

To perform the modular sum (36), we need to ensure that the summation over the
set of right cosets, $\Gamma=\Gamma_{c}\backslash\Gamma_{g}$, is finite.
$\Gamma_{g}$ is the MCG of the asymptotic boundary $\Sigma_{g}$ and
$\Gamma_{c}$ is the subgroup of $\Gamma_{g}$ that leaves the vacuum seed
$Z_{\text{vac}}(\Omega,\bar{\Omega})$ invariant. The finiteness of the set
$\Gamma$ is mathematically equivalent to the finiteness of the orbit of the
vacuum seed $Z_{\text{vac}}$ under the MCG action, namely the finiteness of
the set
$\\{Z_{\text{vac}}\left(\gamma\Omega,\bar{\gamma}\bar{\Omega}\right)|\gamma\in\Gamma_{g}\\}$.
In Section 4.1, we have argued that the vacuum seed $Z_{\text{vac}}$ is given
by the product of the holomorphic and anti-holomorphic vacuum conformal blocks
of the 2D Ising CFT. Therefore, the MCG orbit of the vacuum seed
$Z_{\text{vac}}$ is dictated by the $\Gamma_{g}$ action on the conformal
blocks of the 2D Ising CFT on $\Sigma_{g}$.

$\Sigma_{g}$ is a genus-$g$ Riemann surface. Considering only the holomorphic
vacuum conformal block $\chi_{\text{vac}}$, the 2D Ising CFT has a total of
$N_{g}=2^{g-1}(2^{g}+1)$ holomorphic conformal blocks on $\Sigma_{g}$. They
form an $N_{g}$-dimensional vector space which admits a $\Gamma_{g}$ action:

$\displaystyle\chi_{i}(\gamma\Omega)=\sum_{i^{\prime}=1}^{N_{g}}\Big{(}\rho_{g}(\gamma)\Big{)}_{ii^{\prime}}\chi_{i^{\prime}}(\Omega).$
(38)

Here $\gamma\in\Gamma_{g}$, where $\chi_{i}(\Omega)$ with $i=1,2,..,N_{g}$
denote the $N_{g}$ different holomorphic conformal blocks of the 2D Ising CFT
on the surface $\Sigma_{g}$, and $\rho_{g}(\gamma)\in U(N_{g})$ is an
$N_{g}\times N_{g}$ unitary matrix that depends on $\gamma$ (but not on the
period matrix $\Omega$). In fact, $\rho_{g}$ is a projective representation of
the MCG $\Gamma_{g}$: For any $\gamma,\gamma^{\prime}\in\Gamma_{g}$,
$\rho_{g}(\gamma)\rho_{g}(\gamma^{\prime})$ is equal to
$\rho_{g}(\gamma\gamma^{\prime})$ up to a $U(1)$ phase. The $\Gamma_{g}$
action on the anti-holomorphic conformal blocks of the 2D Ising CFT is
naturally given by the complex-conjugated version of (38). Therefore, we will
only discuss the representation $\rho_{g}$ that dictates the $\Gamma_{g}$
action on the holomorphic conformal blocks in the following discussion.

When viewed as a map from $\Gamma_{g}$ to $U(N_{g})$, $\rho_{g}$ has an image
set $\rho_{g}(\Gamma_{g})\equiv\\{\rho_{g}(\gamma)|\gamma\in\Gamma_{g}\\}$
which is a subset of $U(N_{g})$. In the following, we will prove that
$\rho_{g}(\Gamma_{g})$ is a finite set. Combining (37) and (38), it is
straightforward to see that the finiteness of the set $\rho_{g}(\Gamma_{g})$
directly implies the finiteness of the MCG orbit
$\\{Z_{\text{vac}}\left(\gamma\Omega,\bar{\gamma}\bar{\Omega}\right)|\gamma\in\Gamma_{g}\\}$
and, consequently, leads to the conclusion that the modular sum (36) is
finite.

We will prove the finiteness of $\rho_{g}(\Gamma_{g})$ by contradiction. Let’s
assume that $\rho_{g}(\Gamma_{g})$ is an infinite set. First, we show that
this assumption leads to the consequence that
$\\{\Tr\rho_{g}(\gamma)|\gamma\in\Gamma_{g}\\}$ also has to be an infinite
set. Since $\rho_{g}(\gamma)\in U(N_{g})$, $|\Tr\rho_{g}(\gamma)|\leq N_{g}$.
To show that $\\{\Tr\rho_{g}(\gamma)|\gamma\in\Gamma_{g}\\}$ is an infinite
set, it is sufficient to show that, for any small number $\epsilon>0$, we can
either find (i) a pair of elements $\gamma,\gamma^{\prime}\in\Gamma_{g}$ such
that $0<|\Tr\rho_{g}(\gamma)-\Tr\rho_{g}(\gamma^{\prime})|<\epsilon$ or (ii)
an element $\gamma^{\prime\prime}\in\Gamma_{g}$ such that
$N_{g}-\epsilon<|\Tr\rho_{g}(\gamma^{\prime\prime})|<N_{g}$. First, we start
with a sufficiently small $\epsilon^{\prime}>0$. Since $U(N_{g})$ is a compact
space, the assumption that $\rho_{g}(\Gamma_{g})$ is an infinite set
guarantees the existence of a pair of elements
$\gamma,\gamma^{\prime}\in\Gamma_{g}$ such that
$0<\norm{\rho_{g}(\gamma)-\rho_{g}(\gamma^{\prime})}<\epsilon^{\prime}$ where
$\norm{\cdot}$ represents the Frobenius norm.232323The Frobenius norm
$\norm{A}$ of a matrix $A$ is defined as the square root of the sum of the
absolute squares of its elements, namely $\norm{A}=\sqrt{\Tr(A^{\dagger}A)}$.
$\rho_{g}(\gamma)$ is not identical to $\rho_{g}(\gamma^{\prime})$. But we
still need to distinguish two situations depending on whether
$\rho_{g}(\gamma)$ and $\rho_{g}(\gamma^{\prime})$ differ by only a $U(1)$
phase or not. In first situation where $\rho_{g}(\gamma)$ differs from
$\rho_{g}(\gamma^{\prime})$ by a $U(1)$ phase, the sufficiently small
$\epsilon^{\prime}$ can guarantee that
$0<|\Tr\rho_{g}(\gamma)-\Tr\rho_{g}(\gamma^{\prime})|<\epsilon$. Hence, we
find the pair of elements $\gamma,\gamma^{\prime}$ described in (i). In second
situation where $\rho_{g}(\gamma)$ is not proportional to
$\rho_{g}(\gamma^{\prime})$, we notice $\rho_{g}(\gamma^{-1}\gamma^{\prime})$,
which is equal to $\rho_{g}(\gamma)^{-1}\rho_{g}(\gamma^{\prime})$ up to a
$U(1)$ phase, is then not proportional to the identity operator. Then, with
$\gamma^{\prime\prime}=\gamma^{-1}\gamma^{\prime}$,
$|\Tr\rho_{g}(\gamma^{\prime\prime})|<N_{g}$. However, with a sufficiently
small $\epsilon^{\prime}$, $\rho_{g}(\gamma^{\prime\prime})$ can be
arbitrarily close to the identity operator up a $U(1)$ phase. Therefore, we
have $N_{g}-\epsilon<|\Tr\rho_{g}(\gamma^{\prime\prime})|<N_{g}$. Hence, we
find the element $\gamma^{\prime\prime}$ described in (ii). Now, we can
conclude that the assumption that $\rho_{g}(\Gamma_{g})$ is an infinite set
has a consequence that $\\{\Tr\rho_{g}(\gamma)|\gamma\in\Gamma_{g}\\}$ also
has to be an infinite set.

In the remainder of this subsection, we will show that
$\\{\Tr\rho_{g}(\gamma)|\gamma\in\Gamma_{g}\\}$ in fact cannot be an infinite
set and, hence, that the assumption that $\rho_{g}(\Gamma_{g})$ is an infinite
set is incorrect.

For any $\gamma\in\Gamma_{g}$, $\Tr\rho_{g}(\gamma)$ can be interpreted as a
partition function of the 3D Ising topological quantum field theory (TQFT).
The 3D Ising TQFT is closely related to the 2D Ising CFT. In particular, the
3D Ising TQFT assigns a $N_{g}$-dimensional Hilbert space to the genus-$g$
surface $\Sigma_{g}$ whose basis vectors are in one-to-one correspondence with
the holomorphic conformal blocks of the 2D Ising CFT on $\Sigma_{g}$
MooreSeiberg . The details of this correspondence will be reviewed in the next
subsection. A $\Gamma_{g}$ action $\gamma$ on the genus-$g$ surface
$\Sigma_{g}$ induces a unitary transformation within the 3D Ising TQFT Hilbert
space which is exactly given by $\rho_{g}(\gamma)$. $\Tr\rho_{g}(\gamma)$ can
be interpreted as the 3D Ising TQFT partition function $Z_{\rm
iTQFT}(M_{\gamma})$ evaluated on the mapping torus
$M_{\gamma}\equiv\frac{[0,1]\times\Sigma_{g}}{(0,x)\sim(1,\gamma(x))}$. The
mapping torus $M_{\gamma}$ is a 3-manifold obtained from gluing the two
$\Sigma_{g}$ boundary components of the Cartesian product
$[0,1]\times\Sigma_{g}$ with an MCG action $\gamma$ performed on one of the
$\Sigma_{g}$ components. For a general 3-manifold $M^{3}$, the 3D Ising TQFT
partition can be expressed as Walker1991 ; Kirby1991

$\displaystyle Z_{\rm iTQFT}(M^{3})=\sum_{\text{spin structure
}\zeta}e^{\frac{2\pi i}{16}\mu(M^{3},\zeta)},$ (39)

where $\sum_{\zeta}$ represents the summation over all spin structures $\zeta$
on $M^{3}$ and $\mu(M^{3},\zeta)$ is Rokhlin’s $\mu$-invariant242424For
$(M^{3},\zeta)$, it is defined as the signature of the intersection form of
any smooth compact spin 4-manifold with the spin boundary $(M^{3},\zeta)$. of
the 3-manifold $M^{3}$ with the spin structure $\zeta$. The invariant
$\mu(M^{3},\zeta)$ is defined modulo 16 and is always an even integer. For a
general 3-manifold $M^{3}$, the number of spin structures on $M^{3}$ is equal
to $|H^{1}(M^{3},\mathbb{Z}_{2})|=|H_{1}(M^{3},\mathbb{Z}_{2})|$. Here, we are
viewing $H^{1}$ and $H_{1}$ as groups. $|\cdot|$ means the order of the group
in this context. For the mapping torus $M_{\gamma}$ of $\Sigma_{g}$, we can
consider the following long exact sequence (see, e.g., Example 2.48 of
Hatcher2 ),

$\displaystyle...\rightarrow
H_{n}(\Sigma_{g},\mathbb{Z}_{2})\xrightarrow{~{}}H_{n}(\Sigma_{g},\mathbb{Z}_{2})\rightarrow
H_{n}(M_{\gamma},\mathbb{Z}_{2})\rightarrow
H_{n-1}(\Sigma_{g},\mathbb{Z}_{2})\rightarrow...,$ (40)

which implies an upper bound on the number of spin structures on $M_{\gamma}$
that only depends on $g$ but not $\gamma$:

$\displaystyle|H^{1}(M_{\gamma},\mathbb{Z}_{2})|=|H_{1}(M_{\gamma},\mathbb{Z}_{2})|\leq|H_{1}(\Sigma_{g},\mathbb{Z}_{2})|\times|H_{0}(\Sigma_{g},\mathbb{Z}_{2})|.$
(41)

The inequality above is a direct consequence of the
$H_{1}(\Sigma_{g},\mathbb{Z}_{2})\rightarrow
H_{1}(M_{\gamma},\mathbb{Z}_{2})\rightarrow H_{0}(\Sigma_{g},\mathbb{Z}_{2})$
part of the long exact sequence (40).252525With the map
$H_{1}(M_{\gamma},\mathbb{Z}_{2})\rightarrow H_{0}(\Sigma_{g},\mathbb{Z}_{2})$
in (40) viewed as a linear map between vector spaces (over $\mathbb{Z}_{2}$),
the sum of the dimensions of its kernel and its image is equal to the
dimension of $H_{1}(M_{\gamma},\mathbb{Z}_{2})$. The image of the linear map
$H_{1}(M_{\gamma},\mathbb{Z}_{2})\rightarrow
H_{0}(\Sigma_{g},\mathbb{Z}_{2})$, as a vector space over $\mathbb{Z}_{2}$,
has a dimension less than or equal to the dimension of
$H_{0}(\Sigma_{g},\mathbb{Z}_{2})$. From the fact that (40) is exact, the
kernel of the map $H_{1}(M_{\gamma},\mathbb{Z}_{2})\rightarrow
H_{0}(\Sigma_{g},\mathbb{Z}_{2})$ has the same dimension as the image of the
map $H_{1}(\Sigma_{g},\mathbb{Z}_{2})\rightarrow
H_{1}(M_{\gamma},\mathbb{Z}_{2})$ whose dimension is smaller than or equal to
the dimension of the vector space $H_{1}(\Sigma_{g},\mathbb{Z}_{2})$.
Therefore, the dimension of the vector space
$H_{1}(M_{\gamma},\mathbb{Z}_{2})$ is not greater than the sum of the
dimensions of $H_{0}(\Sigma_{g},\mathbb{Z}_{2})$ and of
$H_{1}(\Sigma_{g},\mathbb{Z}_{2})$, which implies (41).  Therefore, according
to (39), for any $\gamma\in\Gamma_{g}$,

$\displaystyle\Tr\rho_{g}(\gamma)=Z_{\rm iTQFT}(M_{\gamma})$
$\displaystyle~{}\in\left\\{\sum_{n=0,2,4,...,14}a_{n}e^{\frac{2\pi
i}{16}n}~{}\Bigg{|}~{}a_{n}\in\mathbb{Z},~{}0\leq
a_{n}\leq|H_{1}(\Sigma_{g},\mathbb{Z}_{2})|\times|H_{0}(\Sigma_{g},\mathbb{Z}_{2})|\right\\}.$
(42)

Notice that the set given in the second line a finite set. Therefore,
$\\{\Tr\rho_{g}(\gamma)|\gamma\in\Gamma_{g}\\}$ cannot be an infinite set,
which is in contradiction to the consequence of the assumption that
$\rho_{g}(\Gamma_{g})$ is an infinite set. Now, we can conclude that
$\rho_{g}(\Gamma_{g})$ has to be a finite subset of $U(N_{g})$. It follows
that the modular sum (36) is finite.

This proof of the finiteness of the modular sum (36) relies on the expression
of the vacuum seed $Z_{\rm vac}$ (37) that we argued for in Section 4.1. In
fact, as long as the vacuum seed $Z_{\rm vac}$ can be written as a product of
a holomorphic and an anti-holomorphic conformal block of the 2D Ising CFT (or
even as a sum of products of this type), the proof given in this subsection is
still applicable and the modular sum (36) is still finite.

###  4.3 Irreducibility of the MCG representation and the modular sum

With the modular sum (36) proven to be finite, the full gravitational
partition function $Z_{\text{grav}}(\Omega,\bar{\Omega})$ is then, by
construction, invariant under any $\Gamma_{g}$ action on the asymptotic
boundary $\Sigma_{g}$. Since a MCG action generally transforms the holomorphic
(anti-holomorphic) vacuum conformal blocks of the 2D Ising CFT into a linear
superposition of all holomorphic (anti-holomorphic) conformal blocks, we
expect the modular sum (36), together with the vacuum seed (37), to yield

$\displaystyle
Z_{\text{grav}}(\Omega,\bar{\Omega})=\sum_{i,i^{\prime}=1}^{N_{g}}B_{ii^{\prime}}\bar{\chi}_{i^{\prime}}(\bar{\Omega})\chi_{i}(\Omega),$
(43)

where $B$ is a $N_{g}\times N_{g}$ matrix. The invariance of
$Z_{\text{grav}}(\Omega,\bar{\Omega})$ under the action of the MCG implies
that

$\displaystyle\rho_{g}(\gamma)^{{\dagger}}B\rho_{g}(\gamma)=B,$ (44)

for any $\gamma\in\Gamma_{g}$. Importantly, as we will prove later in this
subsection, the projective representation $\rho_{g}$ of the MCG $\Gamma_{g}$
is irreducible. As a consequence, by Schur’s lemma, $B$ has to be proportional
to the identity matrix to satisfy (44). Therefore, the full gravitational
partition function satisfies

$\displaystyle
Z_{\text{grav}}(\Omega,\bar{\Omega})\propto\sum_{i=1}^{N_{g}}\bar{\chi}_{i}(\bar{\Omega})\chi_{i}(\Omega).$
(45)

In the following, we will present the proof of the irreducibility of the MCG
representation $\rho_{g}$.

Figure 7:
$a_{1,2,...,g}~{},~{}b_{1,2,...,g}~{},~{}w_{1,2,...,g-1}\in\\{1,\sigma,\psi\\}$.
Fusion rules need to be applied at each trivalent vertex for the fusion
diagram to be admissible.

First, we review the connections between the 2D Ising CFT and 3D Ising TQFT
that will be useful for the proof of the irreducibility of the representation
$\rho_{g}$. On the genus-$g$ surface $\Sigma_{g}$, there are $N_{g}$
holomorphic conformal blocks in the 2D Ising CFT and there are $N_{g}$
orthogonal quantum states in the 3D Ising TQFT. Each of the holomorphic
conformal blocks has a corresponding TQFT quantum state and vice versa. Each
of holomorphic conformal blocks and its corresponding TQFT quantum state can
be represented by an admissible fusion diagram as shown in Figure 7. Each line
in the fusion diagram is labeled by $1$, $\sigma$ or $\psi$. That is to say,
in Figure 7, all the labels $a_{1,2,...,g}$, $b_{1,2,...,g}$ and
$w_{1,2,..,g-1}$ take values in the set $\\{1,\sigma,\psi\\}$. The labels
$\\{1,\sigma,\psi\\}$ should be viewed as the labels for the primary fields in
the 2D (chiral) Ising CFT and, equivalently, also as the labels for the anyons
(or objects or particles) in the 3D Ising TQFT. Note that the lines in the
fusion diagrams are also directed. In general, a directed line carrying an
anyon label $a$ is equivalent to the line with the opposite direction and with
the label $\bar{a}$, namely the label for the anti-particle of $a$. The
directions of all the lines in Figure 7 are chosen merely as a convention. In
fact, in 3D Ising TQFT, each of $1$, $\sigma$ and $\psi$ is its own
antiparticle. Therefore, it should not cause confusion even if we don’t
specify the directions of the lines in a fusion diagram in in the discussion
below. Also, $1$ represents the trivial anyon in the 3D Ising TQFT and the
trivial (identity) primary operator in the 2D Ising CFT. In the fusion
diagram, a line labeled by $1$ can also be erased. Only a so-called admissible
fusion diagram corresponds to a holomorphic conformal block or a TQFT quantum
state on $\Sigma_{g}$. For the fusion diagram in Figure 7 to be admissible in
the 2D Ising CFT or the 3D Ising TQFT, we first need to require $a_{1}=b_{1}$
and $a_{g}=b_{g}$. Moreover, an admissible fusion diagram also requires each
trivalent vertex to be admissible. Each trivalent vertex has two incoming
(outgoing) lines and one outgoing (incoming) line. If the anyons $a$ and $b$
labeling the two incoming (outgoing) lines have a fusion product $a\times b$
that contains the anyon $c$ labeling the one outgoing (incoming) line, the
trivalent vertex is admissible. The full set of fusion rules of the 3D Ising
TQFT (or the 2D Ising CFT) is given by

$\displaystyle 1\times
1=1,~{}~{}~{}1\times\sigma=\sigma,~{}~{}~{}1\times\psi=\psi,$
$\displaystyle\psi\times
1=\psi,~{}~{}~{}\psi\times\sigma=\sigma,~{}~{}~{}\psi\times\psi=1,$ (46)
$\displaystyle\sigma\times
1=\sigma,~{}~{}~{}\sigma\times\psi=\sigma,~{}~{}~{}\sigma\times\sigma=1+\psi.$

One can directly show (see below) that there are $N_{g}$ admissible fusion
diagrams (with different anyon labels $a_{1,2,..,g}$, $b_{1,2,..,g}$ and
$w_{1,2,..,g-1}$) of the form shown in Figure 7, where
$N_{g}=2^{g-1}(2^{g-1}+1)$ is the dimension of the representation $\rho_{g}$
of the MCG discussed above.

We will denote the Ising TQFT quantum state (and its correspond Ising-CFT
conformal block) by the corresponding fusion diagram labels. For example, the
Ising TQFT quantum state associated to the fusion diagram shown in Figure 7
will be denoted as $|\\{a_{i}\\},\\{b_{i}\\},\\{w_{i}\\}\rangle$. Physically,
in the language of 3D TQFT, one can think of an admissible fusion diagram as
describing the world lines of anyon. Therefore, in the discussion below, we
will also refer to a fusion diagram as an anyon diagram. The correspondence
between the state $|\\{a_{i}\\},\\{b_{i}\\},\\{w_{i}\\}\rangle$ and its fusion
diagram can be understood as follows. The state
$|\\{a_{i}\\},\\{b_{i}\\},\\{w_{i}\\}\rangle$ on $\Sigma_{g}$ can be viewed as
generated by the 3D Ising TQFT path integral on a genus-$g$ handlebody $H_{g}$
such that $\partial H_{g}=\Sigma_{g}$, and such that the corresponding fusion
diagram (or anyon diagram) is embedded in the core of $H_{g}$ (in the same
configuration as shown in Figure 7). In particular, there is a “special” state
$|{\rm vac}\rangle\equiv|\\{a_{i}=1\\},\\{b_{i}=1\\},\\{w_{i}=1\\}\rangle$
with all of the labels on the fusion diagram set to be $1$. The state $|{\rm
vac}\rangle$ can be viewed as the result of the Ising TQFT path integral on
the handlebody $H_{g}$ without an anyon diagram inside (remember that anyon
lines labeled by $1$ can be erased). The so-defined TQFT state $|{\rm
vac}\rangle$ corresponds to the holomorphic vacuum conformal block $\chi_{{\rm
vac}}(\Omega)$ of the 2D Ising CFT.

Because of the correspondence between the holomorphic conformal blocks of the
2D Ising CFT and the states on $\Sigma_{g}$ of the 3D Ising TQFT, the MCG
$\Gamma_{g}$ acts on the states $|\\{a_{i}\\},\\{b_{i}\\},\\{w_{i}\\}\rangle$
via the same representation $\rho_{g}$. The $\Gamma_{g}$ action on the Ising-
TQFT states can also be understood as follows. In the picture where the Ising-
TQFT states are generated by the Ising TQFT path integral on a handlebody
$H_{g}$ with an anyon diagram, the MCG action on $\Sigma_{g}=\partial H_{g}$
should be extended to the whole handlebody $H_{g}$. Such an extended action of
$\Gamma_{g}$ deforms the anyon diagram inside $H_{g}$. The deformed anyon
diagram can be rewritten in terms of a linear superposition of anyon diagrams
of the original shape shown in Figure 7 with different anyon labels. That is
to say that when a state $|\\{a_{i}\\},\\{b_{i}\\},\\{w_{i}\\}\rangle$ is
acted on by an element $\gamma\in\Gamma_{g}$ of the MCG, the resulting state
is in general a superposition of many states with different anyon labels in
their fusion diagrams:

$\displaystyle\rho_{g}(\gamma)|\\{a_{i}\\},\\{b_{i}\\},\\{w_{i}\\}\rangle$
$\displaystyle=\sum_{a_{i}^{\prime},b_{i}^{\prime},w_{i}^{\prime}=\\{1,\sigma,\psi\\}}\langle\\{a_{i}^{\prime}\\},\\{b_{i}^{\prime}\\},\\{w_{i}^{\prime}\\}|\rho_{g}(\gamma)|\\{a_{i}\\},\\{b_{i}\\},\\{w_{i}\\}\rangle\
\ ~{}|\\{a_{i}^{\prime}\\},\\{b_{i}^{\prime}\\},\\{w_{i}^{\prime}\\}\rangle.$
(47)

A particularly simple case is when the MCG action is a Dehn twist $\nu_{C}$
along a loop $C$ that is threaded by a single anyon line labeled by $a$ (as is
shown in Figure 8). Such a Dehn twist does not change the shape of the anyon
diagram, the action $\rho_{g}(\nu_{C})$ only yields extra $U(1)$ phase
$e^{i2\pi h_{a}}$ on the state represented by the anyon diagram, where $h_{a}$
depends on the anyon label $a$:

$\displaystyle h_{1}=0,~{}~{}h_{\sigma}=1/16,~{}~{}h_{\psi}=1/2.$ (48)

Here $h_{a}$ can be viewed as the conformal weight of the primary field
labeled by $a$ in the 2D (chiral) Ising CFT. Also, in the 3D Ising TQFT
language, we can view $e^{i2\pi h_{a}}$ as the topological spin of the anyon
labeled by $a$.

Figure 8: The Dehn twist along the non-contractible loop $C$ only yields a
$U(1)$ phases $e^{i2\pi h_{a}}$ that depends on the label $a$.

In the following, we will show that $\rho_{g}$ is an irreducible projective
representation of the MCG $\Gamma_{g}$. In fact, the irreducibility of
$\rho_{g}$ is equivalent to the statement that the $\mathbb{C}$-linear matrix
algebra $\mathbb{C}[\rho_{g}]$ generated by $\rho_{g}\left(\Gamma_{g}\right)$
(through addition and matrix multiplication) is identical to the full matrix
algebra $\mathbb{M}_{N_{g}}$ of all $N_{g}\times N_{g}$ complex matrices,
namely $\mathbb{C}[\rho_{g}]\cong\mathbb{M}_{N_{g}}$. Obviously,
$\mathbb{C}[\rho_{g}]\subseteq\mathbb{M}_{N_{g}}$. Therefore, what we need to
prove is that $\mathbb{M}_{N_{g}}\subseteq\mathbb{C}[\rho_{g}]$. The strategy
of the proof is to explicitly construct all the operators of the form
$|\\{a_{i}\\},\\{b_{i}\\},\\{w_{i}\\}\rangle\
\langle\\{a_{i}^{\prime}\\},\\{b_{i}^{\prime}\\},\\{w_{i}^{\prime}\\}|$ within
$\mathbb{C}[\rho_{g}]$.

Figure 9: The Dehn twists along the loops $C_{1,2,..,g}$,
$C^{\prime}_{1,2,..,g}$ and $C^{\prime\prime}_{1,2,..,g-1}$ can distinguish
all of the states $|\\{a_{i}\\},\\{b_{i}\\},\\{w_{i}\\}\rangle$

We will first construct the projection operators
$|\\{a_{i}\\},\\{b_{i}\\},\\{w_{i}\\}\rangle\langle\\{a_{i}\\},\\{b_{i}\\},\\{w_{i}\\}|$,
which will be denoted as $P_{\\{a_{i}\\},\\{b_{i}\\},\\{w_{i}\\}}$ in the
following discussion, for any state
$|\\{a_{i}\\},\\{b_{i}\\},\\{w_{i}\\}\rangle$. For this purpose, we can focus
on the set of non-intersecting loops $C_{1,2,..,g}$, $C^{\prime}_{1,2,..,g}$
and $C^{\prime\prime}_{1,2,..,g-1}$ shown in Figure 9. The Dehn twists
$\nu_{C_{i}}$, $\nu_{C_{i}^{\prime}}$ and $\nu_{C_{i}^{\prime\prime}}$ along
each of these loops commute with each other. A state
$|\\{a_{i}\\},\\{b_{i}\\},\\{w_{i}\\}\rangle$ with a fixed set of labels
$a_{1,2,..,g}$, $b_{1,2,..,g}$ and $w_{1,2,..,g-1}$ is a simultaneous
eigenstate of all such Dehn twists:

$\displaystyle\rho_{g}(\nu_{C_{j}})|\\{a_{i}\\},\\{b_{i}\\},\\{w_{i}\\}\rangle=e^{i2\pi
h_{a_{j}}}|\\{a_{i}\\},\\{b_{i}\\},\\{w_{i}\\}\rangle,~{}~{}~{}~{}~{}{\rm
for}~{}j=1,2,...,g$
$\displaystyle\rho_{g}(\nu_{C^{\prime}_{j}})|\\{a_{i}\\},\\{b_{i}\\},\\{w_{i}\\}\rangle=e^{i2\pi
h_{b_{j}}}|\\{a_{i}\\},\\{b_{i}\\},\\{w_{i}\\}\rangle,~{}~{}~{}~{}~{}{\rm
for}~{}j=1,2,...,g$ (49)
$\displaystyle\rho_{g}(\nu_{C^{\prime\prime}_{j}})|\\{a_{i}\\},\\{b_{i}\\},\\{w_{i}\\}\rangle=e^{i2\pi
h_{w_{j}}}|\\{a_{i}\\},\\{b_{i}\\},\\{w_{i}\\}\rangle,~{}~{}~{}~{}~{}{\rm
for}~{}j=1,2,...,g-1.$

Since $e^{i2\pi h_{1}}$, $e^{i2\pi h_{\sigma}}$, $e^{i2\pi h_{\psi}}$ are all
different, one can use the set of Dehn twists $\nu_{C_{i}}$,
$\nu_{C_{i}^{\prime}}$ and $\nu_{C_{i}^{\prime\prime}}$ to fully distinguish
all the states $|\\{a_{i}\\},\\{b_{i}\\},\\{w_{i}\\}\rangle$. Building on
this, we can construct the following projection operators associated with any
loop $C$ and an anyon label $1$, $\sigma$, or $\psi$ within
$\mathbb{C}[\rho_{g}]$:

$\displaystyle
P_{1}(C)\equiv\frac{1}{16}\sum_{n=1}^{16}\rho_{g}\left(\nu_{C}^{n}\right),~{}~{}~{}~{}$
$\displaystyle P_{\sigma}(C)\equiv\left(1-e^{2\pi
i/8}\right)^{-1}\left(\mathds{1}-\rho_{g}\left(\nu_{C}^{2}\right)\right),~{}~{}~{}~{}$
(50) $\displaystyle P_{\psi}(C)=\mathds{1}-P_{1}(C)-P_{\sigma}(C),$

where $\nu_{C}\in\Gamma_{g}$ represents the Dehn twist along the loop $C$ and
$\mathds{1}$ represents the $N_{g}\times N_{g}$ identity matrix. Choosing $C$
to be $C_{j}$, $C_{j}^{\prime}$ or $C_{j}^{\prime\prime}$, we see that

$\displaystyle
P_{a}(C_{j})|\\{a_{i}\\},\\{b_{i}\\},\\{w_{i}\\}\rangle=\delta_{a,a_{j}}|\\{a_{i}\\},\\{b_{i}\\},\\{w_{i}\\}\rangle,~{}~{}~{}~{}~{}{\rm
for}~{}j=1,2,...,g$ $\displaystyle
P_{b}(C_{j}^{\prime})|\\{a_{i}\\},\\{b_{i}\\},\\{w_{i}\\}\rangle=\delta_{b,b_{j}}|\\{a_{i}\\},\\{b_{i}\\},\\{w_{i}\\}\rangle,~{}~{}~{}~{}~{}{\rm
for}~{}j=1,2,...,g$ (51) $\displaystyle
P_{w}(C_{j}^{\prime\prime})|\\{a_{i}\\},\\{b_{i}\\},\\{w_{i}\\}\rangle=\delta_{w,w_{j}}|\\{a_{i}\\},\\{b_{i}\\},\\{w_{i}\\}\rangle,~{}~{}~{}~{}~{}{\rm
for}~{}j=1,2,...,g-1,$

where $a_{i},b_{i},w_{i}\in\\{1,\sigma,\psi\\}$. Any projection operator
$P_{\\{a_{i}\\},\\{b_{i}\\},\\{w_{i}\\}}$ onto a given state
$|\\{a_{i}\\},\\{b_{i}\\},\\{w_{i}\\}\rangle$ can then be written as a product
of $P_{a}(C_{j})$, $P_{b}(C_{j}^{\prime})$ and $P_{w}(C_{j}^{\prime\prime})$.
Therefore, all projection operators $P_{\\{a_{i}\\},\\{b_{i}\\},\\{w_{i}\\}}$
belong to $\mathbb{C}[\rho_{g}]$.

Next, we will show that all the operators of the form $|{\rm
vac}\rangle\langle\\{a_{i}\\},\\{b_{i}\\},\\{w_{i}\\}|$ can be constructed
within $\mathbb{C}[\rho_{g}]$. Upon inspection we observe that in any
admissible fusion diagram of the form shown in Figure 7, the labels $w_{i}$
for $i=1,2,...,g-1$ can only take values $1$ or $\psi$. We will first focus on
the case with $w_{i}=1$ for all $i=1,2,...,g-1$. In this case, an admissible
diagram further requires $a_{i}=b_{i}$ for all $i=1,2,...,g$. Therefore, the
relevant states in this case are of the form
$|\\{a_{i}\\},\\{b_{i}=a_{i}\\},\\{w_{i}=1\\}\rangle$, which will be denoted
by $|\\{a_{i}\\}\rangle$ in short hand in the following discussion. The anyon
diagram of $|\\{a_{i}\\}\rangle$, after we have erased all the lines carrying
label $1$, is simply a disjoint union of anyon loops labeled by
$a_{1,2,...,g}$. To construct an operator of the form $|{\rm
vac}\rangle\langle\\{a_{i}\\}|$ in $\mathbb{C}[\rho_{g}]$, it is sufficient to
find an MCG element $\gamma$ such that $\langle{\rm
vac}|\rho_{g}(\gamma)|\\{a_{i}\\}\rangle\neq 0$ which allows us to write
$|{\rm vac}\rangle\langle\\{a_{i}\\}|$ as $|{\rm vac}\rangle\langle{\rm
vac}|\rho_{g}(\gamma)|\\{a_{i}\\}\rangle\langle\\{a_{i}\\}|$ up to a non-zero
multiplicative constant. Remember that we have already constructed the
operators $|{\rm vac}\rangle\langle{\rm vac}|$ and
$|\\{a_{i}\\}\rangle\langle\\{a_{i}\\}|$ within $\mathbb{C}[\rho_{g}]$.
Therefore, we only need to find the suitable MCG element $\gamma$.

In principle, the choice of $\gamma$ can depend on the state
$|\\{a_{i}\\}\rangle$. Interestingly, we can show that there is a specific MCG
element $\gamma_{0}\in\Gamma_{g}$ that works for all $|\\{a_{i}\\}\rangle$.
The MCG element $\gamma_{0}$ can be identified as follows. Consider the
disjoint union of two copies of a genus-$g$ handlebody $H_{g}$ and
$H_{g}^{\prime}$ whose boundaries are given by two identical copies
$\Sigma_{g}$ and $\Sigma_{g}^{\prime}$ of the same Riemann surface, i.e.,
$\Sigma_{g}=\partial H_{g}$ and $\Sigma_{g}^{\prime}=\partial H_{g}^{\prime}$.
In general, we can perform a MCG action $\gamma\in\Gamma_{g}$ on
$\Sigma_{g}^{\prime}$ and then glue it to $\Sigma_{g}$. This procedure glues
the two genus-$g$ handlebodies $H_{g}$ and $H_{g}^{\prime}$ into a single
closed 3-manifold that depends on the choice of $\gamma$. There exists an
element $\gamma_{0}$ such that the resulting closed 3-manifold is the 3-sphere
$S^{3}$. We will show that $\langle{\rm
vac}|\rho_{g}(\gamma_{0})|\\{a_{i}\\}\rangle\neq 0$ for any states
$|\\{a_{i}\\}\rangle$. Again, consider the setup with two copies $H_{g}$ and
$H^{\prime}_{g}$ of the genus-$g$ handlebody. Performing the 3D Ising-TQFT
path integral on $H_{g}$ (without any anyon diagram) yields the state $|{\rm
vac}\rangle$ on its boundary $\partial H_{g}=\Sigma_{g}$. Now, we embeded the
anyon diagram of $|\\{a_{i}\\}\rangle$, which is a collection of disjoint
anyon loops labeled by $a_{1,2,..,g}$ respectively, in $H_{g}^{\prime}$. The
TQFT path integral on $H_{g}^{\prime}$ then yields the state
$|\\{a_{i}\\}\rangle$ on its boundary $\partial
H_{g}^{\prime}=\Sigma_{g}^{\prime}$. When $\Sigma^{\prime}_{g}$ is acted on by
$\gamma_{0}$ and then glued to $\Sigma_{g}$, we obtain a 3D Ising TQFT path
integral on $S^{3}$ together with the anyon diagram that was originally
embedded in $H_{g}^{\prime}$. The result of such a path integral is exactly
$\langle{\rm vac}|\rho_{g}(\gamma_{0})|\\{a_{i}\\}\rangle$. Since the anyon
diagram involved here is a disjoint union of anyon loops labeled by
$a_{1,2,..,g}$ respectively, the Ising TQFT path integral on $S^{3}$ with such
anyon diagrams is definitely non-vanishing. Therefore,

$\displaystyle\langle{\rm vac}|\rho_{g}(\gamma_{0})|\\{a_{i}\\}\rangle\neq 0,$
(52)

for any choice of $a_{1,2,...,g}$. Consequently, we can conclude that the
operators of the form $|{\rm vac}\rangle\langle\\{a_{i}\\}|$ all belong to
$\mathbb{C}[\rho_{g}]$. By Hermitian conjugation, the any operator of the form
$|\\{a_{i}\\}\rangle\langle{\rm vac}|$ also belongs to $\mathbb{C}[\rho_{g}]$.

Figure 10: Local configurations of fusion diagrams with some of the $w_{i}$
labels taking the value $\psi$

Now, we are ready to construct the operators $|{\rm
vac}\rangle\langle\\{a_{i}\\},\\{b_{i}\\},\\{w_{i}\\}|$ with some of the
$w_{i}$ labels equal to $\psi$. When some of the $w_{i}$ labels equal to
$\psi$, the anyon diagram associated to
$|\\{a_{i}\\},\\{b_{i}\\},\\{w_{i}\\}\rangle$ must be in one of the
configurations shown in Figure 10 in the vicinity of the diagram where the
$w_{i}$ labels take the value $\psi$. In the Ising TQFT, we have the following
linear relations between the diagrams

$\displaystyle\raisebox{-0.4pt}{\includegraphics[width=56.9055pt]{fig-
FDmove1.jpg}}~{}~{}\propto~{}~{}\left(\raisebox{-0.4pt}{\includegraphics[width=56.9055pt]{fig-
FDmove2.jpg}}-\raisebox{-0.4pt}{\includegraphics[width=56.9055pt]{fig-
FDmove3.jpg}}\right)=\raisebox{-0.4pt}{\includegraphics[width=56.9055pt]{fig-
FDmove4.jpg}},$ (53)

which can help us relate an anyon diagram with some of the $w_{i}$ labels
equal to $\psi$ to another diagram with less of the $w_{i}$ labels equal to
$\psi$. For example, the leftmost configuration shown in Figure 10 obeys

$\displaystyle\raisebox{-0.4pt}{\includegraphics[width=85.35826pt]{fig-
TubeDeformed-01.jpg}}~{}~{}\propto~{}~{}\raisebox{-0.4pt}{\includegraphics[width=85.35826pt]{fig-
TubeDeformed-02.jpg}}~{}~{}=~{}~{}\raisebox{-0.4pt}{\includegraphics[width=85.35826pt]{fig-
TubeDeformed-03.jpg}},$ (54)

where the relation between the first two diagrams is a graphical
representation of the relation (53). The last equality in (54) means that a
Dehn twist along the loop $\tilde{C}$ can transform the rightmost diagram
shown in (54), before it was acted on by the Dehn twist, to the diagram shown
in the middle of the same equation. Remember that the Dehn twist along
$\tilde{C}$ on the surface should be extended into the interior of the
handlebody leading to the transformation from the third diagram to the second
in (54). Thus, Equation (54) shows an example to use Dehn twists to relate a
diagram with a $w_{i}$ label equal to $\psi$ to another diagram without such a
$w_{i}$ label. A similar relation can also be obtained for the second
configuration shown in Figure 10:

$\displaystyle\raisebox{-0.4pt}{\includegraphics[width=85.35826pt]{fig-
TubeDeformed-
Long-01.jpg}}~{}~{}\propto~{}~{}\raisebox{-0.4pt}{\includegraphics[width=85.35826pt]{fig-
TubeDeformed-Long-02.jpg}},$ (55)

where a Dehn twist along $\tilde{\tilde{C}}$ is performed. In fact, similar
procedures can be carried out on all of the configurations shown in Figure 10
(and their generalizations that are not depicted). Therefore, all of the
states $|\\{a_{i}\\},\\{b_{i}\\},\\{w_{i}\\}\rangle$ with some $w_{i}$ labels
equal to $\psi$ can be obtained from the states the states without such
$w_{i}$ labels, i.e., the states $|\\{a_{i}\\}\rangle$, by applying one or a
sequence of Dehn twists of the type shown above. Consequently, all the
operators $|{\rm vac}\rangle\langle\\{a_{i}\\},\\{b_{i}\\},\\{w_{i}\\}|$ and
$|\\{a_{i}\\},\\{b_{i}\\},\\{w_{i}\\}\rangle\langle{\rm vac}|$ can be obtained
from multiplying the operators of the form $|{\rm
vac}\rangle\langle\\{a_{i}\\}|$ or $|\\{a_{i}\\}\rangle\langle{\rm vac}|$ with
the unitary operators associated to the proper set of Dehn twists.

Having constructed all of the operators $|{\rm
vac}\rangle\langle\\{a_{i}\\},\\{b_{i}\\},\\{w_{i}\\}|$ and
$|\\{a_{i}\\},\\{b_{i}\\},\\{w_{i}\\}\rangle\langle{\rm vac}|$ (regardless of
the value of the $w_{i}$ labels) within $\mathbb{C}[\rho_{g}]$, we can simply
obtain via matrix multiplication operators of the more general form
$|\\{a_{i}^{\prime}\\},\\{b_{i}^{\prime}\\},\\{w_{i}^{\prime}\\}\rangle\langle\\{a_{i}\\},\\{b_{i}\\},\\{w_{i}\\}|$,
which form a complete basis for the full matrix algebra $\mathbb{M}_{N_{g}}$,
within $\mathbb{C}[\rho_{g}]$. At this point, we have completed the proof for
$\mathbb{C}[\rho_{g}]\cong\mathbb{M}_{N_{g}}$ and, hence, for the
irreducibility of the (projective) representation $\rho_{g}$ of the MCG for a
general $g$.

###  4.4 Duality to 2D Ising CFT

In Section 4.1, we proposed the expression (37) for the vacuum seed in terms
of the product of the holomorphic and anti-holomorphic vacuum conformal blocks
of the 2D Ising CFT. Based on this proposed vacuum seed, we proved the
finiteness of the “gravitational” modular sum (36) in Section 4.2 and obtained
the final expression (45) of the gravitational partition function $Z_{\rm
grav}$ up to a multiplicative constant in Section 4.3. We need to emphasize
that, in our discussion, the result (45) is purely a consequence of our
arguments for the vacuum seed $Z_{\rm vac}$ which were made from the gravity
bulk perspective, as well as of the mathematical results that we proved
including the finiteness of $\rho_{g}(\Gamma_{g})$ and the irreducibility of
the MCG representation $\rho_{g}$. In fact, even if $Z_{\rm vac}$ is not of
the form (37), as long as it be written as a product of a holomorphic and an
anti-holomorphic conformal block of the 2D Ising CFT (or even as a sum of
products of this type), we can still conclude the finiteness of the modular
sum (36) and further obtain the same expression (45) for $Z_{\rm grav}$, based
on our mathematical results, i.e., the finiteness of $\rho_{g}(\Gamma_{g})$
and the irreducibility of $\rho_{g}$.

The right hand side of (45) can also be naturally identified with the (full)
partition function of the 2D Ising CFT on the Riemann surface $\Sigma_{g}$
with period matrix $\Omega$. We therefore conclude that, at Brown-Henneaux
central charge $c=1/2$, and for genus $g$, the full gravitational partition
function $Z_{\rm grav}(\Omega,\bar{\Omega})$ with a genus-$g$ asymptotic
boundary $\Sigma_{g}$ is always proportional to the partition function of the
2D Ising CFT $Z_{\rm Ising}(\Omega,\bar{\Omega})$ on $\Sigma_{g}$:

$\displaystyle Z_{\rm
grav}(\Omega,\bar{\Omega})\propto\sum_{i=1}^{N_{g}}\chi_{i}(\Omega)\bar{\chi}_{i}(\bar{\Omega})=Z_{\rm
Ising}(\Omega,\bar{\Omega}).$ (56)

Figure 11: A schematic picture for pinching off a long cylinder.

At this point, we would like to come back to our proposed expression (37) for
the vacuum seed. In Section 4.1, we have already provided physical arguments
that suggest that (37) is a natural expression for the vacuum seed. Now, we
would like to further substantiate this proposal (37) by commenting on the
resulting gravitational partition function $Z_{\rm grav}$ (56). (56) is a
sensible result from the following perspectives. Firstly, the gravitational
partition function (56) for arbitrary genus $g$ is compatible with and is the
natural extension of the genus-one result obtained in Castro . Secondly, the
gravitational partition function (56), in the “pinching limits”, is self-
consistent and is consistent with the genus-one result obtained in Castro .
The pinching limit we focus on here is the limit of the period matrix $\Omega$
of the asymptotic boundary $\Sigma_{g}$ in which some part of the asymptotic
boundary $\Sigma_{g}$ is stretched into a very long cylinder and eventually
can be effectively viewed as pinched off. Figure 11 is a schematic picture for
pinching off a long cylinder. In the gravity context, such a pinching limit
has previously only been investigated, to the best of our knowledge, in semi-
classical gravity Yin1 ; Pinching . With the gravitational partition function
given by (56), we can now study the pinching limit of strongly coupled gravity
with Brown-Henneaux central charge $c=1/2$. In the pinching limit,
intuitively, we expect the genus of the asymptotic boundary to be effectively
reduced by 1. Hence, we expect a reduction of a gravitational partition
function with a genus-$g$ boundary to one with a genus-$(g-1)$ boundary. This
physical intuition is indeed consistent with (56), since the partition
function of 2D Ising CFT on a genus-$g$ surface indeed reduces to that on a
genus-$(g-1)$ surface in the pinching limit DVV .

Starting with a genus-$g$ asymptotic boundary, we can take successive pinching
limits such that the genus of the resulting asymptotic boundaries eventually
reduces to $g=1$. In this case, (56) implies that the gravitational partition
function eventually reduces, up an overall multiplicative constant, to the
product of genus-one partition functions of the 2D Ising CFT. This result is
again consistent with Castro . Here, we have provided general arguments for
the behavior of the gravitational partition function in the pinching limits
using (56) and using the behavior of the 2D Ising CFT partition in the same
limit. In Appendix C, we provide an example of analytic studies of the
pinching limit of the gravitational partition function with genus-two
asymptotic boundaries. Having provided arguments that substantiate the result
(56) (and thereby its starting point (37)), we would like to conclude that
based on a natural choice for the vacuum seed, we establish duality between 3D
AdS quantum gravity at Brown-Henneaux central charge $c=1/2$ and the 2D Ising
CFT using the all-genus partition functions.

Besides the gravitational partition function, the mathematical result that
$\rho_{g}$ is an irreducible projective representation of the MCG $\Gamma_{g}$
of the Riemann surface $\Sigma_{g}$ for any $g$ also has an interesting
implication purely for the 2D Ising CFT. It was proven in Cappelli  that, up
to a multiplicative constant, there is only one unique modular invariant
partition function that can be constructed using 2D Ising CFT conformal blocks
on a genus-one surface. To the best of our knowledge, there is no
generalization of such a proof to higher-genus surfaces in previous works. Our
result that $\rho_{g}$ is an irreducible representation of $\Gamma_{g}$
implies that, up to a multiplicative constant, for any fixed genus $g$, there
is always a unique partition function constructed from Ising-CFT conformal
blocks that is invariant under the MCG $\Gamma_{g}$ action on a genus-$g$
surface.

###  4.5 Difficulty in extending beyond the 2D Ising CFT

Solely based on the consideration of gravitational partition functions with
genus-1 asymptotic boundary, Castro et al. Castro  argued that, for Brown-
Henneaux central charge $c<1$, the only 2D CFTs that can be dual to pure
Einstein gravity in AdS3 at the corresponding Brown-Henneaux central charge
$c$ are the Ising and the Tricritical Ising CFTs of central charges $c=1/2$
and $c=7/10$. Our results obtained in the previous subsections on the all-
genus partition functions has established duality between 3D AdS quantum
gravity at Brown-Henneaux central charge $c=1/2$ and the 2D Ising CFT.

However, as we will show later in this subsection, the consideration of
higher-genus partition functions at $c=7/10$ reveals a fundamental difficulty
in establishing duality between 3D gravity at Brown-Henneaux central charge
$c=7/10$ and the Tricritical Ising CFT.

At Brown-Henneaux central charge $c=7/10$, we can follow the same reasoning as
in Section 4.1 to argue that the corresponding gravitational vacuum seed at
$c=7/10$ with a genus-$g$ asymptotic boundary $\Sigma_{g}$ should be
identified as the vacuum conformal block of the 2D Tricritical Ising CFT on
$\Sigma_{g}$. Hence, the modular sum (36) at $c=7/10$ is dictated by the MCG
$\Gamma_{g}$ representation $\rho_{g}^{\prime}$ that governs how the
holomorphic conformal blocks of the 2D Tricritical Ising CFT transform under
the action of $\Gamma_{g}$. In complete analogy with the connection between
the 2D Ising CFT and the 3D Ising TQFT, the information about the MCG
representation $\rho_{g}^{\prime}$ (which is associated with the holomorphic
conformal blocks of the 2D Tricritical Ising CFT) is fully contained in the 3D
Tricritical Ising TQFT, which can be mathematically described, equivalently,
by the chiral Tricritical Ising Modular Tensor Category (MTC) MooreSeiberg ;
Kirillov .

By inspection, this MTC contains a sub-Modular Tensor Category called
$\mathrm{Fib}$, with the so-called Fibonacci Fusion Rules262626The MTC
$\mathrm{Fib}$ has only two labels ($=$“simple objects” or “particles” or
“anyons” types) which, when denoted by $1$ and $x$, possess the Fusion Rules
$x\times x=1+x$, $1\times x=x\times 1=x$, $1\times 1=1$. Strictly speaking,
the MTC Fib that is needed here is the conjugate of the MTC Fib typically used
in the literature WangBook . (see, e.g., WangBook ). It follows from a very
general Theorem by Müger Muger  that the Tricritical Ising MTC at $c=7/10$
must then be the tensor product of the sub MTC $\mathrm{Fib}$ and the MTC
associated to the chiral Ising CFT. This factorization implies that the MCG
representation $\rho_{g}^{\prime}$ given by the Tricritical Ising MTC must be
a tensor product of an MCG representation given by the MTC $\mathrm{Fib}$ and
an MCG representation given by the MTC of the 2D chiral Ising CFT. Each of
these MCG representations mentioned here can be viewed as a map from the MCG
to a unitary group.

Finally, a fundamental Theorem by Freedman, Larsen and Wang Freedman  states
that the MCG representations given by the MTC $\mathrm{Fib}$ has an infinite
image set.272727In fact, this image set is dense in a unitary group, a result
that, as is well known, is related to the fundamental importance of the
$\mathrm{Fib}$ MTC for the subject of fault-tolerant quantum computation. It
then immediately follows that the image set of $\rho_{g}^{\prime}$ must also
be infinite, i.e., the set $\rho_{g}^{\prime}(\Gamma_{g})$, where $\Gamma_{g}$
is the Mapping Class Group of the genus-$g$ Riemann surface, is an infinite
set. This result implies that at Brown-Henneaux central charge $c=7/10$, the
modular sum of the gravitational partition function in (36) cannot be defined
for genus $g\geq 2$ because the sum occurring in this equation has an infinite
number of terms and cannot be naturally regularized, as discussed in Footnote
4.

###### Acknowledgements.

We thank Jing-Yuan Chen, Scott Collier, Xi Dong, Michael H. Freedman, Ori J.
Ganor, Thomas Hartman, Alexei Y. Kitaev, Thilo Kuessner, Xiao-Liang Qi, Shu-
Heng Shao, Douglas Stanford, Yong-Shi Wu and Xi Yin for helpful discussions on
various topics. H.-Y. S. thanks Semeon Artamonov, Filip Kos and Christopher
Paciorek for help with parallel computing. C.-M. J. is supported by the Gordon
and Betty Moore Foundations EPiQS Initiative through Grant GBMF4304. Z.-X. L.
and H.-Y. S. are supported in part by the National Science Foundation under
Grant No. NSF PHY-1748958 and by the Heising-Simons Foundation. Z. W. is
partially supported by NSF grant FRG-1664351. This work was supported in part
by the National Science Foundation under Grant No. DMR-1309667 (A.W.W.L.).
This work used the Savio computational cluster resource provided by the
Berkeley Research Computing program at the University of California, Berkeley.

##  Appendix A Genus one modular sum revisited

This appendix presents some new results concerning the genus-one case, which
aim to explain the mathematical meaning of the factor of eight in (19). We
will first introduce several necessary concepts.

As discussed in Bantay , for any 2D rational CFT $\mathcal{C}$ with a finite
set $\mathscr{I}$ of primaries, the field extension $F$ of $\mathbb{Q}$ by
adjoining the all matrix elements of the modular $\mathcal{S}$ matrix as in
(99) de Boer ; Gannon  is a subfield of a cyclotomic field
$\mathbb{Q}[\zeta_{n}]$ for some positive integer $n$, where $\zeta_{n}\equiv
e^{2\pi i/n}$ is the primitive $n^{\textbf{th}}$ root of unity, by the
Kronecker-Weber theorem in number theory. Following the terminology in
algebraic number theory, the smallest $n$ for which
$F\subseteq\mathbb{Q}[\zeta_{n}]$ is called the conductor of $\mathcal{C}$
(also defined in GukovVafa ), and can be shown Bantay  to be equal to the
order $N$ of modular $\mathcal{T}$ matrix. For the Ising CFT of interest to us
here, the modular $\mathcal{T}$ matrix is listed in (99).

Another important player for us is the kernel $\mathscr{K}$ of the linear
representation of $SL(2,\mathbb{Z})$, defined as the set of modular
transformations represented by the identity matrix,

$\mathscr{K}=\left\\{\gamma\in
SL(2,\mathbb{Z})|\,M_{ij}(\gamma)=\delta_{ij}\right\\},$ (57)

where $i,j\in\mathscr{I}$, and $M_{ij}(\gamma)$ is the representation matrix
of $\gamma$ transforming between characters:

$\chi_{i}\left(\gamma\cdot\tau\right)=\sum_{j}M_{ij}(\gamma)\chi_{j}(\tau).$
(58)

$\mathscr{K}$ can be shown Bantay ; NgSch  to be a congruence subgroup of
level $N$, whose meaning will be clear soon.

Now one can consider the index of the principal congruence subgroup
$\Gamma(N)$ of $SL(2,\mathbb{Z})$ of level $N$ inside the kernel
$\mathscr{K}$, i.e., the quantity $|\mathscr{K}:\Gamma(N)|$. $\Gamma(N)$ is
defined as282828For the principal congruence subgroup of a Siegel modular
group $Sp(2g,\mathbb{Z})$ of level $N$, the definition is the group of
diagonal matrices with entries being 1 mod $N$ Koecher ; Krieg .

$\Gamma(N)=\left\\{\begin{pmatrix}a&b\\\ c&d\\\ \end{pmatrix}\in
SL(2,\mathbb{Z})\Bigg{|}\,a,d\equiv 1\,(\text{mod}\,N),\,b,c\equiv
0\,(\text{mod}\,0)\right\\},$ (59)

the kernel of the homomorphism $\pi_{N}:SL(2,\mathbb{Z})\rightarrow
SL(2,\mathbb{Z}_{N})$ induced by the reduction modulo $N$ homomorphism
$\mathbb{Z}\rightarrow\mathbb{Z}_{N}$. With the above definition, a subgroup
of $SL(2,\mathbb{Z})$ is called a congruence subgroup of level $n$ if there
exists $N\geq 1$ such that it contains the principal congruence subgroup
$\Gamma(N)$, and and $n$ is the smallest such $N$.

Bantay then showed Bantay , using solely the knowledge of the modular
$\mathcal{S}$ matrix, that the conductor of the 2D Ising CFT is $N=48$, and he
further proved292929Bantay proved that the index $|\mathscr{K}:\Gamma(N)|$
equals the order of the image of $\mathscr{K}$ under a group homomorphism
$\mu_{N}:SL(2,\mathbb{Z})\rightarrow SL(2,\mathbb{Z})/\Gamma(N)$.  that the
index $|\mathscr{K}:\Gamma(48)|=64$.

On the other hand, the formula for the index of $\Gamma(N)$ inside
$SL(2,\mathbb{Z})$ is

$|SL(2,\mathbb{Z}):\Gamma(N)|=N^{3}\prod_{p|N}\left(1-\frac{1}{p^{2}}\right),$
(60)

where the product is (as indicated) over all prime numbers $p$ that divide the
level $N$. The above equals 73728 for $N=48$.

Then by the Lagrange’s theorem in group theory, we have

$|SL(2,\mathbb{Z}):\Gamma(48)|=|SL(2,\mathbb{Z}):\mathscr{K}|\cdot|\mathscr{K}:\Gamma(48)|,$
(61)

so one would naively expect that the index $|SL(2,\mathbb{Z}):\mathscr{K}|$
for the 2D Ising CFT should be $73728/64=1152$.

However, there is a subtlety: There exist three distinct ways of lifting a
projective representation of $SL(2,\mathbb{Z})$ to a linear representation
(i.e., three distinct “linearizations”), see for example Wang1 ; TenerWang .
Bantay considered a linear representation in Bantay , while we are focusing on
projective ones because a TQFT (in our case as discussed in Section 4.2, the
3D Ising TQFT, whose algebraic theory is described by the Ising MTC Wang1 )
gives rise to a projective representation of the mapping class group of a
Riemann surface (compare with (33) and (B)), partially due to the non-
degeneracy axiom for the modular $\mathcal{S}$ matrix of the TQFT Muger ;
Turaev ; Kirillov . This is also consistent with the (projective)
transformations of Jacobi theta functions under $SL(2,\mathbb{Z})$ mentioned
in Appendix B below. After linearization of the projective representation, the
order of the image of the $SL(2,\mathbb{Z})$ generator $T$ changes from $16$
to $48$. Taking into account this factor of three, we finally arrive at the
index $384=1152/3$, which agrees with the result from our Mathematica
code.303030The indices of the $SL(2,\mathbb{Z})$ subgroups which preserve only
one of the three characters $\chi_{1,1}$, $\chi_{1,2}$, or $\chi_{2,1}$ are
$384$, $384$ and $48$, respectively. If the characters are required to be
preserved only up to a $U(1)$ phase, then these indices become $24$, $24$ and
$3$, respectively.

Now we notice that $\mathscr{K}$ in Bantay  is not the enhanced symmetry group
$\Gamma_{c}^{[g=1]}$ for genus 1 discussed in Castro , since the latter is
defined as in (17), in a similar but different way than in (57), i.e., only
preserving the vacuum character up to a $U(1)$ phase. $\Gamma_{c}^{[g=1]}$
turns out to be an index-24 subgroup of $SL(2,\mathbb{Z})$, consistent with
29, and in Castro  Bantay’s $\mathscr{K}$ is merely used as an argument
justifying the finiteness of the genus one modular sum.

Up to now, our entire discussion is about genus one. Our final remark is that
the mathematical meaning of the prefactor 384 in (23) in the genus two case is
the analogue of the prefactor $8$ for the genus one partition function in
(19).313131The fact that the numerical prefactor “$384$” in (23) is the same
number as the above-mentioned index “$384$” $=1152/3$ is probably a
coincidence.

The factor $384$ arises because the subgroup $\Gamma_{c}^{[g=2]}$ of the
mapping class group $\Gamma_{g=2}$ is the immediate counterpart of the
subgroup $\Gamma_{c}^{[g=1]}$ of the mapping class group
$\Gamma_{g=1}=SL(2,Z)$, both subgroups being denoted by $\Gamma_{c}$ in (17)
as well as below (20).

##  Appendix B Generators for $Sp(4,\mathbb{Z})$ and the Algorithm

The group $Sp(4,\mathbb{Z})$ is minimally generated by $K$ and $L$ with the
following representations:

$K=\left(\begin{matrix}1&0&0&0\\\ 1&-1&0&0\\\ 0&0&1&1\\\
0&0&0&-1\end{matrix}\right),\quad L=\left(\begin{matrix}0&0&-1&0\\\
0&0&0&-1\\\ 1&0&1&0\\\ 0&1&0&0\end{matrix}\right).$ (62)

They satisfy $K^{2}=L^{12}=\mathbbm{1}_{4}$ and the following six additional
relations Bender ,

$\begin{split}&\left(L^{2}X\right)^{2}=\left(XL^{2}\right)^{2},\quad
L\left(L^{6}X\right)^{2}=\left(L^{6}X\right)^{2}L,\quad\left(KL^{5}\right)^{5}=\left(L^{6}X\right)^{2},\\\
&\left(KL^{7}KL^{5}K\right)L=LX,\quad\left(L^{2}KL^{4}\right)X=X\left(L^{2}KL^{4}\right),\quad\left(L^{3}KL^{3}\right)X=X(L^{3}KL^{3}),\end{split}$
(63)

with $X\equiv KL^{5}KL^{7}K=\mathbbm{1}_{2}\otimes\sigma_{x}$,
$L^{6}=\mathbbm{1}_{2}\otimes\sigma_{z}$, where $\sigma_{i}$ denote the
standard three Pauli matrices. Its generalization to arbitrary genus
$Sp(2g,\mathbb{Z})$ with at most 3 generators and $3g+5$ relations can be
found in Lu . In the following basis of Riemann theta functions

$\displaystyle\vartheta^{1/2}\left[\begin{matrix}0&0\\\
0&0\end{matrix}\right](\Omega),\,\vartheta^{1/2}\left[\begin{matrix}\frac{1}{2}&0\\\
0&0\end{matrix}\right](\Omega),\,\vartheta^{1/2}\left[\begin{matrix}0&0\\\
\frac{1}{2}&0\end{matrix}\right](\Omega),\,\vartheta^{1/2}\left[\begin{matrix}0&0\\\
\frac{1}{2}&\frac{1}{2}\end{matrix}\right](\Omega),\,\vartheta^{1/2}\left[\begin{matrix}0&\frac{1}{2}\\\
0&0\end{matrix}\right](\Omega),$ (64)
$\displaystyle\vartheta^{1/2}\left[\begin{matrix}\frac{1}{2}&0\\\
0&\frac{1}{2}\end{matrix}\right](\Omega),\,\vartheta^{1/2}\left[\begin{matrix}0&\frac{1}{2}\\\
\frac{1}{2}&0\end{matrix}\right](\Omega),\,\vartheta^{1/2}\left[\begin{matrix}0&0\\\
0&\frac{1}{2}\end{matrix}\right](\Omega),\,\vartheta^{1/2}\left[\begin{matrix}\frac{1}{2}&\frac{1}{2}\\\
0&0\end{matrix}\right](\Omega),\,\vartheta^{1/2}\left[\begin{matrix}\frac{1}{2}&\frac{1}{2}\\\
\frac{1}{2}&\frac{1}{2}\end{matrix}\right](\Omega),$

the projective representations of $K$ and $L$ are:

$\mathcal{K}=\begin{pmatrix}1&0&0&0&0&0&0&0&0&0\\\ 0&1&0&0&0&0&0&0&0&0\\\
0&0&0&1&0&0&0&0&0&0\\\ 0&0&1&0&0&0&0&0&0&0\\\ 0&0&0&0&0&0&0&0&1&0\\\
0&0&0&0&0&1&0&0&0&0\\\ 0&0&0&0&0&0&0&0&0&1\\\ 0&0&0&0&0&0&0&1&0&0\\\
0&0&0&0&1&0&0&0&0&0\\\ 0&0&0&0&0&0&1&0&0&0\\\
\end{pmatrix},\quad\mathcal{L}=\begin{pmatrix}0&0&1&0&0&0&0&0&0&0\\\
1&0&0&0&0&0&0&0&0&0\\\ 0&e^{\pi i/8}&0&0&0&0&0&0&0&0\\\ 0&0&0&0&0&0&0&0&e^{\pi
i/8}&0\\\ 0&0&0&1&0&0&0&0&0&0\\\ 0&0&0&0&1&0&0&0&0&0\\\ 0&0&0&0&0&e^{\pi
i/8}&0&0&0&0\\\ 0&0&0&0&0&0&1&0&0&0\\\ 0&0&0&0&0&0&0&1&0&0\\\
0&0&0&0&0&0&0&0&0&e^{3\pi i/8}\\\ \end{pmatrix},$ (65)

where $\mathcal{L}$ is of order 24 and $\mathcal{K}$ is of order 2.
$Z_{\text{vac}}^{\text{cl}}$ in (21) is invariant under the action of
$\mathcal{K}$, just like the torus vacuum seed $Z_{\text{vac}}$ is invariant
under $T$ of $SL(2,\mathbb{Z})$. Additionally, $Z_{\text{vac}}^{\text{cl}}$ is
also invariant under $\mathcal{L}^{6}$.

We ignore the factor det$(C\Omega+D)^{-1}$ in the modular transformation of
Riemann theta functions (33), because it is expected to be absorbed in the
overall quantum factor for the characters (69), similar to the Dedekind eta
function in the torus case BigYellowBook . The phase $\epsilon(\gamma)$ in
(33) has no effect because it is an overall factor independent of $\Omega$,
which drops out when taking the norm in the expression for
$Z_{\text{vac}}^{\text{cl}}.$

Below we present the pseudocode similar to those used in an arbitrary solvable
“word problem” for the MCG Primer . $\mathcal{K}[\,\cdot\,]$ or
$\mathcal{L}[\,\cdot\,]$ means that $\mathcal{K}$ or $\mathcal{L}$ acts on the
period matrix $\Omega$ in all $seed$, $seed1$ and $seed2$.

Algorithm 1

1:Inititalize $\,\mathcal{K},\,\mathcal{L},\,seed1:=\chi_{1,1}$

2:$seed=\mathcal{K}[seed1]$

3:for $n=0$, $n\leq 23$, $n++$ do

4: $temp=\mathcal{L}[seed]$

5: if $temp$ $\notin$ $seed$, then do nothing

6: else $seed=$Append [$seed$, $temp$]

7:$seed2=seed$

8:if Length [Intersection [$seed1$, $seed2$]]$<$Length [$seed2$] then
$seed1=seed2$, and repeat from 2 to 8

9:else stop

10:Print $Z_{\text{grav}}:=$Total [$seed2$]

Finally, we comment on the “translational” subgroup $\Gamma_{\infty}^{[g=2]}$
of $Sp(4,\mathbb{Z})$. The group gets its name from its genus-one counterpart,
where $\Gamma_{\infty}^{[g=1]}$ is generated by the translation
$T:\tau\rightarrow\tau+1$. (The superscripts $\cdot^{[g=1]}$ and
$\cdot^{[g=2]}$ specify the corresponding genus.) As in the genus one case,
the group $\Gamma_{\infty}^{[g=2]}$ is the “classical analogue” of
$\Gamma_{c}^{[g=2]}$. There is no canonical choice for
$\Gamma_{\infty}^{[g=2]}$ on genus two surfaces, but one possibility is
generated (not necessarily minimally) by

$\displaystyle\tilde{S}$
$\displaystyle=\left(L^{6}X\right)^{3},\quad\tilde{T}=XKL^{6}X,$ (66)
$\displaystyle T_{1}$ $\displaystyle=XT_{2}X,\quad
T_{2}=L^{-1}XL^{-2}XL^{-2},\quad T_{3}=L^{8}KL^{4}X\tilde{S},$

with the same $X$ as before. Here $T_{1}$, $T_{2}$ and $T_{3}$ respectively
shift the entries $\Omega_{11}$, $\Omega_{22}$ and $\Omega_{12}$ by 1; each of
$\tilde{S}$ and $\tilde{T}$ acting on $\Omega$ as in (28) performs the
conjugation $\Omega\rightarrow M\Omega M^{-1}$, where $M$ denotes $S$ or $T$
of $SL(2,\mathbb{Z})$ Yin1 . In the same basis (LABEL:eq:basis), the
10-dimensional projective representations of the generators listed in (66)
are:

$\tilde{\mathcal{T}}_{1}=\begin{pmatrix}0&0&1&0&0&0&0&0&0&0\\\ 0&e^{\pi
i/4}&0&0&0&0&0&0&0&0\\\ 1&0&0&0&0&0&0&0&0&0\\\ 0&0&0&0&0&0&0&1&0&0\\\
0&0&0&0&0&0&1&0&0&0\\\ 0&0&0&0&0&e^{\pi i/4}&0&0&0&0\\\ 0&0&0&0&1&0&0&0&0&0\\\
0&0&0&1&0&0&0&0&0&0\\\ 0&0&0&0&0&0&0&0&e^{\pi i/4}&0\\\
0&0&0&0&0&0&0&0&0&e^{\pi i/4}\\\
\end{pmatrix},\quad\tilde{\mathcal{T}}_{2}=\begin{pmatrix}0&0&0&0&0&0&0&1&0&0\\\
0&0&0&0&0&1&0&0&0&0\\\ 0&0&0&1&0&0&0&0&0&0\\\ 0&0&1&0&0&0&0&0&0&0\\\
0&0&0&0&e^{\pi i/4}&0&0&0&0&0\\\ 0&1&0&0&0&0&0&0&0&0\\\ 0&0&0&0&0&0&e^{\pi
i/4}&0&0&0\\\ 1&0&0&0&0&0&0&0&0&0\\\ 0&0&0&0&0&0&0&0&e^{\pi i/4}&0\\\
0&0&0&0&0&0&0&0&0&e^{\pi i/4}\\\ \end{pmatrix},$
$\tilde{\mathcal{T}}_{3}=\begin{pmatrix}1&0&0&0&0&0&0&0&0&0\\\
0&0&0&0&0&1&0&0&0&0\\\ 0&0&1&0&0&0&0&0&0&0\\\ 0&0&0&1&0&0&0&0&0&0\\\
0&0&0&0&0&0&1&0&0&0\\\ 0&1&0&0&0&0&0&0&0&0\\\ 0&0&0&0&1&0&0&0&0&0\\\
0&0&0&0&0&0&0&1&0&0\\\ 0&0&0&0&0&0&0&0&0&-i\\\ 0&0&0&0&0&0&0&0&-i&0\\\
\end{pmatrix},\,\,\tilde{\mathcal{S}}=\begin{pmatrix}1&0&0&0&0&0&0&0&0&0\\\