# Poincaré Series, 3d Gravity and Averages of Rational CFT

Viraj Meruliya  Sunil Mukhi  and Palash Singh

###### Abstract

We investigate the Poincaré series approach to computing 3d gravity partition
functions dual to Rational CFT. For a single genus-1 boundary, we show that
for certain infinite sets of levels, the SU(2)k WZW models provide unitary
examples for which the Poincaré series is a positive linear combination of two
modular-invariant partition functions. This supports the interpretation that
the bulk gravity theory (a topological Chern-Simons theory in this case) is
dual to an average of distinct CFT’s sharing the same Kac-Moody algebra. We
compute the weights of this average for all seed primaries and all relevant
values of $k$. We then study other WZW models, notably SU($N$)1 and SU(3)k,
and find that each class presents rather different features. Finally we
consider multiple genus-1 boundaries, where we find a class of seed functions
for the Poincaré sum that reproduces both disconnected and connected
contributions – the latter corresponding to analogues of 3-manifold
“wormholes” – such that the expected average is correctly reproduced.

##  1 Introduction and background

The problem of computing the partition function of pure Einstein gravity with
AdS3 boundary conditions was addressed in [1] using the method of Poincaré
sums (earlier work on these sums in related contexts can be found in [2], in
this context see also [3, 4]). One considers Euclidean gravity on manifolds
whose asymptotic boundary is a torus and assumes a semi-classical summation
formula. This is physically motivated by the existence of an
SL(2,$\mathbb{Z}$) family of Euclidean black holes, all of which have a torus
boundary, that are assumed to be the relevant saddle-points of a gravity path
integral. The result is a Poincaré sum whose first term is the contribution
from the “vacuum” spacetime, namely Euclidean AdS3, while the remaining terms
are modular transforms of this contribution.

Subsequently the results of [1] were streamlined and generalised in [5, 6]
among other works. One of the important generalisations was the fact that the
Poincaré series need not start with the vacuum contribution. Instead one can
start with the partition function for any “seed” primary and compute the
Poincaré sum over this. More generally the contributions from different seeds
can be added, leading to a large family of candidate AdS3 gravity partition
functions. Unfortunately the results in [1, 6] have certain undesirable
features that have proved very difficult, if not impossible, to eliminate even
allowing for such generalised seeds.

Following [1], it was proposed in [7] to examine whether a similar Poincaré
series approach could be used with a different starting point: the existence
of unitary minimal-model RCFT’s with central charge $\mathbf{c}<1$ and the
fact that their modular-transformation matrices $\gamma(a,b,c,d)$ are known
explicitly. One may define the gravity duals of such theories, though they
will not be semi-classical, by treating the identity character of the RCFT as
a “seed” primary and assuming its partition function to be the analogue of a
“vacuum contribution” to a gravity partition function. The rest of the
partition function is then obtained by summing a Poincaré series, which in
this case is a finite sum. The result of [7] is essentially a no-go statement
about unitary $(m,m+1)$ minimal models: beyond the first two models, the
gravity partition functions are not equal to CFT partition functions. Instead
one finds linear combinations of physical CFT partition functions in a few
cases, and for many values of $m$ one also gets linear combinations that
include unphysical modular invariants (in the sense of having negative
coefficients in their $q,{\bar{q}}$ expansion). Thus, it was concluded that
most unitary minimal models cannot be interpreted as being dual to AdS3
gravity111More recently it has been argued in [8] that studying the Poincaré
sum approach with genus-two boundaries provides additional constraints that
exclude even the second ($m=4$) minimal model..

In recent years there has been a realisation that in low dimensions, AdS
gravity may be dual not to a single CFT but to a weighted ensemble of CFT’s
[9, 10, 11, 12]. As briefly noted in [11], the result of [7] for unitary
minimal models might be an example of the same phenomenon. This motivates us
to examine the Poincaré sum for various classes of RCFT and try to understand
whether it can be interpreted in terms of either a single RCFT dual or an
average over finitely many RCFT’s. In the former case one can claim a (non-
semi-classical) AdS/CFT duality, while in the latter one can try to interpret
the dual as a weighted average over different physical CFT’s. The latter
interpretation requires in particular that the weights appearing in the
average be non-negative, so one has to look for classes where this is the
case. Next, one may go beyond the case of a single toroidal boundary since
additional consistency conditions can arise by considering 3-manifolds with
multiple disconnected torus boundaries, as well as higher-genus boundaries. In
particular, with multiple genus-1 boundaries the interpretation as an average
of CFT’s should work with the same weights as for single boundaries. We will
find examples that satisfy all of these requirements222We will not consider
higher-genus boundaries in this paper and will only comment briefly on this at
the end.. We expect that this study could teach us valuable lessons, in
relatively simple cases, of how averaging over CFT’s works in the AdS/CFT
correspondence.

Compared with the general (irrational) case, the Poincaré sum for RCFT is a
finite sum and has to be performed over a coset $\Gamma_{c}\backslash\Gamma$
where $\Gamma=$ SL(2,$\mathbb{Z}$) and $\Gamma_{c}$ is the finite-index
subgroup that keeps the seed contribution to the partition function invariant.
This needs to be worked out case by case and quickly becomes quite tedious.
Also the precise definition of $\Gamma_{c}$ depends on the seed rather than
being just a property of the characters. Hence we resort to a more practical
approach: we carry out our Poincaré sum over the coset
$\Gamma(N_{0})\backslash\Gamma$ where $\Gamma$ is the principal congruence
subgroup of SL(2,$\mathbb{Z}$), while $N_{0}$ is an integer that we will
define (it is similar, but not equal, to the “conductor” $N$ of an RCFT
defined in [13]). This is a single group for each set of characters,
independent of which primary is used as the seed, and is easily found just be
evaluating $N_{0}$. It is a subgroup of the true invariance group $\Gamma_{c}$
of the given seed partition function, and since it has finite index in
$\Gamma_{c}$ (which in turn has finite index in $\Gamma$) we will just get the
desired result of the Poincaré sum with an overall factor that can be removed
by a normalisation.

For the SU($N$)k WZW models the criteria for which one finds different numbers
of RCFT in the sum depend on $N$ and the level $k$. Depending on these values
we find precise criteria under which the Poincaré sum yields the partition
function of a single RCFT or a linear combination of physical RCFT’s. There
are infinitely many values of $k$ for which each of these possibilities is
realised. Where the Poincaré sum yields a sum of distinct partition functions,
we will find a method to compute the weights of this average as a function of
$N,k$, and work it out in the simplest cases where the average is over two
CFT’s. We will work things out in considerable detail for the unitary SU(2)k
WZW models, and subsequently discuss aspects of SU($N$)k for $N\geq 3$
333While discussing the “gravity duals” of WZW models, we will make some
tentative comments on what this should mean.. We also revisit the minimal
(unitary and non-unitary) series, for which the analysis is quite similar to
SU(2)k in many aspects.

We then move on to the Poincaré sum for multiple genus-1 boundaries. We will
find that for carefully chosen seeds, one finds answers such that the
partition function is an average over products of CFT partition functions with
the same weights, for any number of boundaries. In the irrational case this
type of setup implies the existence of “wormhole” contributions from manifolds
with multiple boundaries [14, 9, 15, 16] and we propose analogous “wormhole”
seeds in the RCFT context.

Let us now briefly review the approach of [1, 6] (henceforth referred to as
MWK) to the calculation of the semi-classical partition function of AdS3
Einstein gravity, as well as its generalisation to the RCFT case in [7]. The
MWK approach sums a seed over a set of manifolds denoted $M_{\gamma}$ where
$\gamma\in\Gamma=$ SL(2,$\mathbb{Z}$). Of these, $M_{\mathbf{1}}$ is thermal
AdS3 and the others are Euclidean BTZ black holes. The most obvious starting
point is to take thermal AdS3 as the seed. Because the BTZ black holes are
SL(2,$\mathbb{Z}$) transforms of thermal AdS3, we have:

$Z_{\gamma}(\tau,{\bar{\tau}})=Z_{\mathbf{1}}(\tau_{\gamma},{\bar{\tau}}_{\gamma})$
(1.1)

where:

$\gamma=\begin{pmatrix}a&b\\\
c&d\end{pmatrix}\in\Gamma,\qquad\tau_{\gamma}=\frac{a\tau+b}{c\tau+d}$ (1.2)

Thus one would naively write the gravity partition function as a Poincaré sum:

$Z(\tau,{\bar{\tau}})=\sum_{\gamma\in\Gamma}Z_{\gamma}(\tau,{\bar{\tau}})=\sum_{\gamma\in\Gamma}Z_{\mathbf{1}}(\tau_{\gamma},{\bar{\tau}}_{\gamma})$
(1.3)

However, SL(2,$\mathbb{Z}$) elements of the form:

$\gamma=\begin{pmatrix}1&n\\\ 0&1\end{pmatrix}$ (1.4)

leave $Z_{\mathbf{1}}$ invariant, and these generate a group isomorphic to
$\mathbb{Z}$. So the sum above is replaced by a Poincaré sum over coset
representatives:

$Z(\tau,{\bar{\tau}})=\sum_{\gamma\in\mathbb{Z}\backslash\Gamma}Z_{\mathbf{1}}(\tau_{\gamma},{\bar{\tau}}_{\gamma})$
(1.5)

The rest of the calculation proceeds by evaluating $Z_{\mathbf{1}}$ using the
action and symmetries of thermal AdS3, and then evaluating the sum. After
regularisation the result is finite but unfortunately has several undesirable
features including negativity of coefficients that should correspond to
degeneracies of states. A relatively recent work [17] noted a large-scale
occurence of negative coefficients in the $q$-series. One class of remedies
that has been proposed [6, 18, 19] is to cure the negativity by generalising
from the identity seed to a sum over suitably selected primary seeds. A
different remedy [20] is based on a relation (via Kaluza-Klein reduction) to
the analogous problem for AdS2, which suggests one should sum over additional
3-manifolds – however the compatibility of this proposal with modular
invariance is not yet clear.

To find gravity duals for rational CFT, the approach pioneered in [7], starts
from a slightly different perspective. This work considers the unitary
Virasoro minimal models with central charge $0<\mathbf{c}<1$. At such small
values of $\mathbf{c}$ the gravity dual (if such a concept has a meaning)
would have an AdS radius $\ell_{\rm AdS}=\frac{2G_{N}\mathbf{c}}{3}\sim G_{N}$
which is far from the semi-classical regime. So one gives up the idea of
summing over manifolds that are classical solutions. In its place, one
considers the partition function $Z_{\rm Id}(\tau,{\bar{\tau}})=|\chi_{\rm
Id}|^{2}$ coming from secondaries over the identity primary (after null
vectors are removed). Defining $\Gamma_{c}\subset\Gamma$ as the subgroup of
$\Gamma$ that preserves $Z_{\rm Id}$, one evaluates the Poincaré sum:

$Z(\tau,{\bar{\tau}})=\sum_{\gamma\in\Gamma_{c}\backslash\Gamma}Z_{\rm
Id}(\tau_{\gamma},{\bar{\tau}}_{\gamma})$ (1.6)

Unlike the semi-classical case this sum is finite, because for RCFT
$\Gamma_{c}$ is a finite-index subgroup of $\Gamma$ and therefore the coset
has a finite number of representative elements.

When applied to the $(m,m+1)$ Virasoro minimal models, the results of [7] are
as follows. For $m=3,4$ there are unique modular invariants built out of the
characters, and the Poincaré series Eq. (1.6) reproduces these up to an
overall factor (this is the unique possibility consistent with modular
invariance). But for $m\geq 5$ the Poincaré series for the identity seed
leads, in the first few cases, to a linear combination of the allowed physical
CFT partition functions [21, 22]. At larger values of $m$ one starts to
encounter unphysical partition functions (those whose coefficients in the
$q,{\bar{q}}$ series are either non-integral when the identity state is
normalised to unity, or non-positive, or both) and these appear in the
Poincaré sum 444We note in passing that the existence of modular-invariant
partition functions with negative integral coefficients seems to be a non-
holomorphic version of the “quasi-characters” extensively studied in [23, 24,
25]. It seems that negative integrality is a rather widespread feature in the
context of both holomorphic vector-valued modular forms and non-holomorphic
modular invariants..

In what follows, we examine the Poincaré series approach for rational CFT,
focusing primarily on SU($N$)k WZW models but also revisiting the case of
Virasoro minimal models. In Section 2 we explicitly evaluate the Poincaré sum
for a number of models, working successively with SU(2)k, SU($N$)1 and SU(3)k.
For the first two cases we develop an approach to compute the weights of the
resulting average over CFT’s as a function of $k,N$ respectively and identify
infinite families of models (and seed primaries) for which these weights are
non-negative. Thereafter we revisit the minimal models and compute the weights
of the average for cases where there are two physical invariants. In Section 3
we study the Poincaré series for multiple genus-1 boundaries, focusing on
SU(2)k. The conclusions are in Section 4, while the Appendices provide some
basic data on the relevant CFT’s as well as some additional tables with the
results of our calculations.

##  2 Poincaré Sums for a Single Genus-1 Boundary

###  2.1 Generic RCFT

Let us start by defining the Poincaré sum for an arbitrary RCFT. We start with
a set of holomorphic characters
$\chi_{0}(\tau),\chi_{1}(\tau),\allowbreak\cdots,\chi_{P-1}(\tau)$. These form
a vector-valued modular form (VVMF) under $\Gamma=$ SL(2,$\mathbb{Z}$). Thus,
under a modular transformation:

$\gamma=\begin{pmatrix}a&b\\\ c&d\end{pmatrix}\in\Gamma$ (2.1)

we have:

$\tau\to\tau_{\gamma}=\frac{a\tau+b}{c\tau+d}$ (2.2)

and:

$\chi_{i}(\tau_{\gamma})=\sum_{j=0}^{P-1}(M_{\gamma})_{ij}\chi_{j}(\tau)$
(2.3)

for some matrix $M_{\gamma}$. The above relation can be written more briefly
as $\chi(\tau_{\gamma})=M_{\gamma}\,\chi(\tau)$.

Each character has the behaviour:

$\chi_{i}(\tau)=q^{\alpha_{i}}\big{(}a_{0}^{(i)}+a_{1}^{(i)}q+a_{2}^{(i)}q^{2}+\cdots\big{)}$
(2.4)

for some rational exponents $\alpha_{i}$ and non-negative integers
$a_{n}^{(i)}$. In a unitary theory, $\alpha_{0}$ is the smallest of the
$\alpha_{i}$ and is identified with $-\frac{\mathbf{c}}{24}$ where
$\mathbf{c}$ is the central charge. The remaining $\alpha_{i}$ are then
identified with $h_{i}-\frac{\mathbf{c}}{24}$ where $h_{i}$ are the
holomorphic conformal dimensions.

Now we pick one of the characters, not necessarily the identity, and consider:

$Z_{ii}(\tau,{\bar{\tau}})\equiv|\chi_{i}(\tau)|^{2}$ (2.5)

We think of this as the partition function corresponding to the “spinless seed
primary” $i$ of dimension $(h_{i},h_{i})$. It is not modular-invariant by
itself. We can also allow non-diagonal primaries with dimensions
$(h_{i},h_{j})$ for $j\neq i$ as long as $h_{j}-h_{i}$ is an integer. The
corresponding seed partition function would be:

$Z_{ij}(\tau,{\bar{\tau}})\equiv{\overline{\chi}}_{i}({\bar{\tau}})\chi_{j}(\tau)+{\rm
cc}$ (2.6)

Next we take $Z_{\rm seed}$ to be any of the various seed primaries as above,
or a linear combination of them. The relevant Poincaré series is then:

$Z(\tau,{\bar{\tau}})\equiv\sum_{\gamma\in\Gamma_{c}\backslash\Gamma}Z_{\rm
seed}(\tau_{\gamma},{\bar{\tau}}_{\gamma})$ (2.7)

where $\Gamma_{c}$ is the subgroup of $\Gamma$ that leaves $Z_{\rm seed}$
invariant.

The result of the sum in Eq. (2.7) is manifestly modular invariant. Thus it
must be a linear combination of all allowed modular invariants for the given
set of characters. A basis for these is:

$Z_{J}(\tau,{\bar{\tau}})={\overline{\chi}}({\bar{\tau}})I_{J}\,\chi(\tau)\equiv\sum_{i,j=0}^{P-1}{\overline{\chi}}_{i}(I_{J})_{ij}\chi_{j}$
(2.8)

were $I_{J}$ is a basis of matrices that lead to modular-invariants $Z_{J}$
and $J$ runs over the number of independent matrices of this type. It follows
that:

$Z(\tau,{\bar{\tau}})=\sum_{J}c_{J}Z_{J}(\tau,{\bar{\tau}})$ (2.9)

where $c_{J}$ are some basis-dependent coefficients. Such invariants sometimes
have all positive coefficients in their $q,{\bar{q}}$ expansion. Then they can
correspond to the partition function of a CFT, but only if they satisfy some
additional requirements such as having a unique vacuum state. In this case we
refer to them as “physical”. Alternatively they may have some negative Fourier
coefficients, in which case they cannot correspond to a CFT.

Thus the possible results of performing a Poincaré sum are of three types: (I)
when the index $J$ only takes one value and the invariant corresponds to a
physical RCFT, we get the partition function of that RCFT, (II) when the index
$J$ ranges over multiple modular invariants that all correspond to physical
RCFT’s, we get a linear combination of partition functions of these different
RCFT’s, (III) when the index $J$ runs over both physical and unphysical
invariants, we get a result that cannot be interpreted as an average over
RCFT. Within type (II) there are two sub-classes: if all the $c_{J}\geq 0$,
the linear combination can be interpreted as a weighted average. This case can
be called type (IIa). On the other hand if at least one of the $c_{J}$ is
negative then the averaging interpretation is no longer tenable, this is
called (IIb). This classification applies independently for any $Z_{\rm
seed}$, which, as indicated above can be associated to a single seed primary
or a sum over seed primaries.

In view of our discussion above, we will consider types (I) and (IIa) to be
physically acceptable, with the latter corresponding to an average over RCFT’s
with definite weights, while types (IIb) and (III) will be rejected as
unphysical . Our primary goal is to explore the type (IIa) behaviour in
considerable detail in many examples, as well as to find general formulae for
the coefficients $c_{J}$ in families of theories where such behaviour holds.

It remains to find the appropriate coset to sum over when performing a
Poincaré sum for RCFT. As indicated above, it is sufficient to search for a
finite-index subgroup of $\Gamma$ that leaves the seed partition function
$Z_{\rm seed}$ invariant. This may be smaller than the largest possible such
group, that we have earlier called $\Gamma_{c}$, but this does not matter
since the result for the Poincaré sum will be the same up to an overall
factor.

We need a few definitions. The conductor $N$ of an RCFT is defined as the LCM
of the denominators of the exponents $\alpha_{i}$ defined in Eq. (2.4) above.
Then, it is well-known [13] that the principal congruence subgroup $\Gamma(N)$
leaves each character separately invariant555Various congruence subgroups,
among which we will make use of $\Gamma(N)$ and $\Gamma_{1}(N)$ in particular,
are defined in Appendix A.:

$\chi_{i}(\tau_{\gamma})=\chi_{i}(\tau),\quad\gamma\in\Gamma(N)$ (2.10)

In general, $\Gamma(N)$ is a proper subgroup of the kernel of the modular
representation (the largest subgroup of $\Gamma$ that preserves the
characters).

However, this result is too strong for us because we are interested in the
subgroup that preserves $Z_{ij}={\overline{\chi}}_{i}\chi_{j}$ (where
$|h_{i}-h_{j}|$ is integer) rather than $\chi_{i}$ itself. For this, we my
consider the larger subgroup $\Gamma(N_{0})$ where $N_{0}$ is defined as
follows. Picking $\alpha_{0}$ to be the most negative exponent (it will
eventually be identified with $-\frac{\mathbf{c}}{24}$) we define the
conformal dimensions associated to our characters by
$h_{i}=\alpha_{i}-\alpha_{0}$. Now we define $N_{0}$ to be the LCM of the
denominators of the $h_{i}$. We will refer to $N_{0}$ as the semi-conductor.

All elements of $\Gamma(N_{0})$ preserve each character up to an overall
phase, which implies that $\Gamma(N_{0})$ is contained in $\Gamma_{c}$. The
argument is as follows. The proof of [13] that $\Gamma(N)$ preserves all
characters makes use of the standard action of the generators $T,S$ of
SL(2,$\mathbb{Z}$) on the characters. Of these, $T$ acts as a phase:

$T:\quad\chi_{i}\to e^{2\pi i\alpha_{i}}\chi_{i}$ (2.11)

where $\alpha_{i}=-\frac{\mathbf{c}}{24}+h_{i}$. Now we can modify the $T$
generator by defining:

$T^{\prime}=\omega T$ (2.12)

where $\omega=e^{\frac{2\pi i\mathbf{c}}{24}}$. Then $T^{\prime}$ and $S$ also
form a representation of SL(2,$\mathbb{Z}$), and we can find the subgroup that
preserves the characters under this action. By the same arguments as before,
this group will be $\Gamma(N_{0})$. It follows that the subgroup that
preserves characters upto a common phase, in the usual action, is
$\Gamma(N_{0})$. Since the seed partition function is blind to this phase,
$\Gamma(N_{0})$ is therefore a subset of $\Gamma_{c}$. It may be noted that
$N_{0}$ divides the conductor $N$, and therefore $\Gamma(N_{0})$ is a superset
of $\Gamma(N)$.

For diagonal seeds of the form $Z_{ii}=|\chi_{i}|^{2}$ one can find a larger
invariance group than $\Gamma(N_{0})$. For this one uses the fact that $T$ is
not an element of $\Gamma(N_{0})$. However it multiplies each character by a
(distinct) phase, so it preserves every seed partition function. $T$ generates
a hyperbolic subgroup of $\Gamma$, of which the independent elements that are
not in $\Gamma(N_{0})$ are $T,T^{2},\cdots,T^{N_{0}-1}$. We can now combine
$T$ with $\Gamma(N_{0})$ and find the resulting group, which turns out to be
$\Gamma_{1}(N_{0})$ (this is defined in Appendix A). This follows from the
inclusions $\Gamma(N_{0})\subset\Gamma_{1}(N_{0})$ and
$T^{n}\in\Gamma_{1}(N_{0})$, from which we see that the combination has at
least $N_{0}$ times the number of elements of $\Gamma(N_{0})$. Because
$[\Gamma_{1}(N_{0}):\Gamma(N_{0})]=N_{0}$ it also has at most this number of
elements, therefore it must be precisely the congruence subgroup
$\Gamma_{1}(N_{0})$.

In what follows, we will work with both $\Gamma(N_{0})$ and
$\Gamma_{1}(N_{0})$ at different points. The former group has the advantage of
being normal, so left- and right-cosets are the same. Some of the calculations
are also simpler. However the latter is a larger group and therefore has a
smaller index in $\Gamma$. As a result the calculation of the Poincaré sum has
fewer terms and this economy is helpful when working with larger values of
$N_{0}$.

Thus, we finally define the Poincaré sum for an RCFT as:

$Z(\tau,{\bar{\tau}})\equiv\frac{1}{{\cal N}}\sum_{\gamma\in\Gamma_{\rm
sub}\backslash\Gamma}Z_{\rm seed}(\tau_{\gamma},{\bar{\tau}}_{\gamma})$ (2.13)

where $\Gamma_{\rm sub}$ will be either $\Gamma(N_{0})$ or
$\Gamma_{1}(N_{0})$. In each case this will be a finite sum. Here ${\cal N}$
is a normalisation factor such that the result has a leading term normalised
to unity. Though our notation does not show this explicitly, the normalisation
factor will depend on the seed. Note that depending on the choice of
$\Gamma_{\rm sub}$, the values of the coefficients $c_{I}$ in Eq. (2.8) will
be scaled by an overall constant.

Let us now describe the method to derive the coefficients $c_{J}$ (this was
used in [7] for specific minimal models). Consider a general seed for the
Poincaré series. This can be written as:

$Z_{\rm seed}(X,\tau,{\bar{\tau}})={\overline{\chi}}({\bar{\tau}})\,X_{\rm
seed}\,\chi(\tau)$ (2.14)

where $X_{\rm seed}$ is a $P\times P$ matrix. Now under a modular
transformation we have:

$Z_{\rm
seed}(X,\tau_{\gamma},{\bar{\tau}}_{\gamma})=\bar{\chi}({\bar{\tau}}_{\gamma})\,X_{\rm
seed}\,\chi(\tau_{\gamma})=\bar{\chi}({\bar{\tau}})\,(M^{\dagger}_{\gamma}X_{\rm
seed}M_{\gamma})\,\chi(\tau)$ (2.15)

The result of performing a Poincaré sum over $Z_{\rm seed}$ will be denoted
$Z(X,\tau,{\bar{\tau}})$ to highlight its dependence on the seed $X$ (this is
the modular invariant partition function that in other contexts we just called
$Z(\tau,{\bar{\tau}})$). Then:

$Z(X,\tau,{\bar{\tau}})=\sum_{\gamma}Z_{\rm
seed}(X,\tau_{\gamma},{\bar{\tau}}_{\gamma})=\bar{\chi}({\bar{\tau}})\,\Big{(}\sum_{\gamma}M^{\dagger}_{\gamma}X_{\rm
seed}M_{\gamma}\Big{)}\,\chi(\tau)$ (2.16)

Thus, the modular-invariant matrix $\sum_{\gamma}M^{\dagger}_{\gamma}X_{\rm
seed}M_{\gamma}$ determines what linear combinations of invariants will appear
in the final answer. Referring to Eq. (2.8), we have:

$\sum_{\gamma}M^{\dagger}_{\gamma}X_{\rm
seed}M_{\gamma}=\sum_{J=1}^{D}\,c_{J}\,I_{J}$ (2.17)

Now, we can define an inner product on the space of matrices as:

$d_{JK}=\Tr(I_{J}I_{K})$ (2.18)

Using this in the above equation, we have:

$\begin{split}\sum_{J}d_{KJ}\,c_{J}&=\sum_{\gamma}\mbox{Tr}(I_{K}M^{\dagger}_{\gamma}X_{\rm
seed}M_{\gamma})\\\ &=\sum_{\gamma}\mbox{Tr}(I_{K}X_{\rm seed})\\\
&=\left|\Gamma_{\rm sub}\backslash\Gamma\right|\,\mbox{Tr}(I_{K}X_{\rm
seed})\end{split}$ (2.19)

where in the intermediate step we used modular invariance of the $I_{J}$. This
matrix equation can be inverted to give:

$c_{J}=\left|\Gamma_{\rm
sub}\backslash\Gamma\right|\,\sum_{K}d^{-1}_{JK}\Tr(I_{K}X_{\rm seed})$ (2.20)

These are the desired coefficients which, after normalisation, should be
interpreted as the weights with which we average different CFT’s.

###  2.2 SU(2)k WZW Models

We now show that in the unitary SU(2)k WZW models there are infinitely many
realisations of both type (I) and type (IIa) behaviour as described in the
previous section. In the latter case we will be able to compute the
coefficients $c_{J}$ for generic models within such families and classify the
cases where they are non-negative.

Before embarking on the calculation let us briefly remark on the
generalisation from Virasoro to current-algebra minimal models. By generic
rules of the AdS/CFT correspondence, this should mean that the corresponding
gravity theory will have gauge fields for the corresponding Lie algebra, in
this case SU(2). So at first one might think the gravity theory should be
Einstein gravity coupled to a Yang-Mills field. However, it has been suggested
in [11] in a related case, that where a current algebra is present in the CFT,
the dual theory could be pure Chern-Simons theory. The logic is that in the
boundary theory the energy-momentum tensor is not an independent field but a
composite of the currents via the Sugawara construction, hence the bulk should
have no independent metric field, but only gauge fields that are the
counterparts of boundary currents. Since the bulk theory is topological we do
have a general coordinate invariant theory, something like a gravity theory
without gravitons. This in turn is not so strange, given that AdS3 Einstein
gravity anyway has no local graviton degrees of freedom in the bulk.

Applying that logic here, the bulk dual for the SU($N$)k WZW model should be
two copies of pure SU($N$) Chern-Simons theory, one representing left-movers
and the other representing right-movers of the CFT. Denoting the gauge fields
by matrix valued 1-forms as ${\mathbf{A}}={\mathbf{A}}_{\mu}\,dx^{\mu},{\mkern
1.5mu\overline{\mkern-1.5mu\mathbf{A}\mkern-1.5mu}\mkern 1.5mu}={\mkern
1.5mu\overline{\mkern-1.5mu\mathbf{A}\mkern-1.5mu}\mkern
1.5mu}_{\mu}\,dx^{\mu}$, the action is then:

$S_{\rm bulk}=\frac{k}{4\pi}\left(\int\tr({\mathbf{A}}\wedge
d{\mathbf{A}}+{\textstyle\frac{2}{3}}{\mathbf{A}}\wedge{\mathbf{A}}\wedge{\mathbf{A}})-\int\tr({\mkern
1.5mu\overline{\mkern-1.5mu\mathbf{A}\mkern-1.5mu}\mkern 1.5mu}\wedge d{\mkern
1.5mu\overline{\mkern-1.5mu\mathbf{A}\mkern-1.5mu}\mkern
1.5mu}+{\textstyle\frac{2}{3}}{\mkern
1.5mu\overline{\mkern-1.5mu\mathbf{A}\mkern-1.5mu}\mkern 1.5mu}\wedge{\mkern
1.5mu\overline{\mkern-1.5mu\mathbf{A}\mkern-1.5mu}\mkern 1.5mu}\wedge{\mkern
1.5mu\overline{\mkern-1.5mu\mathbf{A}\mkern-1.5mu}\mkern 1.5mu})\right)$
(2.21)

This difference action is well-known to lead to a parity-conserving theory
with an SU($N$)${}_{k}\times$ SU($N$)k current algebra on its boundary, and
has found applicability in the context of multiple membranes in M-theory as
well as more generally in the context of a novel Higgs mechanism in 3d (for a
review and references to the original works, see [26]). The novelty here is
that we are not simply considering the path integral on a fixed 3-manifold,
but performing a Poincaré sum that is to be thought of as the analogue (for
small $\ell_{\rm AdS}$) of a sum over 3-manifolds.

Now we turn to a discussion of $SU(2)_{k}$ WZW models in the present context.
The basic features of the models are summarised in Appendix B and we can
perform the Poincaré sums using this data. The main information required is
the form taken by the $S$ and $T$ matrices when acting on the characters
$\chi_{\lambda}$ (note that we are now labelling the characters by
$\lambda=2j+1$, the multiplicity of the isospin $j$ representation, which goes
from 1 to $k+1$).

We will find that the prime factors of the shifted level $n=k+2$, also known
as the “height”, determine the behaviour of the Poincaré sum in the
corresponding model. It is easily verified that the semi-conductor, defined in
Section 2.1, takes the simple form $N_{0}=4n$, except for $k=1$ in which case
$N_{0}=4$. We started by calculating the Poincaré sums for the seed
corresponding to each individual character for the values $k=1,2,\dots,41$
using SageMath [27]. The results are given in Table LABEL:SU2fulltable in
Appendix C.

In general the results of the Poincaré sum are linear combinations of multiple
modular invariants. As shown in [21, 22], these consist of the following
physical invariants: $Z_{A}$ which exists for all $n$, $Z_{D}$ which exists
for all even $n$, and $Z_{E}$ which exists for $n=12,18,30$, in addition to
unphysical invariants that exist for various values of $n$. Where our answer
involves only physical invariants we write it as
$c_{A}Z_{A}+c_{D}Z_{D}+c_{E}Z_{E}$ where any one or two of the coefficients
$c_{A},c_{D},c_{E}$ may vanish. These coefficients are only defined up to an
overall normalisation in view of the fact that we summed over cosets of
$\Gamma$ by a group $\Gamma(N_{0})\subset\Gamma_{c}$.

We make the following observations about the results in this Table:

  * •

Within a given model, several distinct seeds give the same linear combination
of invariants.

  * •

The vacuum primary seed always leads to a linear combination involving non-
negative integers unless there are unphysical invariants at the given level.

  * •

For $n=4\rho,(\rho\geq 2)$, the ratio of coefficients $\frac{c_{D}}{c_{A}}=1$
for the vacuum seed.

  * •

For $n=4\rho+2,(\rho\geq 1)$, the ratio of coefficients
$\frac{c_{D}}{c_{A}}=-\frac{1}{2}$ for the seed $\chi_{2}$.

  * •

For certain values of $n$ and for specific seeds, one finds only $Z_{A}$ or
only $Z_{D}$ as the answer.

We will now explain these observations and generalise them to arbitrary values
of $n=k+2$. For this we briefly review some details of the results of [21,
22]. Modular invariants made of the SU(2)k characters at any fixed $k$ can be
written as follows:

$Z_{\delta}=\frac{1}{2}\sum_{\lambda,\lambda^{\prime}=1}^{2n}\bar{\chi}_{\lambda}\,\left(\Omega_{\delta}\right)_{\lambda\lambda^{\prime}}\,\chi_{\lambda^{\prime}}$
(2.22)

where $\delta$ is a positive integer that divides $n$. The range of indices on
the characters and the matrices $\Omega_{\delta}$ has been doubled, with a
constraint: they are defined modulo $2n$ and obey the relations
$\chi_{\lambda}=\chi_{\lambda+2n}=-\chi_{-\lambda}$. These relations imply in
particular that $\chi_{n}=0$.

The matrices $\Omega_{\delta}$ are as follows. For a given $\delta$ we define
$\alpha=[\delta,n/\delta]$ (the GCD of the two integers) and $\omega^{2}=1$
mod $\frac{4n}{\alpha^{2}}$. Then:

$\left(\Omega_{\delta}\right)_{\lambda\lambda^{\prime}}=\begin{cases}0,\hskip
96.73918pt\alpha\nmid\lambda\mbox{ or }\alpha\nmid\lambda^{\prime}\\\
\sum_{\xi=0}^{\alpha-1}\delta_{\lambda^{\prime},\,\omega\lambda+2\xi
n/\alpha},\quad\mbox{otherwise}\end{cases}$ (2.23)

Not all the $Z_{\delta}$ are linearly independent. For example, for $\delta=n$
we have $\alpha=1,\,\omega=1$, so
$(\Omega_{n})_{\lambda\lambda^{\prime}}=\delta_{\lambda\lambda^{\prime}}$,
while for $\delta=1$ we have $\alpha=1,\,\omega=-1$, so
$(\Omega_{1})_{\lambda\lambda^{\prime}}=\delta_{\lambda,-\lambda^{\prime}}$.
This means that $Z_{n}=-Z_{1}$. In general, $\Omega_{\delta}$ and
$\Omega_{n/\delta}$ will have the same value of $\alpha$ but $\omega$ changes
sign to $-\omega$. Hence the non-zero matrix elements of $\Omega_{n/\delta}$
can be written as:

$\left(\Omega_{n/\delta}\right)_{\lambda\lambda^{\prime}}=\sum_{\xi=0}^{\alpha-1}\delta_{\lambda^{\prime},\,-\omega\lambda+2\xi
n/\alpha}\quad(\alpha|\lambda\mbox{ and }\alpha|\lambda^{\prime})$ (2.24)

Changing variables by $\xi=\alpha-\xi^{\prime}$ we have that

$\left(\Omega_{n/\delta}\right)_{\lambda\lambda^{\prime}}=\sum_{\xi^{\prime}=1}^{\alpha}\delta_{\lambda^{\prime},\,-\omega\lambda-2\xi^{\prime}n/\alpha+2n}=\sum_{\xi^{\prime}=0}^{\alpha-1}\delta_{\lambda^{\prime},\,-\omega\lambda-2\xi^{\prime}n/\alpha}\quad(\alpha|\lambda\mbox{
and }\alpha|\lambda^{\prime})$ (2.25)

So, when we compute the partition function, we find:

$Z_{n/\delta}=\frac{1}{2}\sum_{\lambda=1}^{2n}\sum_{\xi^{\prime}=0}^{\alpha-1}\bar{\chi}_{\lambda}\,\chi_{-\omega\lambda-2\xi^{\prime}n/\alpha}=-\frac{1}{2}\sum_{\lambda=1}^{2n}\sum_{\xi^{\prime}=0}^{\alpha-1}\bar{\chi}_{\lambda}\,\chi_{\omega\lambda+2\xi^{\prime}n/\alpha}=-Z_{\delta}$
(2.26)

What we saw above for the case of $\delta=1,n$ was a special case of this
linear relation.

To write a formula for the number of invariants, define the divisor function:

$\sigma(n)=\sum_{\delta|n}1$ (2.27)

which counts the number of positive integers that divide $n$. Now if
$\sigma(n)$ is even then each divisor $\delta$ has a distinct counterpart
$\frac{n}{\delta}$ and in this case there are $\frac{\sigma(n)}{2}$
invariants. However if $\sigma(n)$ is odd then $\delta$ and $\frac{n}{\delta}$
must be the same for one value of $\delta$, say $\delta=m$. This means that
$n=m^{2}$. Using the relation (2.26), we have that $Z_{m}=-Z_{m}$, so
$Z_{m}=0$. Therefore, for odd values of $\sigma(n)$, we have only
$(\sigma(n)-1)/2$ linearly independent partition functions.

To summarise, the number of independent modular invariants is given by:

$\mbox{No. of linearly independent
}Z_{\delta}=\begin{cases}\frac{\sigma(n)}{2},\hskip 22.76228pt\sigma(n)\mbox{
even}\\\ \frac{\sigma(n)-1}{2},\quad\sigma(n)\mbox{ odd}\\\ \end{cases}$
(2.28)

We now consider those heights $n$ that give rise to $1,2$ or 3 modular
invariants. In the latter two cases one must additionally ensure that there
are no unphysical invariants. In these cases the result of the Poincaré sum is
either a CFT or an average over CFT’s. Once there are four or more invariants,
at least one is necessarily unphysical and the averaging interpretation no
longer works.

#### One Physical Invariant

First let us ask when there is a unique modular invariant. From (2.28), we
know that this happens when $\sigma(n)=2$, which means $n=p$ for some prime
$p$, or $\sigma(n)=3$ in which case $n=p^{2}$ for some prime $p$. In these
cases the unique modular invariant function is the one corresponding to
$\Omega_{n}$ (defined in Eq. (2.23)) and it is the diagonal invariant $Z_{A}$.
When this is the case, the Poincaré sums that we compute must necessarily be
proportional to $Z_{A}$, regardless of the seed. The central charges for these
cases are of the form:

$\mathbf{c}=3-\frac{6}{p}\quad\hbox{or}\quad\mathbf{c}=3-\frac{6}{p^{2}}$
(2.29)

for some prime $p$. In Table LABEL:ppsq we list the first few examples of
models which fall under this category. The models are labelled by the height
$n=k+2$.

Table 1: Poincaré sums for $SU(2)_{k}$ WZW models with unique modular invariant Level $(n,\mathbf{c},N_{0})$ |  Poincaré sum
---|---
$(3,1,12)$ |  $24\,Z_{A}$
$(4,\frac{3}{2},16)$ |  $32\,Z_{A}$
$(5,\frac{9}{5},20)$ |  $36\,Z_{A}$
$(7,\frac{15}{7},28)$ |  $24\,Z_{A}$
$(9,\frac{7}{3},36)$ |  $54\,Z_{A}$
$(11,\frac{27}{11},44)$ |  $54\,Z_{A}$
$(13,\frac{33}{13},52)$ |  $84\,Z_{A}$
$(17,\frac{45}{17},68)$ |  $108\,Z_{A}$
$(19,\frac{51}{19},76)$ |  $120\,Z_{A}$
$(23,\frac{63}{23},92)$ |  $144\,Z_{A}$
$(25,\frac{69}{25},100)$ |  $150\,Z_{A}$
$(29,\frac{81}{29},116)$ |  $180\,Z_{A}$
$(31,\frac{87}{31},124)$ |  $192\,Z_{A}$
$(37,\frac{105}{37},148)$ |  $228\,Z_{A}$
$(41,\frac{117}{41},164)$ |  $252\,Z_{A}$
$(43,\frac{123}{43},172)$ |  $264\,Z_{A}$
|

Thus, these Poincaré sums are candidate partition functions for a
gravitational theory dual to the diagonal SU(2)k theory. It is intriguing
that, unlike the Virasoro minimal models, here we find an infinite family of
unitary models that pass the first test to be dual to a gravity theory.

#### Two Physical Invariants

Next we consider the case when there are exactly two modular invariants. From
(2.28) we see that this is the case when $\sigma(n)=4$ or $5$. For the former
case, we have $n=pq$ or $n=p^{3}$ where $p,q$ are primes, while for the latter
we have $n=p^{4}$. Since the diagonal invariant $Z_{A}$ always exists, this
tells us there is exactly one more invariant for all these values of $n$.
However, the second invariant is not necessarily physical. In fact, [21, 22]
tells us that whenever there are two physical invariants, they must be as
follows:

$Z_{A}={\overline{\chi}}\,\Omega_{n}\chi,\qquad
Z_{D}={\overline{\chi}}(\Omega_{2}+\Omega_{n})\chi$ (2.30)

Therefore we must consider only those sub-cases of $n=pq,p^{3},p^{4}$ for
which 2 divides $n$ (so that $\Omega_{2}$ can exist). These consist of one
infinite family with $n=2p$ and two sporadic cases with $n=8,16$. Clearly
these cases have no unphysical invariants. In Table LABEL:2p we list several
examples of models which fall into this category, displaying the result of the
Poincaré sum for all possible diagonal seeds of the form:

$(X_{\rm
seed})_{\lambda,\lambda^{\prime}}=\delta_{\lambda\lambda_{0}}\delta_{\lambda^{\prime}\lambda_{0}}$
(2.31)

(there can also be non-diagonal seeds corresponding to primaries with spin,
these will come up later). The identity seed corresponds to $\lambda_{0}=1$.
In the Table, the seed will be labelled by its value of $\lambda_{0}$.

Table 2: Poincaré Sums for $SU(2)_{k}$ WZW model with two modular invariants Data $(n,\mathbf{c},N_{0})$ |  Value of $\lambda_{0}$ |  Poincaré sum
---|---|---
$(6,2,20)$ |  $1,5$ |  $8\,(4Z_{A}+Z_{D})$
|  $2,4$ |  $32\,(2Z_{A}-Z_{D})$
|  $3$ |  $48\,Z_{D}$
$(10,\frac{12}{5},40)$ |  $1,3,7,9$ |  $24\,(2Z_{A}+Z_{D})$
|  $2,4,6,8$ |  $48\,(2Z_{A}-Z_{D})$
|  $5$ |  $96\,Z_{D}$
$(14,\frac{18}{7},56)$ |  $1,3,5,9,11,13$ |  $8\,(8Z_{A}+5Z_{D})$
|  $2,4,6,8,10,12$ |  $64\,(2Z_{A}-Z_{D})$
|  $7$ |  $144\,Z_{D}$
$(22,\frac{30}{11},88)$ |  ${\rm odd},\neq 11$ |  $24\,(4Z_{A}+3Z_{D})$
|  even |  $96\,(2Z_{A}-Z_{D})$
|  $11$ |  $240\,Z_{D}$
$(26,\frac{36}{13},104)$ |  ${\rm odd},\neq 13$ |  $8\,(14Z_{A}+11Z_{D})$
|  even |  $112\,(2Z_{A}-Z_{D})$
|  $13$ |  $288\,Z_{D}$
$(34,\frac{48}{17},136)$ |  ${\rm odd},\neq 17$ |  $24\,(6Z_{A}+5Z_{D})$
|  even |  $144\,(2Z_{A}-Z_{D})$
|  $17$ |  $384\,Z_{D}$
$(38,\frac{54}{19},152)$ |  ${\rm odd},\neq 19$ |  $8\,(20Z_{A}+17Z_{D})$
|  even |  $160\,(2Z_{A}-Z_{D})$
|  $19$ |  $432\,Z_{D}$
|  |

We see in Table LABEL:2p that the linear combinations appearing for the case
when the vacuum is the seed primary have positive coefficients for $Z_{A}$ and
$Z_{D}$. From the perspective of AdS/CFT, this means that the gravity
partition function could plausibly be interpreted as an average of the
partition functions of an ensemble of physical CFTs. Here the ensemble has
just 2 elements, a consequence of our choice of symmetry algebra. If we write
the Poincare sum as $c_{A}Z_{A}+c_{D}Z_{D}$, then the relative size of the
coefficients $c_{A}$ and $c_{D}$, provides the measure for how to average over
the two theories.

Following the discussion in the previous section, we now derive the
coefficients $c_{A},c_{D}$, namely the linear combinations of $Z_{A}$ and
$Z_{D}$ that appear in the last column of Table LABEL:2p for any seed. This
gives a successful prediction of all the results in Table LABEL:2p and extends
them to $n=2p$ for arbitrary prime $p$.

Referring to Eq. (2.20), we see that the matrix $d_{JK}$ is a $2\times 2$
matrix in the present case, and is readily inverted. First, we need to find
the relevant matrices $I_{J}$. Recall from the discussion around Eq. (2.30)
that $\Omega_{n}$ and $\Omega_{2}$ exist. Let us now look at the modular
invariant functions that we get from these matrices. We start with
$\Omega_{n}$, so $\alpha=1$ and $\omega=1$ and the invariant is:

$Z_{n}=\frac{1}{2}\sum_{\lambda,\lambda^{\prime}=1}^{2n}\bar{\chi}_{\lambda}(\Omega_{n})_{\lambda\lambda^{\prime}}\chi_{\lambda^{\prime}}=\frac{1}{2}\sum_{\lambda,\lambda^{\prime}=1}^{2n}\bar{\chi}_{\lambda}\delta_{\lambda\lambda^{\prime}}\chi_{\lambda^{\prime}}=\frac{1}{2}\sum_{\lambda=1}^{2n}\absolutevalue{\chi_{\lambda}}^{2}=\sum_{\lambda=1}^{n-1}\absolutevalue{\chi_{\lambda}}^{2}$
(2.32)

where we have used $\chi_{\lambda}=-\chi_{-\lambda}=\chi_{\lambda+2n}$. Thus
we see that $\Omega_{n}$ when acting on the restricted space
($1\leq\lambda\leq n-1$) still operates as the identity matrix. Hence
$(I_{n})_{\lambda\lambda^{\prime}}=\delta_{\lambda\lambda^{\prime}}$, where
$\lambda,\lambda^{\prime}$ takes values from $1$ to $n-1$. Next we consider
$\Omega_{2}$. Here $\alpha=[2,p]=1$, since $n=2p$ and $p>2$. We need to chose
$\omega^{2}=1$ mod $8p$. For this, we can chose $\omega=2p-1=n-1$ which is
allowed since $\omega^{2}=4(p-1)p+1=1$ mod $8p$. The associated modular
invariant is:

$\begin{split}Z_{2}&=\frac{1}{2}\sum_{\lambda,\lambda^{\prime}=1}^{2n}\bar{\chi}_{\lambda}(\Omega_{2})_{\lambda\lambda^{\prime}}\chi_{\lambda^{\prime}}=\frac{1}{2}\sum_{\lambda,\lambda^{\prime}=1}^{2n}\bar{\chi}_{\lambda}\,\delta_{\lambda^{\prime},(n-1)\lambda}\,\chi_{\lambda^{\prime}}\\\
&=\frac{1}{2}\sum_{\lambda=1}^{2n}\bar{\chi}_{\lambda}\,\chi_{(n-1)\lambda}=\frac{1}{2}\sum_{\lambda\,{\rm
even}}^{2n}\bar{\chi}_{\lambda}\,\chi_{(n-1)\lambda}+\frac{1}{2}\sum_{\lambda\,{\rm
odd}}^{2n}\bar{\chi}_{\lambda}\,\chi_{(n-1)\lambda}\end{split}$ (2.33)

Now, if $\lambda$ is even then in the first sum we can shift the subscript
$\chi_{(n-1)\lambda}$ by some multiple of $2n$ and bring it to
$\chi_{-\lambda}=-\chi_{\lambda}$. Similarly, for the case of odd $\lambda$,
we can shift the subscript to bring it to $\chi_{n-\lambda}$. Therefore we
have:

$\begin{split}Z_{2}&=-\frac{1}{2}\sum_{\lambda\,{\rm
even}}^{2n}\absolutevalue{\chi_{\lambda}}^{2}+\frac{1}{2}\sum_{\lambda\,{\rm
odd}}^{2n}\bar{\chi}_{\lambda}\,\chi_{n-\lambda}\\\ &=-\sum_{\lambda\,{\rm
even}}^{n-1}\absolutevalue{\chi_{\lambda}}^{2}+\sum_{\lambda{\rm
odd}}^{n-1}\bar{\chi}_{\lambda}\,\chi_{n-\lambda}\\\ \end{split}$ (2.34)

So the matrix elements for $I_{2}$ can be written:

$(I_{2})_{\lambda\lambda^{\prime}}=\begin{cases}-\delta_{\lambda\lambda^{\prime}}~{},\quad\lambda\mbox{
even}\\\ \delta_{\lambda^{\prime},n-\lambda}~{},\hskip 9.95863pt\lambda\mbox{
odd}\end{cases}$ (2.35)

Note that there are $\frac{n-1}{2}=p-1$ even values and $\frac{n}{2}=p$ odd
values for $\lambda$. Also, the only diagonal term for odd $\lambda$ occurs at
$\lambda=\frac{n}{2}=p$.

With this information we are now equipped to compute the matrix $d_{JK}$ of
inner products defined in Eq. (2.18), where $J,K$ take the two values $2,n$.
We have:

$\begin{split}\mbox{Tr}(I_{n}^{2})&=2p-1\\\ \mbox{Tr}(I_{n}\,I_{2})&=2-p\\\
\mbox{Tr}(I_{2}^{2})&=2p-1\end{split}$ (2.36)

Now Eq. (2.19) becomes:

$\begin{pmatrix}2p-1&2-p\\\ 2-p&2p-1\end{pmatrix}\begin{pmatrix}c_{n}\\\
c_{2}\end{pmatrix}=\absolutevalue{\Gamma_{\rm
sub}\backslash\Gamma}\begin{pmatrix}\mbox{Tr}(I_{n}X_{{\rm seed}})\\\
\mbox{Tr}(I_{2}X_{\rm seed})\end{pmatrix}$ (2.37)

where we have taken $\sum_{\gamma}M^{\dagger}_{\gamma}X_{\rm
seed}M_{\gamma}=c_{n}I_{n}+c_{2}I_{2}$. Solving the above equation, we get:

$\begin{split}\begin{pmatrix}c_{n}\\\
c_{2}\end{pmatrix}&=\frac{\absolutevalue{\Gamma_{\rm
sub}\backslash\Gamma}}{3(p^{2}-1)}\begin{pmatrix}2p-1&p-2\\\
p-2&2p-1\end{pmatrix}\begin{pmatrix}\Tr(I_{n}X_{\rm seed})\\\
\mbox{Tr}(I_{2}X_{\rm seed})\end{pmatrix}\\\
&=\frac{\absolutevalue{\Gamma_{\rm
sub}\backslash\Gamma}}{3(p^{2}-1)}\begin{pmatrix}(2p-1)\mbox{Tr}(I_{n}X_{\rm
seed})+(p-2)\mbox{Tr}(I_{2}X_{\rm seed})\\\ (p-2)\mbox{Tr}(I_{n}X_{\rm
seed})+(2p-1)\mbox{Tr}(I_{2}X_{\rm seed})\end{pmatrix}\\\ \end{split}$ (2.38)

This is the general expression for the result of a Poincaré sum for an
arbitrary seed at level $k=n-2=2p-2$.

Let us now look at all possible individual cases of $X_{\rm seed}$ and see how
they explain the data that we found in Table LABEL:2p. We will use the fact
that $Z_{n}=Z_{A}$ and $Z_{2}=Z_{D}-Z_{A}$, from which it follows that
$c_{A}=c_{n}-c_{2},c_{D}=c_{2}$.

  * •

$(X_{\rm seed})_{\lambda\lambda^{\prime}}=\delta_{\lambda
1}\delta_{\lambda^{\prime}1}$. In this case, $\mbox{Tr}(I_{n}X_{\rm seed})=1$
and $\mbox{Tr}(I_{2}X_{\rm seed})=0$. This means that $c_{n}=2p-1$ and
$c_{2}=p-2$. So the Poincaré sum is proportional to:

$(2p-1)Z_{n}+(p-2)Z_{2}=(p+1)Z_{A}+(p-2)Z_{D}$ (2.39)

The ratio of the two coefficients is
$\frac{c_{D}}{c_{A}}=\frac{p-2}{p+1}=\frac{n-4}{n+2}=\frac{k-2}{k+4}$. In the
limit $p\rightarrow{\infty}$, we have $\frac{c_{D}}{c_{A}}\rightarrow{1}$. One
can also verify that all diagonal seeds of the form $(X_{\rm
seed})_{\lambda,\lambda^{\prime}}=\delta_{\lambda,\lambda_{0}}\delta_{\lambda^{\prime},\lambda_{0}}$,
where $\lambda_{0}$ is odd but not equal to $p$, give the same result.

  * •

$(X_{\rm seed})_{\lambda\lambda^{\prime}}=\delta_{\lambda
2}\delta_{\lambda^{\prime}2}$. This gives $\mbox{Tr}(I_{n}X_{\rm seed})=1$ and
$\mbox{Tr}(I_{2}X_{\rm seed})=-1$ and hence $c_{n}=p+1$ and $c_{2}=-(p+1)$. So
the Poincaré sum is proportional to:

$(p+1)Z_{n}-(p+1)Z_{2}=2(p+1)Z_{A}-(p+1)Z_{D}$ (2.40)

One can also verify that all diagonal seeds of the form $(X_{\rm
seed})_{\lambda,\lambda^{\prime}}=\delta_{\lambda,\lambda_{0}}\delta_{\lambda^{\prime},\lambda_{0}}$,
where $\lambda_{0}$ is even, give the same result.

  * •

$(X_{\rm seed})_{\lambda\lambda^{\prime}}=\delta_{\lambda
p}\delta_{\lambda^{\prime}p}$. This gives $\mbox{Tr}(I_{n}X_{\rm seed})=1$ and
$\mbox{Tr}(I_{2}X_{\rm seed})=1$ and hence $c_{n}=3(p-1)$ and $c_{2}=3(p-1)$.
In this case the Poincaré sum is proportional to:

$(3p-3)Z_{n}+(3p-3)Z_{2}=(3p-3)Z_{D}$ (2.41)

Thus for models with $k+2=2p$, if we start with the seed $\chi_{p}$ then the
Poincaré sum gives us only the $Z_{D}$ invariant.

  * •

$(X_{\rm seed})_{\lambda\lambda^{\prime}}=\delta_{\lambda
1}\delta_{\lambda^{\prime},2p-1}\implies\mbox{Tr}(I_{n}X_{\rm seed})=0$ and
$\mbox{Tr}(I_{2}X_{\rm seed})=1$. This means that $c_{n}=p+1$ and
$c_{2}=-(p+1)$. So, the Poincaré sum is

$P_{1,2p-1}\propto(p-2)Z_{n}+(2p-1)Z_{2}=-(p+1)Z_{A}+(2p-1)Z_{D}$ (2.42)

this expression holds true for any odd $\lambda\,(\neq p)$ because
$\mbox{Tr}(I_{n}X_{\rm seed})=0$ and $\mbox{Tr}(I_{2}X_{\rm seed})=1$ still
holds true.

One can now verify that the above expressions exactly match the linear
combinations appearing in Table LABEL:2p. In Table 3 we summarise the above
results by listing the values of $c_{A},c_{D}$ in each category.

Seed | $c_{A},c_{D}$
---|---
$\lambda_{0}=\lambda_{0}^{\prime}={\rm odd}\neq p$ | $p+1,p-2$
$\lambda_{0}=\lambda_{0}^{\prime}=$ even | $2(p+1),-(p+1)$
$\lambda_{0}=\lambda_{0}^{\prime}=p$ | $(0,3p-3)$
$\lambda_{0}={\rm odd}\neq p,\lambda_{0}^{\prime}=2p-\lambda_{0}$ | $-(p+1),2p-1$
Table 3: General result, two modular invariants. The seeds are labelled by
$\lambda_{0},\lambda_{0}^{\prime}$ where $(X_{\rm
seed})_{\lambda\lambda^{\prime}}=\delta_{\lambda\lambda_{0}}\delta_{\lambda^{\prime}\lambda_{0}^{\prime}}$

So far we have worked with individual (and diagonal) seeds. More generally we
can consider linear combinations of different seeds. For future use, we would
like to find the seed that will give us an arbitrary general linear
combination $\alpha Z_{A}+\beta Z_{D}$. Since there are only two terms in the
combination but more than two possible seeds, this can clearly always be done.
Let us therefore find the most general seed $Z_{\rm seed}(\alpha,\beta)$ that
satisfies:

$\sum_{\gamma}Z_{\rm seed}(\alpha,\beta)=\alpha Z_{A}+\beta Z_{D}$ (2.43)

The general solution (now also including possible non-diagonal seeds) is:

$Z_{\rm
seed}(\alpha,\beta)=a_{1}\\!\\!\\!\sum_{\begin{subarray}{c}\lambda\,{\rm
odd},\,\neq
p\end{subarray}}\\!\\!\\!b_{\lambda}\absolutevalue{\chi_{\lambda}}^{2}+a_{2}\\!\\!\sum_{\begin{subarray}{c}\lambda\,\rm
even\end{subarray}}\\!\\!c_{\lambda}\absolutevalue{\chi_{\lambda}}^{2}+a_{3}\absolutevalue{\chi_{p}}^{2}+a_{4}\sum_{\lambda\,{\rm
odd},=1}^{p-2}d_{\lambda}({\overline{\chi}}_{\lambda}\chi_{2p-\lambda}+{\rm
c.c.})$ (2.44)

where:

$\begin{split}&\hskip
56.9055pt\sum_{\lambda}b_{\lambda}=\sum_{\lambda}c_{\lambda}=\sum_{\lambda}d_{\lambda}=1\\\
&a_{3}=\frac{(2p-1)\alpha+(p+1)\beta}{3(p^{2}-1)}-(a_{1}+a_{2}),\quad
2a_{4}=a_{1}+2a_{2}-\frac{\alpha}{p+1}\end{split}$ (2.45)

It is easily verified that the Poincaré sum over (2.44) gives $\alpha
Z_{A}+\beta Z_{D}$ as desired. Also, special values of the coefficients
$a_{i},b_{\lambda}$ reproduce the single-seed results: for example, choosing
$a_{1}=1,a_{2}=a_{3}=a_{4}=0$ and $b_{\lambda_{0}}=1$ for some odd
$\lambda_{0}$ reproduces the answer $(p+1)Z_{A}+(p-2)Z_{D}$, and similarly for
the other cases.

#### Three Physical Invariants

Finally we turn to the cases with three physical invariants, corresponding to
$\sigma(n)=6,7$. Here a fresh analysis is not necessary. From [21, 22] we find
that there are precisely three invariants, all physical, only for $k=10,16$.
Let us write $Z_{A/D/E}={\overline{\chi}}\,\Omega_{A/D/E}\chi$. Then, in the
first case we have the matrices $\Omega_{2},\Omega_{3},\Omega_{12}$ from which
we get $\Omega_{A}=\Omega_{12},\Omega_{D}=\Omega_{12}+\Omega_{2}$ and
$\Omega_{E}=\Omega_{12}+\Omega_{3}+\Omega_{2}$, the last one corresponding to
the $E_{6}$ invariant. In the second case the matrices are
$\Omega_{2},\Omega_{3},\Omega_{18}$ from which one gets
$\Omega_{A}=\Omega_{18},\Omega_{D}=\Omega_{18}+\Omega_{2}$ and
$\Omega_{E}=\Omega_{18}+\Omega_{3}+\Omega_{2}$, of which the last one
corresponds to the $E_{7}$ invariant. For $k=28$ we instead get four matrices
$\Omega_{2},\Omega_{3},\Omega_{5},\Omega_{30}$ but the physical partition
functions comes from just three matrices
$\Omega_{A}=\Omega_{30},\Omega_{D}=\Omega_{30}+\Omega_{2}$ and
$\Omega_{E}=\Omega_{30}+\Omega_{5}+\Omega_{3}+\Omega_{2}$ with the last one
being associated to the $E_{8}$ invariant. However, since there are four
invariants in all but only three physical invariants, there is one independent
unphysical linear combination that can appear in the Poincaré sum. In Table
LABEL:SU2fulltable we see that this unphysical invariant does make an
appearance in the Poincaré sum, so this case cannot be interpreted as an
average over three RCFT’s.

###  2.3 SU($N$)k WZW Models

We now move on to discuss the case of SU($N$)k for $N>2$. We will encounter
both similarities and differences with respect to the SU(2)k case analysed
above. One of the key differences is that all representations of SU(2) are
real or pseudo-real but for SU($N$), $N>3$ there are both complex and real
representations. For each complex representation, the CFT will contain two
primaries (one being the complex conjugate of the other) that share the same
character. This results in a multiplicity of 2 for such representations. If
ignored, this fact can lead to incorrect results, so our strategy will be to
treat a representation and its complex conjugate as distinct at intermediate
stages of the computations, and then identify them at the end.

Using the data in Appendix B one can in principle compute the Poincaré sums to
obtain candidate gravity partition functions for any SU($N$)k, using the
general procedure described earlier. The idea, as before, is to identify
models that have only physical modular invariants, then express the Poincaré
sums in terms of these invariants and attempt to interpret the linear
combination as an average over CFT’s.

While modular invariants for SU(2)k were completely classified in [21, 22],
the case of SU(3)k is considerably more complicated and was carried out later
in [28, 29]. The classification of modular invariants for the SU($N$)1 series
was conjectured in [30] and proved in [31]. Some results for the general case
of SU($N$)k can been found in [32]. In what follows, we will focus on the
families SU($N$)1 and SU(3)k, for which the general classification is well-
understood and reasonably tractable.

####  2.3.1 Poincaré Sums for SU($N$)1

The case of SU($N$)1 shares several features with SU(2)k WZW models. In
particular, we will see below that the modular invariants can be written down
in a similar way as for SU(2)k. In SU($N$)1 there are $N$ allowed
representations including the identity. Using (B.7), we see that the allowed
Dykin labels are given by $\lambda_{i}=\rho_{i}=1$ or
$\lambda_{i}=\rho_{i}+\delta_{b,i}=1+\delta_{b,i}$ for some $b$ between $1$ to
$N-1$. So, we can denote these as $\lambda^{a}$ where
$a\in\\{0,1,\dots,N-1\\}$ and $(\lambda^{a})_{i}=1+\delta_{a,i}$. The
characters can now be labelled by the index $a$.

The central charge and the conformal dimensions are:

$\displaystyle\mathbf{c}$ $\displaystyle=N-1$ (2.46) $\displaystyle h_{a}$
$\displaystyle=\frac{a(N-a)}{2N}$ (2.47)

For $a>0$ the representations labelled by $a$ and $N-a$ are complex conjugates
of each other, due to which the characters labelled by $a$ and $N-a$ are
identical. The fact that $h_{a}=h_{N-a}$ is consistent with this
identification. This leads to the multiplicity 2 in the partition function
that was referred to above.

Following [30, 31], we now list the modular invariants that appear in these
theories. We define the integer $m$ by:

$m=\begin{cases}N\mbox{, if $N$ is odd}\\\ \frac{N}{2}\mbox{, if $N$ is
even}\end{cases}$ (2.48)

This integer plays a somewhat similar (but not identical) role to the height
$n=k+2$ for SU(2)k theories. Now for every divisor $\delta$ of $m$, define
$\alpha\equiv[\delta,m/\delta]$ (the GCD of the integers) and let
$\omega(\delta)=\left(\rho\frac{m}{\alpha\delta}+\sigma\frac{\delta}{\alpha}\right)\mbox{
mod }\frac{N}{\alpha^{2}}$, where we have chosen integers $\rho,\sigma$ such
that $\rho\frac{m}{\alpha\delta}-\sigma\frac{\delta}{\alpha}=1$. This is
always possible because $\frac{m}{\alpha\delta}$ and $\frac{\delta}{\alpha}$
are coprime. We note that $\omega(m/\delta)=-\omega(\delta)\mbox{ mod
}\frac{N}{\alpha^{2}}$. Finally, we define the matrices $\Omega_{\delta}$:

$\left(\Omega_{\delta}\right)_{aa^{\prime}}=\begin{cases}0,\hskip
102.14528pt\alpha\nmid a\mbox{ or }\alpha\nmid a^{\prime}\\\
\sum_{\xi=0}^{\alpha-1}\delta_{a^{\prime},\,\omega(\delta)a+\xi
N/\alpha},\quad\mbox{otherwise}\end{cases}$ (2.49)

The modular invariant partition functions are then:

$Z_{\delta}=\sum_{a=0}^{N-1}\bar{\chi}_{a}\left(\Omega_{\delta}\right)_{aa^{\prime}}\chi_{a^{\prime}}$
(2.50)

Here the indices $a,a^{\prime}$ in the matrix elements and characters will
always be understood as integers modulo $N$. Not all the $Z_{\delta}$ above
are independent, in fact $Z_{\delta}$ and $Z_{m/\delta}$ are equal. This can
be seen from the fact that:

$\begin{split}(\Omega_{m/\delta})_{aa^{\prime}}=\sum_{\xi=0}^{\alpha-1}\delta_{a^{\prime},\,-\omega(\delta)a+\xi
N/\alpha}=\sum_{\xi^{\prime}=0}^{\alpha-1}\delta_{a^{\prime},\,N-\omega(\delta)a-\xi^{\prime}N/\alpha}\quad(\alpha|a\mbox{
and }\alpha|a^{\prime})\\\ \end{split}$ (2.51)

and the corresponding partition function is:

$\begin{split}Z_{m/\delta}&=\sum_{\begin{subarray}{c}a=0\\\
\alpha|a\end{subarray}}^{N-1}\sum_{\xi^{\prime}=0}^{\alpha-1}\bar{\chi}_{a}\,\chi_{N-\omega(\delta)a-\xi^{\prime}N/\alpha}=\sum_{\begin{subarray}{c}a=0\\\
\alpha|a\end{subarray}}^{N-1}\sum_{\xi^{\prime}=0}^{\alpha-1}\bar{\chi}_{a}\,\chi_{\omega(\delta)a+\xi^{\prime}N/\alpha}=Z_{\delta}\\\
\end{split}$ (2.52)

In the above, we used $\xi^{\prime}=\alpha-\xi$ and $\chi_{b}=\chi_{N-b}$.
Thus we have:

$\mbox{Number of linearly independent
}Z_{\delta}=\begin{cases}\frac{\sigma(m)}{2},\hskip 22.76228pt\sigma(m)\mbox{
is even}\\\ \frac{\sigma(m)+1}{2},\quad\sigma(m)\mbox{ is odd}\\\ \end{cases}$
(2.53)

Remarkably, for SU($N$)1 all the $Z_{\delta}$ are not just modular invariant
but also physical – they satisfy positive integrality of coefficients and non-
degeneracy of the vacuum state. This is a major difference from SU(2)k.
Another difference is that while the latter can have at most three physical
invariants namely $Z_{A},Z_{D},Z_{E}$, there is no such restriction for
SU($N$)1 where one can have an arbitrarily large number of physical modular
invariants for sufficiently large and suitably chosen $N$.

We now perform a similar classification as before to find conditions for one
invariant, two invariants and so on. We also derive the linear combinations of
the invariants that will appear by inverting the matrix of inner products
among the modular invariant basis matrices. To do this, we will use all the
available $\Omega_{\delta}$ as the basis for a given model and then compute
the cofficients $c_{\delta}$ corresponding to the modular invariant
$Z_{\delta}$. At the end, we will impose the constraint that
$Z_{m/\delta}=Z_{\delta}$ which equates the label for a representation and its
complex conjugate.

#### One Physical Invariant

These models correspond to what we have called type (I). As can be seen from
(2.53), we have a unique invariant whenever $\sigma(m)=1$ or $2$. This is true
when $m=1$ or $p$, where $p$ is a prime. So altogether we have $N=2,p,2p$
where $p$ is prime. In Table LABEL:sunone.1 we list some examples of models of
this type.

Table 4: Poincaré sums for SU($N$)1 WZW models with a unique modular invariant. SU($N$)1 |  |
---|---|---
$(N,c,N_{0})$ |  Seed Primary $(\chi_{\lambda})$ |  Poincaré sum
$(2,1,4)$ |  $0,1$ |  $3\,Z_{p}$
$(3,2,6)$ |  $0$ |  $6\,Z_{p}$
|  $1,2$ |  $3\,Z_{p}$
$(4,3,8)$ |  $0,2$ |  $8\,Z_{p}$
|  $1,3$ |  $4\,Z_{p}$
$(5,4,10)$ |  $0$ |  $12\,Z_{p}$
|  $1,2,3,4$ |  $6\,Z_{p}$
$(6,5,12)$ |  $0,3$ |  $12\,Z_{p}$
|  $1,2,4,5$ |  $6\,Z_{p}$
$(7,6,14)$ |  $0$ |  $18\,Z_{p}$
|  $1,2,3,4,5,6$ |  $9\,Z_{p}$
$(10,9,20)$ |  $0,5$ |  $24\,Z_{p}$
|  $1,2,3,4,6,7,8,9$ |  $12\,Z_{p}$
$(19,18,38)$ |  $0$ |  $54\,Z_{p}$
|  $1,2,\cdots,18$ |  $27\,Z_{p}$
|  |

Just like the SU(2)k examples with unique modular invariants listed in Table
LABEL:ppsq, the cases in Table LABEL:sunone.1 provide candidate gravity
partition functions dual to a unique SU($N$)1 WZW model.

#### Two physical invariants

Now, we consider the case where there are exactly two modular invariants. From
(2.53), we see that here $\sigma(m)=3$ or $4$ which is true when
$m=p^{2},\,p^{3},\,pq$, where $p,q$ are primes and $m>5$. So the relevant
values of $N$ are $N=p^{2},\,p^{3},pq,2p^{2},\,2p^{3}$ or $2pq$.

All these values of $N$ describe models which fall into the type (IIa)
category. Unlike SU(2)k, we do not need to restrict further to sub-families as
all of them are physical. In Table LABEL:sunone.2 we list examples of models
having $m=p^{2}$, i.e. $N=p^{2}$ or $2p^{2}$, along with the result of the
Poincaré sum for each seed primary. In these models the possible values of the
integers $\delta$ that divide $m$ are $1,p,p^{2}$ and correspondingly there
are three modular invariants that we label $Z_{1},Z_{p},Z_{p^{2}}$. We have
$Z_{1}=Z_{p^{2}}$ resulting from complex conjugation, as described above, so
the results can only be linear combinations of $Z_{p},Z_{p^{2}}$.

Table 5: Poincaré sums for $SU(N)_{1}$ with two invariants, for $N=p^{2},2p^{2}$ $SU(N)_{1}$ |  |
---|---|---
$(N,c,N_{0})$ |  Seed Primary $(\chi_{\lambda})$ |  Poincaré sum
$(8,7,16)$ |  $0,4$ |  $4(4Z_{p^{2}}+Z_{p})$
|  $0\neq\mbox{ mod }2$ |  $8(2Z_{p^{2}}-Z_{p})$
|  $0=\mbox{ mod }2,\,\neq 0,4$ |  $12Z_{p}$
$(9,8,18)$ |  $0$ |  $6(3Z_{p^{2}}+Z_{p})$
|  $0\neq\mbox{ mod }3$ |  $4.5(3Z_{p^{2}}-Z_{p})$
|  $0=\mbox{ mod }3,\,\neq 0$ |  $1.5(3Z_{p^{2}}+7Z_{p})$
$(18,17,36)$ |  $0,9$ |  $12(3Z_{p^{2}}+Z_{p})$
|  $0\neq\mbox{ mod }3$ |  $9(3Z_{p^{2}}-Z_{p})$
|  $0=\mbox{ mod }3,\,\neq 0,9$ |  $3(3Z_{p^{2}}+7Z_{p})$
$(25,24,50)$ |  $0$ |  $12(5Z_{p^{2}}+2Z_{p})$
|  $0\neq\mbox{ mod }5$ |  $7.5(5Z_{p^{2}}-Z_{p})$
|  $0=\mbox{ mod }5,\,\neq 0$ |  $4.5(5Z_{p^{2}}+7Z_{p})$
$(49,48,98)$ |  $0$ |  $18(7Z_{p^{2}}+3Z_{p})$
|  $0\neq\mbox{ mod }7$ |  $10.5(7Z_{p^{2}}-Z_{p})$
|  $0=\mbox{ mod }7,\,\neq 0$ |  $1.5(35Z_{p^{2}}+43Z_{p})$
$(50,49,100)$ |  $0,25$ |  $24(5Z_{p^{2}}+2Z_{p})$
|  $0\neq\mbox{ mod }5$ |  $15(5Z_{p^{2}}-Z_{p})$
|  $0=\mbox{ mod }5,\,\neq 0,25$ |  $9(5Z_{p^{2}}+7Z_{p})$
|  |

These Poincaré sums provide candidate gravity partition functions where the
dual consists of an average over the two CFTs. The precise linear combinations
of the physical CFT partition function that we obtain can be derived using
methods similar to those for $SU(2)_{k}$. We illustrate this for the same sub-
families $N=p^{2}$ and $N=2p^{2}$, that were explicitly studied above. One can
easily generalise this derivation to any of the other two-invariant families.

#### Two invariants: Coefficients for $N=p^{2}$

Here, $m=N=p^{2}$ and from the discussion of the available modular invariant
partition functions around (2.50), the relevant matrices are
$\Omega_{p^{2}},\,\Omega_{1},$ and $\Omega_{p}$. Their matrix elements are
given by:

$(\Omega_{p^{2}})_{aa^{\prime}}=\delta_{a^{\prime}a}\,,\quad(\Omega_{1})_{aa^{\prime}}=\delta_{a^{\prime},-a}\,,\quad(\Omega_{p})_{aa^{\prime}}=\begin{cases}0,\hskip
59.75095ptp\nmid a\mbox{ or }p\nmid a^{\prime}\\\
\sum_{\xi=0}^{p-1}\delta_{a^{\prime},\xi p},\quad\mbox{otherwise}\end{cases}$
(2.54)

For the case of $\Omega_{p}$, we have used the fact that $\alpha\equiv[p,p]=p$
and so $N/\alpha^{2}=1$. Hence, $\omega(p)=0$. The modular invariant functions
that we get from these matrices are:

$\begin{split}Z_{p^{2}}&=\sum_{a,a^{\prime}=0}^{N-1}\bar{\chi}_{a}\left(\Omega_{p^{2}}\right)_{aa^{\prime}}\chi_{a^{\prime}}=\sum_{a,a^{\prime}=0}^{N-1}\bar{\chi}_{a}\delta_{a^{\prime}a}\chi_{a^{\prime}}=\sum_{a=0}^{N-1}\absolutevalue{\chi_{a}}^{2}\\\
Z_{1}&=\sum_{a,a^{\prime}=0}^{N-1}\bar{\chi}_{a}\left(\Omega_{1}\right)_{aa^{\prime}}\chi_{a^{\prime}}=\sum_{a,a^{\prime}=0}^{N-1}\bar{\chi}_{a}\delta_{a^{\prime},-a}\chi_{a^{\prime}}=\sum_{a=0}^{N-1}\bar{\chi}_{a}\chi_{-a}\\\
Z_{p}&=\sum_{a,a^{\prime}=0}^{N-1}\bar{\chi}_{a}\left(\Omega_{p}\right)_{aa^{\prime}}\chi_{a^{\prime}}=\sum_{\begin{subarray}{c}a,a^{\prime}=0\\\
p|a\,,\,p|a^{\prime}\end{subarray}}^{N-1}\sum_{\xi=0}^{p-1}\bar{\chi}_{a}\,\delta_{a^{\prime},\xi
p}\,\chi_{a^{\prime}}\\\ &\qquad=\sum_{\begin{subarray}{c}a=0\\\
p|a\end{subarray}}^{N-1}\bar{\chi}_{a}\sum_{\xi=0}^{p-1}\chi_{\xi
p}=\left|\sum_{a=0,\,p|a}^{N-1}\chi_{a}\right|^{2}\end{split}$ (2.55)

From the fact that $\chi_{-a}=\chi_{N-a}=\chi_{a}$, we see that
$Z_{1}=Z_{p^{2}}$. However, as explained above, we treat them as independent
for now and identify them at the end.

We see that the theories corresponding to $Z_{p}$ are holomorphically
factorised, or meromorphic, CFT’s. This can be understood as follows. The
central charge of $SU(N)_{1}$ with $N=p^{2}$ is $p^{2}-1$. It is well-known
that for any prime $p$, this number is divisible by $24$. Now $\mathbf{c}$
being a multiple of $24$ is a necessary condition to encounter meromorphic
(one-character) CFT’s, and these are what the $Z_{p}$ are giving us. As an
example, the $Z_{p}$ invariant for $SU(25)_{1}$ is entry No. $67$ in the table
of [33], in which all meromorphic CFT with $\mathbf{c}=24$ are classified.

We can now compute the matrix $d_{JK}$ of inner products. Its elements are:

$\displaystyle\mbox{Tr}(\Omega_{p^{2}}^{2})=p^{2}\,,\quad\mbox{Tr}(\Omega_{1}^{2})=p^{2}\,,\quad\mbox{Tr}(\Omega_{p}^{2})=p^{2}$
(2.56) $\displaystyle\mbox{Tr}(\Omega_{p^{2}}\Omega_{1})=$
$\displaystyle\mbox{Tr}(\Omega_{1})=1\,,\quad\mbox{Tr}(\Omega_{p^{2}}\Omega_{p})=\mbox{Tr}(\Omega_{p})=p\,,\quad\mbox{Tr}(\Omega_{1}\Omega_{p})=p$
(2.57)

The linear combinations that appear in the Poincaré sums can now be computed:

$\begin{split}\begin{pmatrix}c_{p^{2}}\\\ c_{1}\\\
c_{p}\end{pmatrix}&=\absolutevalue{\Gamma_{\rm
sub}\backslash\Gamma}\begin{pmatrix}p^{2}&1&p\\\ 1&p^{2}&p\\\
p&p&p^{2}\end{pmatrix}^{-1}\begin{pmatrix}\mbox{Tr}(\Omega_{p^{2}}X_{\rm
seed})\\\ \mbox{Tr}(\Omega_{1}X_{\rm seed})\\\ \mbox{Tr}(\Omega_{p}X_{\rm
seed})\end{pmatrix}\\\ &=\frac{\absolutevalue{\Gamma_{\rm
sub}\backslash\Gamma}}{p^{2}(p^{2}-1)}\begin{pmatrix}p^{2}&0&-p\\\
0&p^{2}&-p\\\
-p&-p&1+p^{2}\end{pmatrix}\begin{pmatrix}\mbox{Tr}(\Omega_{p^{2}}X_{\rm
seed})\\\ \mbox{Tr}(\Omega_{1}X_{\rm seed})\\\ \mbox{Tr}(\Omega_{p}X_{\rm
seed})\end{pmatrix}\\\ &=\frac{\absolutevalue{\Gamma_{\rm
sub}\backslash\Gamma}}{p^{2}(p^{2}-1)}\begin{pmatrix}p^{2}\,\mbox{Tr}(\Omega_{p^{2}}X_{\rm
seed})-p\,\mbox{Tr}(\Omega_{p}X_{\rm seed})\\\
p^{2}\,\mbox{Tr}(\Omega_{1}X_{\rm seed})-p\,\mbox{Tr}(\Omega_{p}X_{\rm
seed})\\\ -p\,\mbox{Tr}(\Omega_{p^{2}}X_{\rm
seed})-p\,\mbox{Tr}(\Omega_{1}X_{\rm
seed})+(1+p^{2})\mbox{Tr}(\Omega_{p}X_{\rm seed})\end{pmatrix}\\\ \end{split}$
(2.58)

Let us now consider different choices of seeds:

  * •

$(X_{\rm seed})_{a,a^{\prime}}=\delta_{a,0}\delta_{a^{\prime},0}$. We have
$\mbox{Tr}(\Omega_{p^{2}}X_{\rm seed})=\mbox{Tr}(\Omega_{1}X_{\rm
seed})=\mbox{Tr}(\Omega_{p}X_{\rm seed})=1$ so the Poincaré sum is
proportional to:

$p(p-1)(Z_{p^{2}}+Z_{1})+(p-1)^{2}Z_{p}=2(p-1)\left(p\,Z_{p^{2}}+\frac{(p-1)}{2}Z_{p}\right)$
(2.59)

To get the second equality we have used $Z_{p^{2}}=Z_{1}$.

  * •

$(X_{\rm seed})_{a,a^{\prime}}=\delta_{a,1}\delta_{a^{\prime},1}$. We have
$\mbox{Tr}(\Omega_{p^{2}}X_{\rm seed})=1\,,\mbox{Tr}(\Omega_{1}X_{\rm
seed})=0\,,\mbox{Tr}(\Omega_{p}X_{\rm seed})=0$ and so the Poincaré sum is
proportional to:

$p^{2}Z_{p^{2}}-pZ_{p}=p(p\,Z_{p^{2}}-Z_{p})$ (2.60)

This is true for any seed primary $\absolutevalue{\chi_{a}}^{2}$ where $p\nmid
a$.

  * •

$(X_{\rm seed})_{a,a^{\prime}}=\delta_{a,p}\delta_{a^{\prime},p}$. We have
$\mbox{Tr}(\Omega_{p^{2}}X_{\rm seed})=1\,,\mbox{Tr}(\Omega_{1}X_{\rm
seed})=0\,,\mbox{Tr}(\Omega_{p}X_{\rm seed})=1$ so in this case the Poincaré
sum is proportional to:

$(p^{2}-p)Z_{p^{2}}-p\,Z_{1}+(p^{2}-p+1)Z_{p}=p(p-2)Z_{p^{2}}+(p^{2}-p+1)Z_{p}$
(2.61)

This is true for any seed primary $\absolutevalue{\chi_{a}}^{2}$ where $p\mid
a$ unless $a=0$.

These results can now be compared with the explicit examples in Table
LABEL:sunone.2 that correspond to $N=p^{2}$ and we find perfect agreement.

From this example we learn the general lesson that Poincaré sums will mix the
partition function of a meromorphic CFT (as long as it has a Kac-Moody
algebra) with the diagonal invariant of the same Kac-Moody algebra. This
potentially leads to many more cases that can be investigated, a point to
which he hope to return in the future.

#### Two invariants: Coefficients for $N=2p^{2}$

Here, $m=\frac{N}{2}=p^{2}$, so again the relevant matrices are
$\Omega_{p^{2}},\,\Omega_{1},$ and $\Omega_{p}$. The matrix elements are:

$(\Omega_{p^{2}})_{aa^{\prime}}=\delta_{a^{\prime}a}\,,\quad(\Omega_{1})_{aa^{\prime}}=\delta_{a^{\prime},-a}\,,\quad(\Omega_{p})_{aa^{\prime}}=\begin{cases}0,\hskip
59.75095ptp\nmid a\mbox{ or }p\nmid a^{\prime}\\\
\sum_{\xi=0}^{p-1}\delta_{a^{\prime},a+2\xi
p},\quad\mbox{otherwise}\end{cases}$ (2.62)

For the case of $\Omega_{p}$, we have used the fact that $\alpha=[p,p]=p$,
$N/\alpha^{2}=2$ and so we take $\omega(p)=1$. The modular invariant functions
that we get from these matrices are

$\displaystyle Z_{p^{2}}$
$\displaystyle=\sum_{a,a^{\prime}=0}^{N-1}\bar{\chi}_{a}\left(\Omega_{p^{2}}\right)_{aa^{\prime}}\chi_{a^{\prime}}=\sum_{a,a^{\prime}=0}^{N-1}\bar{\chi}_{a}\delta_{a^{\prime}a}\chi_{a^{\prime}}=\sum_{a=0}^{N-1}\absolutevalue{\chi_{a}}^{2}$
(2.63) $\displaystyle Z_{1}$
$\displaystyle=\sum_{a,a^{\prime}=0}^{N-1}\bar{\chi}_{a}\left(\Omega_{1}\right)_{aa^{\prime}}\chi_{a^{\prime}}=\sum_{a,a^{\prime}=0}^{N-1}\bar{\chi}_{a}\delta_{a^{\prime},-a}\chi_{a^{\prime}}=\sum_{a=0}^{N-1}\bar{\chi}_{a}\chi_{-a}$
(2.64) $\displaystyle Z_{p}$
$\displaystyle=\sum_{a,a^{\prime}=0}^{N-1}\bar{\chi}_{a}\left(\Omega_{p}\right)_{aa^{\prime}}\chi_{a^{\prime}}=\sum_{\begin{subarray}{c}a,a^{\prime}=0\\\
p|a\,,\,p|a^{\prime}\end{subarray}}^{N-1}\sum_{\xi=0}^{p-1}\bar{\chi}_{a}\,\delta_{a^{\prime},a+2\xi
p}\,\chi_{a^{\prime}}=\absolutevalue{\sum_{\begin{subarray}{c}l=0\end{subarray}}^{p-1}\chi_{2lp}}^{2}+\absolutevalue{\sum_{\begin{subarray}{c}l=0\end{subarray}}^{p-1}\chi_{(2l+1)p}}^{2}$
(2.65)

Again $Z_{1}=Z_{p^{2}}$, but $Z_{p}$ is no longer a meromorphic CFT – rather,
it is a two-character CFT.

The elements of the matrix of inner products $d_{JK}$ are:

$\displaystyle\mbox{Tr}(\Omega_{p^{2}}^{2})=2p^{2}\,,\quad\mbox{Tr}(\Omega_{1}^{2})=2p^{2}\,,\quad\mbox{Tr}(\Omega_{p}^{2})=2p^{2}$
(2.66) $\displaystyle\mbox{Tr}(\Omega_{p^{2}}\Omega_{1})=$
$\displaystyle\mbox{Tr}(\Omega_{1})=2\,,\quad\mbox{Tr}(\Omega_{p^{2}}\Omega_{p})=\mbox{Tr}(\Omega_{p})=2p\,,\quad\mbox{Tr}(\Omega_{1}\Omega_{p})=2p$
(2.67)

and the linear combinations in the Poincaré sums are:

$\displaystyle\begin{split}\begin{pmatrix}c_{p^{2}}\\\ c_{1}\\\
c_{p}\end{pmatrix}&=\absolutevalue{\Gamma_{\rm
sub}\backslash\Gamma}\begin{pmatrix}2p^{2}&2&2p\\\ 2&2p^{2}&2p\\\
2p&2p&2p^{2}\end{pmatrix}^{-1}\begin{pmatrix}\mbox{Tr}(\Omega_{p^{2}}X_{\rm
seed})\\\ \mbox{Tr}(\Omega_{1}X_{\rm seed})\\\ \mbox{Tr}(\Omega_{p}X_{\rm
seed})\end{pmatrix}\\\ &=\frac{\absolutevalue{\Gamma_{\rm
sub}\backslash\Gamma}}{2p^{2}(p^{2}-1)}\begin{pmatrix}p^{2}&0&-p\\\
0&p^{2}&-p\\\
-p&-p&1+p^{2}\end{pmatrix}\begin{pmatrix}\mbox{Tr}(\Omega_{p^{2}}X_{\rm
seed})\\\ \mbox{Tr}(\Omega_{1}X_{\rm seed})\\\ \mbox{Tr}(\Omega_{p}X_{\rm
seed})\end{pmatrix}\\\ &=\frac{\absolutevalue{\Gamma_{\rm
sub}\backslash\Gamma}}{2p^{2}(p^{2}-1)}\begin{pmatrix}p^{2}\,\mbox{Tr}(\Omega_{p^{2}}X_{\rm
seed})-p\,\mbox{Tr}(\Omega_{p}X_{\rm seed})\\\
p^{2}\,\mbox{Tr}(\Omega_{1}X_{\rm seed})-p\,\mbox{Tr}(\Omega_{p}X_{\rm
seed})\\\ -p\,\mbox{Tr}(\Omega_{p^{2}}X_{\rm
seed})-p\,\mbox{Tr}(\Omega_{1}X_{\rm
seed})+(1+p^{2})\mbox{Tr}(\Omega_{p}X_{\rm seed})\end{pmatrix}\\\ \end{split}$
(2.68)

For the various different seeds, this gives:

  * •

$(X_{\rm seed})_{aa^{\prime}}=\delta_{a0}\delta_{a^{\prime}0}$. We have
$\mbox{Tr}(\Omega_{p^{2}}X_{\rm seed})=\mbox{Tr}(\Omega_{1}X_{\rm
seed})=\mbox{Tr}(\Omega_{p}X_{\rm seed})=1$ and the Poincaré sum is
proportional to:

$p(p-1)(Z_{p^{2}}+Z_{1})+(p-1)^{2}Z_{p}=2(p-1)\left(p\,Z_{p^{2}}+\frac{(p-1)}{2}Z_{p}\right)$
(2.69)

where we again used $Z_{p^{2}}=Z_{1}$. This will also be true of the seed
$\absolutevalue{\chi_{p^{2}}}$.

  * •

$(X_{\rm seed})_{aa^{\prime}}=\delta_{a1}\delta_{a^{\prime}1}$. We have
$\mbox{Tr}(\Omega_{p^{2}}X_{\rm seed})=1\,,\mbox{Tr}(\Omega_{1}X_{\rm
seed})=0\,,\mbox{Tr}(\Omega_{p}X_{\rm seed})=0$ and the Poincaré sum is
proportional to:

$p^{2}Z_{p^{2}}-pZ_{p}=p(p\,Z_{p^{2}}-Z_{p})$ (2.70)

This is true for any seed primary $\absolutevalue{\chi_{a}}^{2}$ where $p\nmid
a$.

  * •

$(X_{\rm seed})_{aa^{\prime}}=\delta_{ap}\delta_{a^{\prime}p}$. We have
$\mbox{Tr}(\Omega_{p^{2}}X_{\rm seed})=1\,,\mbox{Tr}(\Omega_{1}X_{\rm
seed})=0\,,\mbox{Tr}(\Omega_{p}X_{\rm seed})=1$ and the Poincaré sum is
proportional to

$(p^{2}-p)Z_{p^{2}}-p\,Z_{1}+(p^{2}-p+1)Z_{p}=p(p-2)Z_{p^{2}}+(p^{2}-p+1)Z_{p}$
(2.71)

This is true for any seed primary $\absolutevalue{\chi_{a}}^{2}$ where $p\mid
a$ unless $a=0,p^{2}$.

These predictions agree with the results of explicit computations, of which a
few cases are displayed in Table LABEL:sunone.2.

####  2.3.2 Poincaré Sums for SU(3)k

Next we turn to the case of SU(3)k. These WZW models are considerably more
complicated than those for SU(2)k and SU($N$)1. The classification of modular
invariants in this case has been carried out by Gannon [28, 29] but not quite
with the same methods as those employed in the classifications for SU(2)k and
SU($N$)1. In particular we do not have the analogue of the matrices
$\Omega_{\delta}$ discussed above in Eqs.(2.23), (2.49). This makes it more
difficult to make general predictions about the result of the Poincaré sums,
though evaluating these sums for any specific case is straightforward.

The classification of [28, 29] provides a list of physical invariants that
appear at different levels. It is found that two invariants called
$Z_{A},Z_{D}$ exist for all heights $n=k+3$ (with $k>1$), in contrast to the
$SU(2)$ case where $Z_{D}$ existed only for even $n$. There are also four
exceptional invariants denoted $Z_{E}$ which appear at $n=5,9,21$ (with two
distinct invariants appearing at $n=9)$.

We calculated the Poincaré sums for the seed corresponding to each individual
character for the values $k=1,2,\dots,16$ using SageMath. As in the
$SU(2)_{k}$ case, we found that the Poincaré sums are generally linear
combinations of multiple modular invariants. Moreover these combinations often
contain unphysical invariants that we have denoted $Z_{new}$. The results up
to $k=8$ are given in Table LABEL:SU3fulltable.

Table 6: Poincaré sums for $SU(3)_{k}$. Level $(n,c,N_{0})$ |  Seed primary $(\lambda_{1}\lambda_{2})$ |  Poincaré sum
---|---|---
$(4,2,12)$ |  $(11)$ |  $24Z_{A}$
|  $(12)$ |  $12Z_{A}$
$(5,\frac{16}{5},15)$ |  $(11),(22)$ |  $24Z_{A}$
|  $(12),(13)$ |  $12Z_{A}$
$(6,4,18)$ |  $(11)$ |  $18Z_{A}$
|  $(12),(13),(23)$ |  $4.5(3Z_{A}-Z_{D})$
|  $(14)$ |  $4.5(Z_{A}+Z_{D})$
|  $(22)$ |  $18Z_{D}$
$(7,\frac{32}{7},21)$ |  $(11),(22),(33)$ |  $12(Z_{A}+Z_{D})$
|  $(12),(24)$ |  $3(9Z_{A}-7Z_{D})$
|  $(13),(14),(15),(23)$ |  $6(Z_{A}+Z_{D})$
$(8,5,24)$ |  $(11),(33)$ |  $4(2Z_{A}+2Z_{D}+Z_{E})$
|  $(12),(15)$ |  $4(4Z_{A}-2Z_{D}-Z_{E})$
|  $(13),(34)$ |  $6(2Z_{A}-2Z_{D}+Z_{E})$
|  $(14)$ |  $12Z_{E}$
|  $(16),(23)$ |  $2(2Z_{A}+2Z_{D}+Z_{E})$
|  $(22)$ |  $16(Z_{A}+Z_{D}-Z_{E})$
|  $(24),(25)$ |  $8(Z_{A}+Z_{D}-Z_{E})$
$(9,\frac{16}{3},27)$ |  $(11),(22),(44)$ |  $3(6Z_{A}+Z_{D})$
|  $(12),(13),(15),(16),(23),$ |  $4.5(3Z_{A}-Z_{D})$
|  $(24),(26),(34),(35)$ |
|  $(14),(17),(25)$ |  $1.5(3Z_{A}+5Z_{D})$
|  $(33)$ |  $27Z_{D}$
$(10,\frac{28}{5},30)$ |  $(11),(33)$ |  $4(3Z_{A}+2Z_{D}+2Z_{new})$
|  $(12),(13),(16),(27)$ |  $0.5(21Z_{A}-13Z_{D}-4Z_{new})$
|  $(14),(25)$ |  $3(3Z_{A}+Z_{D}+4Z_{new})$
|  $(15),(23),(35),(45)$ |  $0.5(27Z_{A}-15Z_{D}+6Z_{new})$
|  $(17),(36)$ |  $3Z_{A}+5Z_{D}-4Z_{new}$
|  $(18),(34)$ |  $2(3Z_{A}+2Z_{D}+2Z_{new})$
|  $(22),(44)$ |  $12(Z_{D}-2Z_{new})$
|  $(24),(26)$ |  $6(Z_{D}-2Z_{new})$
$(11,\frac{64}{11},33)$ |  $(11),(22),(33),(44),(55)$ |  $12(Z_{A}+Z_{D})$
|  $(12),(13),(16),(18),(23),(24),$ |  $3(5Z_{A}-3Z_{D})$
|  $(26),(37),(45),(46)$ |
|  $(14),(15),(17),(19),(25),(27),$ |  $1.5(3Z_{A}+5Z_{D})$
|  $(28),(34),(35),(36)$ |
|  |

Here at height $n=10$, $Z_{new}$ is an unphysical invariant given by:

$\begin{split}Z_{new}&=-|\chi_{12}|^{2}-|\chi_{13}|^{2}+|\chi_{14}|^{2}-|\chi_{16}|^{2}-|\chi_{17}|^{2}-|\chi_{22}|^{2}\\\
&\quad-2|\chi_{24}|^{2}+|\chi_{25}|^{2}-2|\chi_{26}|^{2}-|\chi_{27}|^{2}-|\chi_{36}|^{2}-|\chi_{44}|^{2}\\\
&\quad\\{-{\overline{\chi}}_{11}\chi_{36}+{\overline{\chi}}_{12}\chi_{34}-{\overline{\chi}}_{13}\chi_{18}+{\overline{\chi}}_{15}\chi_{45}\\\
&\quad-{\overline{\chi}}_{16}\chi_{18}+{\overline{\chi}}_{17}\chi_{33}+{\overline{\chi}}_{23}\chi_{35}+{\overline{\chi}}_{27}\chi_{34}+{\rm
c.c.}\\}\end{split}$ (2.72)

Based on the table we make the following observations:

  * •

Within a given model, several distinct seeds give the same linear combination
of modular invariants.

  * •

The vacuum seed always gives a linear combination involving non-negative
weights.

  * •

Unlike the $SU(2)$ case, we have observed that the linear combination for all
$n>8$ (up to 16) contains an unphysical invariant.

  * •

However whenever the height $n$ is a prime $p>5$, the linear combination from
the vacuum seed is proportional to $Z_{A}+Z_{D}$ and moreover does not contain
any unphysical invariants.

It would be interesting to prove the above statements in general. We will not
do so here, instead we will derive the coefficients for the specific case of
$k=4$ and leave a more detailed analysis for future work. From the table we
see that at $k=4$, we only find the $Z_{A}$ and $Z_{D}$ invariant. These
invariants are given as:

$\displaystyle Z_{A}$
$\displaystyle=\sum_{\lambda}\absolutevalue{\chi_{\lambda}}^{2}$ (2.73)
$\displaystyle Z_{D}$
$\displaystyle=\sum_{\lambda}\bar{\chi}_{A^{kt(\lambda)}\lambda}\,\chi_{\lambda}$
(2.74)

where $t(\lambda)=\lambda_{1}-\lambda_{2}$ and
$A\,\lambda=(n-\lambda_{1}-\lambda_{2},\lambda_{1})$. The operator $A$ is of
order three: $A^{3}=1$. We also have the corresponding conjugate versions of
these given by:

$\displaystyle Z_{A}^{C}$
$\displaystyle=\sum_{\lambda}\bar{\chi}_{\lambda}\,\chi_{C\lambda}$ (2.75)
$\displaystyle Z_{D}^{C}$
$\displaystyle=\sum_{\lambda}\bar{\chi}_{A^{kt(\lambda)}\lambda}\,\chi_{C\lambda}$
(2.76)

where $C\lambda$ is the complex conjugate representation to $\lambda$. This
permits us to work out the corresponding $\Omega$ matrices, which turn out to
be:

$(\Omega_{A})_{\lambda,\lambda^{\prime}}=\delta_{\lambda,\lambda^{\prime}},~{}(\Omega_{CA})_{\lambda,\lambda^{\prime}}=\delta_{C\lambda,\lambda^{\prime}},~{}(\Omega_{D})_{\lambda,\lambda^{\prime}}=\delta_{\lambda,A^{kt(\lambda^{\prime})}\lambda^{\prime}},~{}(\Omega_{CD})_{\lambda,\lambda^{\prime}}=\delta_{C\lambda,A^{kt(\lambda^{\prime})}\lambda^{\prime}}$
(2.77)

The relevant traces of products of these matrices are then:

$\begin{split}\mbox{Tr}(\Omega_{A}^{2})&=\mbox{Tr}(\Omega_{CA}^{2})=\mbox{Tr}(\Omega_{D}^{2})=\mbox{Tr}(\Omega_{CD}^{2})=15\\\
\mbox{Tr}(\Omega_{A}\Omega_{CA})&=3,\quad\mbox{Tr}(\Omega_{A}\Omega_{D})=5,\quad\mbox{Tr}(\Omega_{A}\Omega_{CD})=9,\quad\\\
\mbox{Tr}(\Omega_{CA}\Omega_{D})&=9,\quad\mbox{Tr}(\Omega_{CA}\Omega_{CD})=5,\quad\mbox{Tr}(\Omega_{D}\Omega_{CD})=3\end{split}$
(2.78)

Following the usual strategy, we now invert the relevant $4$ matrix to get the
linear combinations:

$\begin{pmatrix}c_{A}\\\ c_{CA}\\\ c_{D}\\\
c_{CD}\end{pmatrix}=\frac{\absolutevalue{\Gamma_{\rm
sub}\backslash\Gamma}}{128}\begin{pmatrix}15a+3b-5c-9d\\\ 3a+15b-9c-5d\\\
-5a-9b+15c+3d\\\ -9a-5b+3c+15d\end{pmatrix}$ (2.79)

where for compactness we have introduced $a=\mbox{Tr}(\Omega_{A}X_{\rm
seed}),~{}b=\mbox{Tr}(\Omega_{CA}X_{\rm seed}),~{}c=\mbox{Tr}(\Omega_{D}X_{\rm
seed}),~{}d=\mbox{Tr}(\Omega_{CD}X_{\rm seed})$.

Let us now consider different choices of seeds:

  * •

$(X_{\rm
seed})_{\lambda\lambda^{\prime}}=\delta_{\lambda,(1,1)}\delta_{\lambda^{\prime},(1,1)}$.
We have then $a=b=c=d=1$ and so the Poincaré sum is proportional to

$4(Z_{A}+Z^{C}_{A}+Z_{D}+Z^{C}_{D})=8(Z_{A}+Z_{D})$ (2.80)

where we got the second form after using $Z_{i}=Z_{i}^{C}$.

  * •

$(X_{\rm
seed})_{\lambda\lambda^{\prime}}=\delta_{\lambda,(1,2)}\delta_{\lambda^{\prime},(1,2)}$.
We have $a=1,b=c=d=0$ and so the Poincaré sum is proportional to:

$15Z_{A}+3Z^{C}_{A}-5Z_{D}-9Z^{C}_{D}=2(9Z_{A}-7Z_{D})$ (2.81)

  * •

$(X_{\rm
seed})_{\lambda\lambda^{\prime}}=\delta_{\lambda,(1,3)}\delta_{\lambda^{\prime},(1,3)}$.
We have $a=d=1,b=c=0$ and so the Poincaré sum is proportional to:

$6Z_{A}-2Z^{C}_{A}-2Z_{D}+6Z^{C}_{D}=4(Z_{A}+Z_{D})$ (2.82)

These results match precisely with the output of the SageMath calculations
presented in the table for this case.

###  2.4 Virasoro Minimal Models

Poincaré sums for minimal models were first discussed in [7]. In this case
too, all the possible modular invariants are classified [21, 22]. Indeed the
results are very similar to those for SU(2)k that we have already encountered.

The characters for minimal models are labelled $\chi_{r,s}$, the range of
these being defined in Appendix B. The identity character is $\chi_{1,1}$.
Modular invariants for the $(m,m^{\prime})$ minimal model, denoted
$Z_{\delta\delta^{\prime}}$, are defined in terms of a matrix
$\Omega_{\delta\delta^{\prime}}$ by:

$Z_{\delta\delta^{\prime}}=\frac{1}{8}\sum_{r_{1},r_{2}=1}^{2m^{\prime}}\sum_{s_{1},s_{2}=1}^{2m}\bar{\chi}_{r_{1},s_{1}}\,\left[\Omega_{\delta\delta^{\prime}}\right]_{(r_{1},s_{1}),(r_{2},s_{2})}\,\chi_{r_{2},s_{2}}$
(2.83)

where $\delta|m$ and $\delta^{\prime}|m^{\prime}$. The parameters $m$ and
$m^{\prime}$ act as the analogues of the height $n$ for SU(2)k, while the
matrices $\Omega_{\delta\delta^{\prime}}$ are simply tensor products of two of
the $\Omega_{\delta}$ that we encountered in the $SU(2)_{k}$ case:

$\left[\Omega_{\delta\delta^{\prime}}\right]_{(r_{1},s_{1}),(r_{2},s_{2})}=(\Omega_{\delta})_{s_{1},s_{2}}(\Omega_{\delta^{\prime}})_{r_{1},r_{2}}$
(2.84)

The characters are given as:

$\chi_{r,s}(\tau)=\frac{1}{\eta(\tau)}\sum_{t=-\infty}^{\infty}\left(q^{\frac{(2tmm^{\prime}+rm-
sm^{\prime})^{2}}{4mm^{\prime}}}-q^{\frac{(2tmm^{\prime}+rm+sm^{\prime})^{2}}{4mm^{\prime}}}\right)$
(2.85)

from which it follows that $\chi_{r,s}=-\chi_{-r,s}=\chi_{r+2m^{\prime},s}$
and similarly for the shifts of $s$. With this, many of the results for SU(2)k
can be adapted.

#### One physical invariant

This is only possible when $\sigma(m)$ and $\sigma(m^{\prime})$ are each
individually equal to $2$ or $3$. This condition is satisfied only if $m$ and
$m^{\prime}$ are either a prime or the square of a prime. This leads to four
possibilities: $(m,m^{\prime})=(p,q),(p^{2},q),(p,q^{2}),(p^{2},q^{2})$, where
$p,q$ are distinct primes. Since the diagonal invariant always exists and is
physical, it follows that whenever there is a unique modular invariant, it is
physical. The corresponding invariant is labelled $Z_{AA}$. However only a
couple of unitary minimal models belong in the families listed above, namely
$(m,m+1)=(3,4)$ and $(4,5)$.

#### Two physical invariants

There will be exactly two modular invariants when one of $m$ or $m^{\prime}$
is of the form $p$ or $p^{2}$, while the other one is $qr,q^{3},q^{4}$, where
$p,q,r$ are primes. However this does not ensure that both are physical. From
our previous discussion we know that physical invariants only occur if
$\Omega_{2}$ contributes, which in turn requires $m^{\prime}\in\\{2q,8,16\\}$.
Meanwile $m=p,p^{2}$ as before. Thus the models that contain precisely two
physical invariants are those with $(m,m^{\prime})=(p,2q),(p^{2},2q)$ where
$p,q>2$, along with $(m,m^{\prime})=(p,8),(p^{2},8),(p,16),(p^{2},16)$. The
corresponding invariants are labelled $Z_{AA},Z_{AD}$. Due to the symmetry
between $m$ and $m^{\prime}$, we identify models where the values of these
integers are interchanged. As already noted in Appendix of [7], there are some
unitary models in this set – namely those with $m=2p$ where $p$ is a prime
such that $2p+1$ is also prime – these are known as Sophie Germain primes.

In Table LABEL:virmin one finds that for the unitary minimal models with
$(m,m+1)$ ranging from $(5,6)$ to $(10,11)$, only the two physical invariants
$Z_{AA}$ and $Z_{AD}$ appear. From the discussion above there are infinitely
many more such examples, both unitary and non-unitary.

Let us see how to derive the linear combination of the two invariants that
appears when we perform the Poincaré sum for the minimal models of type
$(m,m^{\prime})=(m,2q)$ where $m=p$ or $p^{2}$. For this we consider the
matrices $\Omega_{mm^{\prime}}$ and $\Omega_{m2}$ acting on the characters
$\chi_{r,s}$ where $1\leq r\leq m^{\prime}-1$ and $1\leq s\leq m-1$. Recall
that in the formula Eq. (2.83) the range of these indices is temporarily
doubled, with a constraint that ultimately reduces them to their physical
range. For, the case of $\Omega_{mm^{\prime}}$, we have:

$\begin{split}Z_{mm^{\prime}}&=\frac{1}{8}\sum_{r_{1},r_{2}=1}^{2m^{\prime}}\sum_{s_{1},s_{2}=1}^{2m}\bar{\chi}_{r_{1},s_{1}}\,\left[\Omega_{mm^{\prime}}\right]_{(r_{1},s_{1}),(r_{2},s_{2})}\,\chi_{r_{2},s_{2}}\\\
&=\frac{1}{8}\sum_{r_{1},r_{2}=1}^{2m^{\prime}}\sum_{s_{1},s_{2}=1}^{2m}\bar{\chi}_{r_{1},s_{1}}\,\delta_{s_{1},s_{2}}\delta_{r_{1},r_{2}}\,\chi_{r_{2},s_{2}}=\frac{1}{2}\sum_{r_{1}=1}^{m^{\prime}-1}\sum_{s_{1}=1}^{m-1}\absolutevalue{\chi_{r_{1},s_{1}}}^{2}\end{split}$
(2.86)

Thus when acting on $\chi_{r,s}$, the matrix
$(I_{mm^{\prime}})_{(r_{1},s_{1}),(r_{2},s_{2})}=\delta_{r_{1},r_{2}}\delta_{s_{1},s_{2}}$.
Next, we look at the case of $\Omega_{m2}$:

$\begin{split}Z_{m2}&=\frac{1}{8}\sum_{r_{1},r_{2}=1}^{2m^{\prime}}\sum_{s_{1},s_{2}=1}^{2m}\bar{\chi}_{r_{1},s_{1}}\,\left[\Omega_{m2}\right]_{(r_{1},s_{1}),(r_{2},s_{2})}\,\chi_{r_{2},s_{2}}\\\
&=\frac{1}{4}\sum_{s_{1}=1}^{m-1}\sum_{r_{1}}^{2m^{\prime}}\bar{\chi}_{r_{1},s_{1}}\chi_{(m^{\prime}-1)r_{1},s_{1}}\\\
&=\frac{1}{4}\sum_{s_{1}=1}^{m-1}\left(-\sum_{r_{1}\,{\rm
even}=1}^{2m^{\prime}}\absolutevalue{\chi_{r_{1},s_{1}}}^{2}+\sum_{r_{1}\,{\rm
odd}=1}^{2m^{\prime}}\bar{\chi}_{r_{1},s_{1}}\chi_{m^{\prime}-r_{1},s_{1}}\right)\\\
\end{split}$ (2.87)

So the matrix $I_{m2}$ is:

$[I_{m2}]_{(r_{1},s_{1}),(r_{2},s_{2})}=\delta_{s_{1},s_{2}}\times\begin{cases}-\delta_{r_{1},r_{2}},\hskip
28.45274ptr_{1}\mbox{ even}\\\
~{}~{}\delta_{r_{2},m^{\prime}-r_{1}},\quad\,r_{1}\mbox{ odd}\end{cases}$
(2.88)

Now we compute the inner products for these two matrices to get, for
$m^{\prime}=2q$:

$\begin{split}\mbox{Tr}(I_{mm^{\prime}}^{2})&=(m-1)(2q-1)\\\
\mbox{Tr}(I_{mm^{\prime}}I_{m2})&=(m-1)(2-q)\\\
\mbox{Tr}(I_{m2}^{2})&=(m-1)(2q-1)\end{split}$ (2.89)

We see that all the inner products are the same as in (2.36) upto an overall
factor of $(m-1)$ and so we see that the coefficients will be given by solving
the same equation as in (2.37). Due to the relation
$(r,s)\sim(m^{\prime}-r,m-r)$, at first it might seem that there is some
ambiguity in choosing the $X_{\rm seed}$. However this is not the case. If we
want the seed to be $\absolutevalue{\chi_{r_{0},s_{0}}}^{2}$, it is easily
verified that the two choices:

$\begin{split}(X_{\rm
seed})_{(r,s),(r^{\prime},s^{\prime})}&=\delta_{r,r_{0}}\delta_{r^{\prime},r_{0}}\delta_{s,s_{0}}\delta_{s^{\prime},s_{0}}\\\
(\tilde{X}_{\rm
seed})_{(r,s),(r^{\prime},s^{\prime})}&=\delta_{r,m^{\prime}-r_{0}}\delta_{r^{\prime},m^{\prime}-r_{0}}\delta_{s,m-s_{0}}\delta_{s^{\prime},m-s_{0}}\end{split}$
(2.90)

give the same result.

The result of this computation is that the linear combinations of physical
invariants appearing in the Poincaré sums for $(m,2q)$ minimal models (where
$m=p$ or $p^{2}$) are exactly the same as those that were obtained in the
computations of Poincaré sums for the case of $SU(2)_{k}$ where $k+2=n=2q$.
This can be verified for the examples in Table LABEL:virmin.

#### Three physical invariants

Minimal models can have at most three physical invariants. This happens when
the conditions are satisfied to admit an exceptional invariant, and requires
$m$ to be 12,18 or 30 and $m^{\prime}$ is any odd number (or the equivalent
with $m\leftrightarrow m^{\prime}$). However almost all such cases have
additional unphysical invariants. To have precisely three physical and no
unphysical invariants, the total number of invariants should also be 3. This
will happen when $m=12,18$ and $m^{\prime}=p,p^{2}$ for prime $p$. While there
is an exceptional invariant at $m=30$, it is easy to verify that the total
number of invariants in that case is at least 4.

Applying this to the unitary models, we find that the cases $(11,12)$ and
$(12,13)$ have precisely three physical invariants. At $(13,14)$ we again have
only two invariants. Starting with $(14,15)$ we start to encounter three
invariants but only for $(17,18),(18,19)$ are there no more. So for $m\geq
14,m\neq 17,18$, and other than the cases discussed in the previous
subsection, there is at least one unphysical invariant. This agrees with the
results found in [7] and provides a more complete picture.

##  3 Poincaré Sums for Multiple Genus-1 Boundaries

###  3.1 Generic RCFT

In this Section we consider RCFT’s on 3-manifolds with multiple genus-1
boundaries. This is crucial for the interpretation that a bulk gravity is dual
to an average over CFT’s. Suppose we have $N_{B}>1$ genus-1 boundaries having
independent modular parameters $\tau^{A},A=1,2,\cdots,N_{B}$. A generic seed
would be made up of linear combinations of terms like:

$Z_{i_{1}j_{1}}(\tau^{1})Z_{i_{2}j_{2}}(\tau^{2})\cdots
Z_{i_{N_{B}}j_{N_{B}}}(\tau^{N_{B}})$ (3.1)

with each factor defined as in Eqs. (2.5) or (2.6). Thereafter the Poincaré
sum is carried out over independent modular transforms of each $\tau^{A}$
666Alternatively one could identify the modular parameters of all the
boundaries and carry out a single Poincaré sum over them. We do not consider
this case here.. Since we allow linear combinations of seeds as above, the
final Poincaré sum is not a simple product of Poincaré sums over the
contribution from each boundary. Now, once we have determined the measure for
averaging over CFT’s in the single-boundary case, the same measure should
apply in the multi-boundary case. This requires that:

$Z(\tau^{1},\tau^{2},\cdots,\tau^{N_{B}})=\sum_{J}c_{J}\,Z_{J}(\tau^{1})\,Z_{J}(\tau^{2})\cdots
Z_{J}(\tau^{N_{B}})$ (3.2)

for every $N_{B}$, where the LHS is the result of the multiple Poincaré sum
described above and the RHS has the same coefficients $c_{J}$ that appeared in
Eq. (2.8). As emphasised in a different class of examples [11], it is not at
all obvious that this can hold for all $N_{B}>1$. We now turn to an
investigation of this criterion.

To start with, consider a 3-manifold with two boundaries $\Sigma$ and
$\Sigma^{\prime}$ of genus $g$ and $g^{\prime}$, each having its own moduli.
If the dual of the bulk gravity theory is a single CFT, then we expect that
the one- and two-boundary partition functions should be:

$Z_{\rm grav}(\Sigma)=Z_{\rm CFT}(\Sigma),\quad Z_{\rm
grav}(\Sigma,\Sigma^{\prime})=Z_{\rm CFT}(\Sigma)\,Z_{\rm
CFT}(\Sigma^{\prime})$ (3.3)

If the dual of the bulk theory is not a single CFT but rather an ensemble with
a probability distribution over physical CFTs, then the gravity partition
function will be an average over different CFT partition functions. For one
boundary we have seen that:

$Z_{\rm grav}(\Sigma)=\langle Z_{\rm
CFT}(\Sigma)\rangle=\sum_{J}c_{J}\,Z_{J}(\Sigma),\quad\sum_{J}c_{J}=1$ (3.4)

where the sum is over physical CFTs. The $J$’th CFT occurs with probability
$c_{J}$ 777Here the $c_{J}$ are normalised such that their sum is unity,
though we have not changed the notation to reflect this. It should be evident
that the normalised $c_{J}$ are equal to the un-normalised ones divided by
$\sum_{J}c_{J}$.. Once we have the probability distribution $\\{c_{J}\\}$, the
two-boundary partition function should be:

$Z_{\rm grav}(\Sigma,\Sigma^{\prime})=\langle Z_{\rm
CFT}(\Sigma,\Sigma^{\prime})\rangle=\sum_{i}c_{J}\,Z_{J}(\Sigma)\,Z_{J}(\Sigma^{\prime})$
(3.5)

with the same probabilities $c_{J}$ as in (3.4). The above equation (3.5) has
an obvious generalisation to an arbitrary number of boundaries, once we know
the $c_{J}$ and $Z_{J}(\Sigma)$. To show that indeed the dual of bulk gravity
theory is an ensemble of CFTs one should be able to compute the LHS and RHS of
(3.5) independently and show that they match.

Computation of the LHS is complicated for the following reason: one class of
multi-boundary manifolds is simply the sum of disconnected single-boundary
manifolds. For this choice the Poincaré sum is simply the product of
independent Poincaré sums over the disconnected components, leading to the
result $(\sum_{J}c_{J}Z_{J}(\Sigma))^{2}$. This is of course not the desired
answer. Extra contributions will come from manifolds that connect two or more
boundaries in all possible ways. These can be thought of as wormholes. If we
knew the analogue of a “vacuum” wormhole (corresponding to some particular
seed for the Poincaré sum) we would know how to compute these contributions
and could verify Eq. (3.5)and its $N$-boundary generalisations.

Lacking this information, we will take a different approach. We will assume
that the contribution of the fully disconnected 3-manifold, each with one
genus-1 boundary, is described by a product of independent Poincaré sums that
start with the identity seed, following results in the previous sections. Now
we seek a rule for the wormhole contribution corresponding to manifolds with
all numbers of boundaries $>1$. We then combine these in all possible ways to
generate the full $N_{B}$-boundary result. Then we try to find a seed such
that the Poincaré sum over it reproduces the desired contribution. We now
apply this to the case of SU(2)k.

###  3.2 SU(2)k WZW Models

We have seen that the SU(2)k WZW models with $n=k+2=2p$ (with prime $p>2$)
provides an infinite class of models having two physical partition functions.
For these models we will be able to find a simple and elegant rule for the
“wormhole” seeds that precisely reproduces the expected answer. While the
details we present below are for SU(2)k, this procedure works in the same way
for all models with two invariants.

Let us label each boundary torus with its corresponding independent modulus
$\tau^{(i)}$. The full partition function should be modular invariant with
respect to each of the moduli. To perform the gravity calculation, we compute
a Poincaré sum analogous to the torus partition function:

$Z_{\rm
grav}(\tau^{(1)},\tau^{(2)},\dots,\tau^{(N)})=\sum_{\gamma^{(1)},\dots,\gamma^{(N)}}Z_{\rm
seed}(\gamma^{(1)}\tau^{(1)},\gamma^{(2)}\tau^{(2)},\dots,\gamma^{(N)}\tau^{(N)})$
(3.6)

From Eq. (2.39) we find that the normalised probabilities, starting with the
identity seed, are:

$c_{A}=\frac{p+1}{2p-1},\quad c_{D}=\frac{p-2}{2p-1}$ (3.7)

We would like to show that Eq. (3.6) is equivalent to:

$\begin{split}Z_{\rm grav}(\tau^{(1)},\tau^{(2)},\dots,\tau^{(N)})&=\langle
Z_{\rm CFT}(\tau^{(1)},\tau^{(2)},\dots,\tau^{(N)})\rangle\\\
&=c_{A}\prod_{i=1}^{N}\,Z_{A}(\tau^{(i)})+c_{D}\prod_{i=1}^{N}\,Z_{D}(\tau^{(i)})\\\
\end{split}$ (3.8)

To simplify the notation we will write $Z_{A/D}(\tau^{(i)})$ as
$Z_{A/D}^{(i)}$ in what follows. Also for any function that depends on all the
$\tau^{(i)}$, we will not write the argument explicitly.

Let us start with the case of two boundaries. Then the postulated answer:

$Z_{\rm grav}=c_{A}\,Z_{A}^{(1)}Z_{A}^{(2)}+c_{D}\,Z_{D}^{(1)}Z_{D}^{(2)}$
(3.9)

should come from a sum over manifolds with the two genus-one surfaces as their
boundaries. In this case there can be two distinct types of manifold: a pair
of disconnected manifolds each with a torus boundary, and a single connected
manifold with two disjoint genus-1 boundaries:

$Z_{\rm grav}=Z_{\rm grav}^{\rm disconn.}+Z_{\rm grav}^{\rm conn.}$ (3.10)

From our previous analysis of the single-boundary case, we know that:

$Z_{\rm grav}^{\rm
disconn.}=\left(c_{A}Z_{A}^{(1)}+c_{D}Z_{D}^{(1)}\right)\left(c_{A}Z_{A}^{(2)}+c_{D}Z_{D}^{(2)}\right)$
(3.11)

It follows that the connected contribution should be:

$\begin{split}Z_{\rm grav}^{\rm
conn.}&=\left(c_{A}Z_{A}^{(1)}Z_{A}^{(2)}+c_{D}Z_{D}^{(1)}Z_{D}^{(2)}\right)-\left(c_{A}Z_{A}^{(1)}+c_{D}Z_{D}^{(1)}\right)\left(c_{A}Z_{A}^{(2)}+c_{D}Z_{D}^{(2)}\right)\\\
&=c_{A}\,c_{D}(Z_{A}^{(1)}-Z_{D}^{(1)})(Z_{A}^{(2)}-Z_{D}^{(2)})\\\
\end{split}$ (3.12)

Thus the connected part is a product of terms of the form $Z_{A}-Z_{D}$ with
an overall factor depending on the probabilities. We will see that this is the
general form for the connected contribution linking any number of boundaries.

Before going to the general case, let us extend the above calculation to three
genus-one boundaries. Here we expect that:

$\begin{split}Z_{\rm
grav}&=c_{A}\prod_{i=1}^{3}Z_{A}^{(i)}+c_{D}\prod_{i=1}^{3}Z_{D}^{(i)}\\\
\end{split}$ (3.13)

Now we have three distinct types of manifolds to sum over: (i) a sum of three
disconnected pieces each with one boundary, (ii) a manifold that connects any
two of the three boundaries, together with one that has a single boundary,
(iii) manifolds which connect all three boundaries. These cases are
associated, respectively, to the following partitions of $N=3$:
$\\{1,1,1\\},\\{2,1\\},\\{3\\}$. Henceforth we will label different
contributions by partitions of $N$. Thus we can write:

$Z_{\rm grav}=Z_{\rm grav}^{\\{111\\}}+Z_{\rm grav}^{\\{21\\}}+Z_{\rm
grav}^{\\{3\\}}$ (3.14)

Note that some of the partitions occur in multiple ways, for example there are
three ways to realise $\\{2,1\\}$ depending on the choice of boundary that
remains disconnected. The first term on the RHS will be a product over three
copies of (2.39), while the second term will be a sum of products of one copy
of (3.12) and one copy of (2.39), in three different ways. Then the last term
should be:

$\begin{split}Z_{\rm grav}^{\\{3\\}}&=Z_{\rm grav}-Z_{\rm
grav}^{\\{111\\}}-Z_{\rm grav}^{\\{21\\}}\\\
&=c_{A}\prod_{i=1}^{3}Z_{A}^{(i)}+c_{D}\prod_{i=1}^{3}Z_{D}^{(i)}-\prod_{i=1}^{3}(c_{A}Z_{A}^{(i)}+c_{D}Z_{D}^{(i)})\\\
&-c_{A}\,c_{D}\,(Z_{A}^{(1)}-Z_{D}^{(1)})(Z_{A}^{(2)}-Z_{D}^{(2)})(c_{A}Z_{A}^{(3)}+c_{D}Z_{D}^{(3)})\\\
&-c_{A}\,c_{D}\,(Z_{A}^{(1)}-Z_{D}^{(1)})(Z_{A}^{(3)}-Z_{D}^{(3)})(c_{A}Z_{A}^{(2)}+c_{D}Z_{D}^{(2)})\\\
&-c_{A}\,c_{D}\,(Z_{A}^{(2)}-Z_{D}^{(2)})(Z_{A}^{(3)}-Z_{D}^{(3)})(c_{A}Z_{A}^{(1)}+c_{D}Z_{D}^{(1)})\\\
\end{split}$ (3.15)

where the contribution of $Z_{\rm grav}^{\\{21\\}}$ contains three terms, one
for each pair of boundaries that are connected. Upon simplifying we have:

$Z_{\rm
grav}^{\\{3\\}}=c_{A}c_{D}(c_{D}-c_{A})(Z_{A}^{(1)}-Z_{D}^{(1)})(Z_{A}^{(2)}-Z_{D}^{(2)})(Z_{A}^{(3)}-Z_{D}^{(3)})$
(3.16)

so, we see again that the completely connected part is given as product of
terms of the form $Z_{A}-Z_{D}$. This is now easily generalised to $N_{B}$
boundaries. We find:

$Z_{\rm
grav}^{\\{N_{B}\\}}=f(c_{A},N_{B})\prod_{i=1}^{N_{B}}(Z_{A}^{(i)}-Z_{D}^{(i)}),\quad
N_{B}>1$ (3.17)

where $f(c_{A},N_{B})$ is a function of the number of boundaries and the
probability $c_{A}$, which we will compute below (we have used
$c_{D}=1-c_{A})$.

These observations tells us that if we can find a seed for generating $Z_{\rm
grav}^{\\{N_{B}\\}}$ then we can find the seed for the full $N_{B}$-boundary
gravity partition function. For example,

$\begin{split}Z_{\rm seed}^{N_{B}=1}&=Z_{\rm seed}^{\\{1\\}}(\tau^{(1)})\\\
Z_{\rm seed}^{N_{B}=2}&=\prod_{i=1}^{2}Z_{\rm
seed}^{\\{1\\}}(\tau^{(i)})+Z_{\rm seed}^{\\{2\\}}(\tau^{1},\tau^{(2)})\\\
Z_{\rm seed}^{N_{B}=3}&=\prod_{i=1}^{3}Z_{\rm
seed}^{\\{1\\}}(\tau^{(i)})+Z_{\rm seed}^{\\{1\\}}(\tau^{(1)})\,Z_{\rm
seed}^{\\{2\\}}(\tau^{(2)},\tau^{(3)})+Z_{\rm
seed}^{\\{1\\}}(\tau^{(2)})\,Z_{\rm seed}^{\\{2\\}}(\tau^{(1)},\tau^{(3)})\\\
&\quad+Z_{\rm seed}^{\\{1\\}}(\tau^{(3)})\,Z_{\rm
seed}^{\\{2\\}}(\tau^{(1)},\tau^{(2)})+Z_{\rm
seed}^{\\{3\\}}(\tau^{(1)},\tau^{(2)},\tau^{(3)})\\\ \end{split}$ (3.18)

One possible choice for $Z_{\rm seed}^{\\{1\\}}$, as we have seen, is:

$Z_{\rm seed}^{\\{1\\}}\sim\absolutevalue{\chi_{1}}^{2}$ (3.19)

with a suitable normalisation. But as we have seen in Eqs. (2.44),(2.45),
there are more general choices for this seed. Inserting the normalised values
of $c_{A},c_{D}$ for the identity seed, namely
$\frac{p+1}{2p-1},\frac{p-2}{2p-1}$, we find the general answer to be given
by:

$Z^{\\{1\\}}_{\rm seed}=a_{1}\\!\\!\\!\sum_{\begin{subarray}{c}\lambda\,\rm
odd,\,\neq
p\end{subarray}}\\!\\!\\!b_{\lambda}\absolutevalue{\chi_{\lambda}}^{2}+a_{2}\\!\\!\sum_{\begin{subarray}{c}\lambda\,\rm
even\end{subarray}}\\!\\!c_{\lambda}\absolutevalue{\chi_{\lambda}}^{2}+a_{3}\absolutevalue{\chi_{p}}^{2}+a_{4}\sum_{\lambda\,{\rm
odd},=1}^{p-2}d_{\lambda}({\overline{\chi}}_{\lambda}\chi_{2p-\lambda}+{\rm
c.c.})$ (3.20)

with:

$a_{3}=\frac{1}{2p-1}-a_{1}-a_{2},\quad
2a_{4}=a_{1}+2a_{2}-\frac{1}{2p-1},\quad\sum b_{\lambda}=\sum c_{\lambda}=\sum
d_{\lambda}=1$ (3.21)

Clearly $a_{1}=\frac{1}{2p-1},b_{1}=1$ with all other $a_{i},b_{\lambda}=0$ is
a possible solution in this case, leading to Eq. (3.19). But far more general
solutions are allowed.

Let us now try to find a seed that will give us the linear combination
$Z_{A}-Z_{D}$. This is given by Eqs.(2.44),(2.45) with $\alpha=1,\beta=-1$.
Then Eq. (2.45) leads to:

$a_{3}=\frac{p-2}{3(p^{2}-1)}-a_{1}-a_{2},\quad
2a_{4}=a_{1}+2a_{2}-\frac{1}{p+1}$ (3.22)

This time there is no solution with all $a_{i}=0$ other than $a_{1}$. One of
the simplest solutions is:

$a_{1}=\frac{1}{p+1},\quad a_{3}=\frac{1-2p}{3(p^{2}-1)},\quad
a_{2}=a_{4}=0,\quad b_{1}=1$ (3.23)

However as we will shortly see, there are other solutions that are of
interest.

We can now write the connected contribution for $N_{B}$ boundaries:

$Z_{\rm grav}^{\rm
conn.}(\tau^{(1)},\dots,\tau^{(N_{B})})=\sum_{\gamma^{(i)}\in\Gamma(N_{0})\backslash\Gamma}Z_{\rm
seed}^{\rm conn.}(\\{\gamma^{(i)}\tau^{(i)}\\})$ (3.24)

where:

$Z_{\rm seed}^{\rm
conn.}(\\{\tau^{(i)}\\})=f(c_{A},N_{B})\prod_{i=1}^{N_{B}}Z_{\rm
seed}^{A-D}(\tau^{(i)})$ (3.25)

Finally, using (3.24) in expressions of the form (3.18) gives the general seed
for the partition function with $N_{B}$ genus-one boundaries.

It remains to find the form of the function $f(c_{A},N_{B})$. We provide a
recursive algorithm to generate this function, where the answer for a given
$N_{B}$ is given in terms of the answer for all numbers of boundaries
$<N_{B}$. To do this we first introduce a generating function for computing
the moments of a random variable (say $x$):

$P(k)=\langle e^{kx}\rangle=\sum_{n\geq 0}\frac{k^{n}}{n!}\langle
x^{n}\rangle\quad\mbox{and}\quad\mbox{ln }P(k)\coloneqq\sum_{n\geq
1}\frac{k^{n}}{n!}\langle x^{n}\rangle_{c}$ (3.26)

where $\langle x^{n}\rangle$ are the usual moments and $\langle
x^{n}\rangle_{c}$ denote the connected moments. We can use the above equations
to find relations between the two:

$\sum_{n\geq 0}\frac{k^{n}}{n!}\langle x^{n}\rangle=\exp(\sum_{n\geq
1}\frac{k^{n}}{n!}\langle x^{n}\rangle_{c})=\prod_{n\geq 1}\sum_{l\geq
0}\frac{k^{nl}}{l!}\left(\frac{\langle x^{n}\rangle_{c}}{n!}\right)^{l}$
(3.27)

which implies that:

$\langle x^{m}\rangle=m!\sum_{\\{l_{n}\\}}\prod_{n\geq
1}^{m}\frac{1}{l_{n}!}\left(\frac{\langle
x^{n}\rangle_{c}}{n!}\right)^{l_{n}}$ (3.28)

The sum is over the set of $\\{l_{n}\\}$ such that $\sum_{n}nl_{n}=m$, which
are just the partitions of the integer $m$.

To apply this to the present case we replace $x$ by $Z$ and find:

$\langle Z^{m}\rangle=m!\sum_{\\{l_{n}\\}}\prod_{n\geq
1}^{m}\frac{1}{l_{n}!}\left(\frac{\langle
Z^{n}\rangle_{c}}{n!}\right)^{l_{n}}$ (3.29)

Replacing the LHS by the desired average over products of CFT’s, and
separating two terms in the RHS from the rest in an obvious way, we get:

$(c_{A}Z_{A}^{m}+c_{D}Z_{D}^{m})=\langle
Z\rangle^{m}+m!\sum_{\\{l_{n}\\}}^{\prime}\prod_{n\geq
1}^{m-1}\frac{1}{l_{n}!}\left(\frac{\langle
Z^{n}\rangle_{c}}{n!}\right)^{l_{n}}+\langle Z^{m}\rangle_{c}$ (3.30)

This can now be solved to get the connected part:

$\langle
Z^{m}\rangle_{c}=(c_{A}Z_{A}^{m}+c_{D}Z_{D}^{m})-(c_{A}Z_{A}+c_{D}Z_{D})^{m}-m!\sum_{\\{l_{n}\\}}^{\prime}\prod_{n\geq
1}^{m-1}\frac{1}{l_{n}!}\left(\frac{\langle
Z^{n}\rangle_{c}}{n!}\right)^{l_{n}}$ (3.31)

Now using the fact that $\langle Z^{m}\rangle_{c}\,,(n>1)$ is of the form
$f(c_{A},m)(Z_{A}-Z_{D})^{m}$ and comparing the coefficients of $Z_{A}^{m}$,
we finally get:

$f(c_{A},m)=c_{A}(1-c_{A}^{m-1})-m!\sum_{\\{l_{n}\\}}^{\prime}\frac{(c_{A})^{l_{1}}}{l_{1}!}\prod_{n\geq
2}^{m-1}\frac{1}{l_{n}!}\left(\frac{f(c_{A},n)}{n!}\right)^{l_{n}}$ (3.32)

This is the desired recursive formula for the coefficients in Eq. (3.25). Some
examples of the above formula are as follows:

$\begin{split}m=2\\!:\quad f(c_{A},2)&=c_{A}(1-c_{A})\\\ m=3\\!:\quad
f(c_{A},3)&=c_{A}(1-c_{A}^{2})-3(c_{A})^{2}(1-c_{A})=c_{A}(1-c_{A})(1-2c_{A})\\\
m=4\\!:\quad f(c_{A},4)&=c_{A}(1-c_{A})(1-6c_{A}+6c_{A}^{2})\end{split}$
(3.33)

The fact that such a rule exists at all, and has nice properties, is very
encouraging. In principle it could be checked if we can have a better
understanding of the wormhole-type manifolds, though – as already emphasised
here and in [11], the semi-classical notion of manifold does not really apply
in the RCFT context.

We have shown how to write down a general seed for the partition function with
$N_{B}$ genus-1 boundaries. This expressions involved free parameters and
hence give an infinite family of possible seeds. To conclude this section we
now give an example of a particular choice of these parameters which allows us
to write down the full $N_{B}$-boundary seed in a simple compact form. We are
not sure whether there is any physical interpretation for this choice, but
mathematically it is appealing.

We begin with the general form of $Z_{\rm seed}^{\\{1\\}}$, given in Eq.
(3.20). Now we set $b_{1}=c_{2}=1$ and all the remaining
$b_{\lambda}=c_{\lambda}=0$. We also demand that $a_{4}=0$. From Eq. (3.21),
this leads to the constraint $a_{1}+2a_{2}=\frac{1}{2p-1}$, so the
disconnected seed can be written:

$Z^{\\{1\\}}_{\rm
seed}=\left(\frac{1}{2p-1}-2a_{2}\right)\absolutevalue{\chi_{1}}^{2}+a_{2}\absolutevalue{\chi_{2}}^{2}+a_{2}\absolutevalue{\chi_{p}}^{2}$
(3.34)

For the case of $Z_{\rm seed}^{A-D}$, we make a similar demand. Here we will
denote the parameters with a prime to distinguish from the above. We set
$b_{1}^{\prime},c_{2}^{\prime}=1$ with the remaining
$b_{\lambda}^{\prime}=c_{i}^{\prime}=0$. We also set $a^{\prime}_{4}=0$. This
implies $a^{\prime}_{1}+2a^{\prime}_{2}=\frac{1}{p+1}$. Then the corresponding
seed is:

$Z^{A-D}_{\rm
seed}=\left(\frac{1}{p+1}-2a^{\prime}_{2}\right)\absolutevalue{\chi_{1}}^{2}+a^{\prime}_{2}\absolutevalue{\chi_{2}}^{2}+\left(a^{\prime}_{2}-\frac{2p-1}{3(p^{2}-1)}\right)\absolutevalue{\chi_{p}}^{2}$
(3.35)

Thus we have found the required seeds to set up the recursion relation, as
functions of two arbitrary parameters $a_{2},a_{2}^{\prime}$. We now show that
for special values of these parameters, things simplify considerably. Let us
choose:

$a_{2}=\frac{p-2}{3(p-1)(2p-1)}\,,\quad
a_{2}^{\prime}=\frac{p-2}{3(p-1)(p+1)}$ (3.36)

Now using (3.18), (3.34) and (3.36) we find for one boundary:

$Z_{\rm seed}^{N_{B}=1}=Z_{\rm
seed}^{\\{1\\}}=\frac{p+1}{(2p-1)(3p-3)}\left(\absolutevalue{\chi_{1}}^{2}+\frac{p-2}{p+1}\absolutevalue{\chi_{2}}^{2}+\frac{p-2}{p+1}\absolutevalue{\chi_{p}}^{2}\right)$
(3.37)

while for two boundaries, we get:

$\begin{split}Z_{\rm seed}^{N_{B}=2}&=\prod_{i=1}^{2}Z_{\rm
seed}^{\\{1\\}}(\tau^{(i)})+p_{A}(1-p_{A})Z_{\rm
seed}^{A-D}(\tau^{(1)},\tau^{(2)})\\\
&=\frac{p+1}{(2p-1)(3p-3)^{2}}\left(\prod_{i=1}^{2}\left(\absolutevalue{\chi_{1}^{(i)}}^{2}+\left(\frac{p-2}{p+1}\right)\absolutevalue{\chi_{2}^{(i)}}^{2}\right)+\left(\frac{p-2}{p+1}\right)\prod_{i=1}^{2}\absolutevalue{\chi_{p}^{(i)}}^{2}\right)\\\
\end{split}$ (3.38)

From the above, we see a pattern emerging. So one may conjecture that for
$N_{B}$ boundaries, $Z_{\rm seed}$ is:

$Z_{\rm
seed}^{N_{B}}=\frac{p+1}{(2p-1)(3p-3)^{N_{B}}}\left(\prod_{i=1}^{N}\left(\absolutevalue{\chi_{1}^{(i)}}^{2}+\left(\frac{p-2}{p+1}\right)\absolutevalue{\chi_{2}^{(i)}}^{2}\right)+\left(\frac{p-2}{p+1}\right)\prod_{i=1}^{N_{B}}\absolutevalue{\chi_{p}^{(i)}}^{2}\right)$
(3.39)

Now one proves the conjecture by directly evaluating the Poincaré sum over
this seed and finding that it gives the desired $N_{B}$-boundary answer. We
get:

$\begin{split}\sum_{\gamma}Z_{\rm
seed}^{N_{B}}&=\frac{p+1}{(2p-1)(3p-3)^{N_{B}}}\left(\prod_{i=1}^{N_{B}}(3p-3)Z_{A}^{(i)}+\left(\frac{p-2}{p+1}\right)\prod_{i=1}^{N_{B}}(3p-3)Z_{D}^{(i)}\right)\\\
&=\frac{1}{2p-1}\left((p+1)\prod_{i=1}^{N_{B}}Z_{A}^{(i)}+(p-2)\prod_{i=1}^{N_{B}}Z_{D}^{(i)}\right)\\\
&=\langle Z(\tau^{(1)})\cdots Z(\tau^{(N_{B})})\rangle\\\ \end{split}$ (3.40)

Thus we see that (3.39) provides a simple, compact choice of seed which
reproduces exactly the desired result for the $N_{B}$-boundary partition
function. However, as explained above, it is not unique and one can have more
complicated expressions by choosing the parameters differently. It would be
worth investigating whether some additional physical criterion determines the
free parameters to produce this compact result.

##  4 Discussion and Conclusions

We have attempted to resurrect the original attempt by [7] of finding “non-
semi-classical” 3d gravity duals to 2d rational conformal field theories.
While doing so, we have generalised their considerations to SU($N$)k WZW
models, and also extended their results on Virasoro minimal models. Most
important, we have explored the idea that the dual of AdS3 gravity can be an
average of an ensemble of CFT’s rather than a single CFT. The ensembles we
have studied are extremely small, in most cases having just two members, which
makes the problem highly tractable. In the present work, our considerations
have been limited to 3d spacetimes with one or more genus-1 boundaries.

Our principal results are: (i) there are infinitely many unitary WZW models
having single CFT duals, whose partition functions are correctly reproduced by
a Poincaré sum. This is in contrast to the minimal models, for which the same
considerations only lead to two unitary cases with single duals, namely
$(m,m+1)=(3,4)$ and $(4,5)$, (ii) there are infinitely many unitary WZW
models, as well as both unitary and non-unitary minimal models, whose Poincaré
sum gives an average of two RCFT’s. In these cases the coefficient of the
average can be computed for the entire infinite series as simple functions of
certain prime numbers labelling the individual models in the series, (iii) in
the cases where there are two RCFT’s, it is possible to define a Poincaré
series that correctly averages the products of partition functions of each
RCFT over multiple boundaries. To do so, it requires contributions from (non-
semi-classical) manifolds that are both disconnected and that connect multiple
boundaries in all possible ways, the latter being some kind of analogue
wormholes, (iv) there are also cases, notably for SU($N$)1, where the dual can
be an average over arbitrarily many CFT’s.

Our analysis has been most detailed for SU(2)k models. However the results for
minimal models are quite similar. This seems to arise from a kind of
“doubling” relation between the two classes of models, which is well-known
from the classic papers of Cappelli, Itzykson and Zuber [21, 22]. We hope to
return to more general classes of RCFT for which a given set of characters
admits more than one modular invariant.

In the early studies of Poincaré series for gravity [1], it was assumed that
the “seed” for the series is the identity module, corresponding to thermal
AdS3 on the gravity side. Later studies, notably [6], allowed for more general
primary seeds and studied their contribution. The recent work [18] also
invokes additional seeds associated to conical singularities. Thus there is no
real consensus on what should be the seed for a Poincaré series. In the
present work we have similarly taken an agnostic view on this. However, as
seen in many of our tables, the identity seed often seems to return a positive
linear combination of CFT partition functions, while other primary seeds often
do not 888Amusingly this seems to be the opposite of what is found in pure
AdS3 gravity, where the identity seed leads to negative coefficients and other
primary seeds are added to try and cure this.. Related to this is the question
of additional seeds that must be added to account for analogue wormholes in
the multi-boundary case. We have found candidates that work correctly, which
we think is remarkable, but we do not have a first-principles reason why the
wormhole seeds should be what we propose. Progress on this issue would be most
helpful and could illuminate the more general case of pure AdS gravity.

Our considerations have been based on considering Euclidean AdS3 with one or
more genus-1 boundaries. For minimal models, it has been argued [8] that
stronger constraints are obtained by considering, for example, a single
genus-2 boundary. It would be worthwhile to know what constraints can be put
on the dualities described in the present work by considering genus-2 or
higher-genus boundaries.

In the context of Narain lattices which have U(1)2D Kac-Moody algebras, [11]
proposed that the gravity dual is a topological U(1)2D Chern-Simons theory. In
the same spirit, we have suggested that the gravity dual for WZW theories
based on non-abelian SU($N$)k $\times$ SU($N$)k Kac-Moody algebras is a
topological SU($N$)k $\times$ SU($N$)k Chern-Simons theory. However we did not
really make use of this in the present work. Perhaps one can obtain more
information about the dualities discussed by invoking properties of the
“gravity” side.

Among possible future directions, one would be to investigate SU($N$)1 WZW
models in greater detail. Since all the modular invariants are physical, we do
not need to restrict to any special class as was the case for SU(2)k. Also the
number of invariants is unbounded. Moreover, for any given number of
invariants there will, in general, be an infinite family of models within this
category. Another interesting point is that for each such family we can
consider the large-$N$ (large-$\mathbf{c}$) limit, which may permit
considerations that are closer to being semi-classical. We hope to return to
this in future.

Another fascinating direction is to start with meromorphic RCFT. As is well-
known, these correspond to free bosons on an even unimodular lattice,
orbifolds thereof and some generalisations [34, 33]. Such CFT’s typically have
Kac-Moody algebras and their single character is just an integral linear
combination of Kac-Moody characters. This automatically gives us (at least) a
pair of CFT’s: the meromorphic one and the diagonal invariant. Thus when
computing Poincaré sums, one expects to find linear combinations of both
partition functions. This has already been verified in one example, namely
SU(25)1, in the present paper – see the discussion following Eq. (2.55). A
large fraction of meromorphic CFT’s are based on non-simple Kac-Moody algebras
which renders the analysis more novel as well as more complicated.

Recently, the considerations of [7] have been extended to the case of boundary
minimal models [35]. This requires the introduction of Randall-Sundrum branes
[36] associated to Cardy states [37]. It should be possible to generalise
these ideas to the WZW models that we have studied here, and even take the
large-$\mathbf{c}$ limit as suggested above.

Finally, the MLDE approach to classifying rational CFT [38, 39] might be
fruitfully combined with the Poincaré sum method to generate new examples of
RCFT with multiple modular invariants.

## Acknowledgements

We thank Matthias Gaberdiel, Alex Maloney and Rahul Poddar for helpful
discussions, and Tushar Gopalka for collaboration at the early stages of this
project. VM would like to acknowledge the INSPIRE Scholarship for Higher
Education, Government of India. PS would like to acknowledge support from the
Clarendon Fund and the Mathematical Institute, University of Oxford. All of us
are grateful for support from a grant by Precision Wires India Ltd. for String
Theory and Quantum Gravity research at IISER Pune.

## Appendices

##  Appendix A Congruence subgroups

Here we collect a number of definitions and properties of congruence subgroups
of $\Gamma=$ SL(2,$\mathbb{Z}$). We have:

$\begin{split}\Gamma(N)&:=\left\\{\begin{pmatrix}a&b\\\
c&d\end{pmatrix}\in\Gamma\bigg{|}~{}a,d\equiv 1;b,c\equiv 0\hbox{ mod
}N\right\\}\\\ \Gamma_{1}(N)&:=\left\\{\begin{pmatrix}a&b\\\
c&d\end{pmatrix}\in\Gamma\bigg{|}~{}a,d\equiv 1;c\equiv 0\hbox{ mod
}N\right\\}\\\ \Gamma_{0}(N)&:=\left\\{\begin{pmatrix}a&b\\\
c&d\end{pmatrix}\in\Gamma\bigg{|}~{}c\equiv 0\hbox{ mod
}N\right\\}\end{split}$ (A.1)

We have the inclusions:
$\Gamma(N)\subset\Gamma_{1}(N)\subset\Gamma_{0}(N)\subset\Gamma$. Also,
$\Gamma(nN)\subset\Gamma(N)$ for any integer $n>1$, and similarly for the
others. We also have:

$\Gamma^{0}(N):=\left\\{\begin{pmatrix}a&b\\\
c&d\end{pmatrix}\bigg{|}~{}b\equiv 0\hbox{ mod }N\right\\}$ (A.2)

$\Gamma(N)$ is called the principal congruence subgroup, and is a normal
subgroup of $\Gamma$. Any subgroup of $\Gamma$ that contains $\Gamma(N)$ for
some $N$ is called a congruence subgroup, but will in general not be normal.
The smallest such $N$ (corresponing to the largest $\Gamma(N)$) is called the
level of the congruence subgroup.

We can write down the indices of various congruence subgroups in $\Gamma$:

$\begin{split}[\Gamma:\Gamma(N)]&=N^{3}\prod_{p|N}\left(1-\frac{1}{p^{2}}\right)\\\
[\Gamma:\Gamma_{1}(N)]&=N^{2}\prod_{p|N}\left(1-\frac{1}{p^{2}}\right)\\\
[\Gamma:\Gamma_{0}(N)]&=N\prod_{p|N}\left(1+\frac{1}{p}\right)\end{split}$
(A.3)

from which we find the useful relation:

$[\Gamma_{1}(N):\Gamma(N)]=N$ (A.4)

##  Appendix B Basic formulae for WZW and Virasoro minimal models

### SU(2)k WZW models

We will restrict our attention to the unitary series of SU(2)k models. An
integer level $k$ characterises the values of $\mathbf{c},h_{i}$ in the model.
Defining the “height” $n=k+2$, we have:

$\mathbf{c}=3-\frac{6}{n}\,,\quad(n{~{}\rm integer}\geq 3)$ (B.1)

The primaries can be labelled by $\lambda=2j+1$ where $j$ is the isospin, with
$1\leq\lambda\leq n-1$. Their holomorphic conformal dimensions are given as:

$h_{\lambda}=\frac{\lambda^{2}-1}{4n}$ (B.2)

Under modular transformations $T$ and $S$, the characters
$\chi_{\lambda}(\tau)$ go into linear combinations of themselves as follows:

$\begin{split}T_{\lambda,\lambda^{\prime}}&=\delta_{\lambda,\lambda^{\prime}}\exp(2\pi
i\left(h_{\lambda}-\frac{\mathbf{c}}{24}\right))\\\
S_{\lambda,\lambda^{\prime}}&=\sqrt{\frac{2}{n}}\sin(\frac{\pi\lambda\lambda^{\prime}}{n})\\\
\end{split}$ (B.3)

### SU(3)k WZW models

For SU(3)k the representations are labelled by a pair of integers
$\lambda_{1},\lambda_{2}\geq 1$. In terms of these and the height $n=k+3$, we
have the constraint $\lambda_{1}+\lambda_{2}<n$. The central charge and
conformal dimensions are:

$\mathbf{c}=8-\frac{24}{n}\;,\quad
h_{\lambda}=\frac{\lambda_{1}^{2}+\lambda_{2}^{2}+\lambda_{1}\lambda_{2}-3}{3n}$
(B.4)

It follows from the expression for the conformal dimensions that the semi-
conductor in this case has the simple form, $N_{0}=3n$, whereas the total
number of characters that appear at level $k$ is $\frac{n(n-1)}{2}$. Finally,
the modular $T$ and $S$ matrices can be expressed as:

$\begin{split}T_{\lambda,\mu}&=\delta_{\lambda,\mu}\exp(2\pi
i\left(h_{\lambda}-\frac{c}{24}\right))\\\ \\\
S_{\lambda,\mu}&=-\frac{i}{n\sqrt{3}}\left[\exp(\frac{2\pi
i}{3n}\left(2\lambda_{1}\mu_{1}+\lambda_{1}\mu_{2}+\lambda_{2}\mu_{1}+2\lambda_{2}\mu_{2}\right))\right.\\\
&\hskip 45.52458pt+\exp(\frac{2\pi
i}{3n}\left(-\lambda_{1}\mu_{1}-2\lambda_{1}\mu_{2}+\lambda_{2}\mu_{1}-\lambda_{2}\mu_{2}\right))\\\
&\hskip 45.52458pt+\exp(\frac{2\pi
i}{3n}\left(-\lambda_{1}\mu_{1}+\lambda_{1}\mu_{2}-2\lambda_{2}\mu_{1}-\lambda_{2}\mu_{2}\right))\\\
&\hskip 45.52458pt-\exp(\frac{2\pi
i}{3n}\left(-\lambda_{1}\mu_{1}-2\lambda_{1}\mu_{2}-2\lambda_{2}\mu_{1}-\lambda_{2}\mu_{2}\right))\\\
&\hskip 45.52458pt-\exp(\frac{2\pi
i}{3n}\left(2\lambda_{1}\mu_{1}+\lambda_{1}\mu_{2}+\lambda_{2}\mu_{1}-\lambda_{2}\mu_{2}\right))\\\
&\hskip 45.52458pt\left.-\exp(\frac{2\pi
i}{3n}\left(-\lambda_{1}\mu_{1}+\lambda_{1}\mu_{2}+\lambda_{2}\mu_{1}+2\lambda_{2}\mu_{2}\right))\right]\end{split}$
(B.5)

### SU($N$)k WZW models

As in the case of SU(2)k, the level $k$ characterises the values of
$\mathbf{c},h_{i}$ in the model. Defining the height $n=k+N$, we have:

$\mathbf{c}=\frac{k(N^{2}-1)}{N+k}=(N^{2}-1)\left(1-\frac{N}{n}\right)$ (B.6)

For any given level $k$ only a finite number of representations are allowed,
which are selected by the constraint:

$\sum_{i=1}^{N-1}\lambda_{i}<n$ (B.7)

where $\lambda\equiv(\lambda_{1},\lambda_{2},\dots,\lambda_{N-1})$ is the
Dynkin label for SU($N$) 999Note that we are following the convention where
the Dynkin label of the trivial representation is the unit vector
$(1,1,\dots,1)$, instead of the zero vector., which labels the characters. The
number of such representations is given by:

$\\#(\text{at level }k)=\frac{(N+k-1)!}{k!(N-1)!}$ (B.8)

The conformal dimension of the representation labelled by $\lambda$ can be
expressed as:

$h_{\lambda}=\frac{(\lambda-\rho,\lambda+\rho)}{2n}$ (B.9)

where $\rho=(1,1,\dots,1)$, and the inner product is
$(x,y)=x_{i}\kappa_{ij}y_{j}$, where $\kappa$ is the quadratic form matrix of
SU($N$):

$\kappa\equiv\frac{1}{N}\begin{pmatrix}N-1&N-2&N-3&\cdots&2&1\\\
N-2&2(N-2)&2(N-3)&\cdots&4&2\\\ N-3&2(N-3)&3(N-3)&\cdots&6&3\\\
\vdots&\vdots&\vdots&\ddots&\vdots&\vdots\\\ 2&4&6&\cdots&2(N-2)&N-2\\\
1&2&3&\cdots&N-2&N-1\\\ \end{pmatrix}$ (B.10)

The action of $S$ and $T$ modular transformations on the characters are given
as:

$\displaystyle T_{\lambda\lambda^{\prime}}$
$\displaystyle=\delta_{\lambda,\lambda^{\prime}}\exp(2\pi
i\left(h_{\lambda}-\frac{\mathbf{c}}{24}\right))$ (B.11) $\displaystyle
S_{\lambda\lambda^{\prime}}$
$\displaystyle=\frac{i^{N(N-1)/2}}{\sqrt{Nn^{N-1}}}\sum_{w\in
W}\mbox{det}(w)\,\exp(-2\pi i\left(\frac{\lambda\cdot
w(\lambda^{\prime})}{n}\right))$ (B.12)

where $W$ is the Weyl group of $SU(N)$.

### Virasoro minimal models

These models are labeled by two co-prime integers $(m,m^{\prime})$ with
$m^{\prime}>m>2$. The unitary minimal models are the sub-series obtained by
setting $m^{\prime}=m+1$. The central charge in terms of these parameters is:

$c(m,m^{\prime})=1-\frac{6(m-m^{\prime})^{2}}{mm^{\prime}}$ (B.13)

The primaries for a given model are labeled by a pair of integers $(r,s)$
where:

$h_{r,s}=\frac{(mr-m^{\prime}s)^{2}-(m-m^{\prime})^{2}}{4mm^{\prime}}$ (B.14)
$1\leq r\leq m^{\prime}-1\,,\quad 1\leq s\leq
m-1\,,\quad(r,s)\sim(m^{\prime}-r,m-s)$

Due to the above identifications we have a total of $(m-1)(m^{\prime}-1)/2$
number of independent primaries. Hence this is the dimension of the
SL(2,$\,\mathbb{Z}$) representation which will act on the characters
$\chi_{r,s}$.

The $T,S$ matrices in this representation are given as:

$\begin{split}T_{rs,vw}&=\delta_{r,v}\delta_{s,w}\exp(2\pi
i\left(h_{r,s}-\frac{c}{24}\right))\\\\[5.69054pt]
S_{rs,vw}&=\sqrt{\frac{8}{mm^{\prime}}}(-1)^{1+sv+rw}\sin(\frac{\pi
rvm}{m^{\prime}})\sin(\frac{\pi swm^{\prime}}{m})\end{split}$ (B.15)

##  Appendix C Sage calculations for SU(2)k

Table 7: Poincaré sums for SU(2)k Level $(k,c,N_{0})$ |  Seed primary label $\lambda$ |  Poincaré sum
---|---|---
$(1,1,4)$ |  $1,2$ |  $24\,Z_{A}$
$(2,\frac{3}{2},16)$ |  $1,2,3$ |  $32\,Z_{A}$
$(3,\frac{9}{5},20)$ |  $1,2,3,4$ |  $36\,Z_{A}$
$(4,2,20)$ |  odd $,\neq 3$ |  $8\,(4Z_{A}+Z_{D})$
|  even |  $32\,(2Z_{A}-Z_{D})$
|  $3$ |  $48\,Z_{D}$
$(5,\frac{15}{7},28)$ |  $1,\dots,6$ |  $24\,Z_{A}$
$(6,\frac{9}{4},32)$ |  $1,3,4,5,7$ |  $32\,(Z_{A}+Z_{D})$
|  $2,6$ |  $16\,(7Z_{A}-5Z_{D})$
$(7,\frac{7}{3},36)$ |  $1,\dots,8$ |  $54\,Z_{A}$
$(8,\frac{12}{5},40)$ |  odd $,\neq 5$ |  $24\,(2Z_{A}+Z_{D})$
|  even |  $48\,(2Z_{A}-Z_{D})$
|  $5$ |  $96\,Z_{D}$
$(9,\frac{27}{11},44)$ |  $1,\dots,10$ |  $54\,Z_{A}$
$(10,\frac{5}{2},48)$ |  $1,5,7,11$ |  $32(Z_{A}+Z_{D}+Z_{E}^{(10)})$
|  $2,10$ |  $32(4Z_{A}-2Z_{D}-Z_{E}^{(10)})$
|  $3,6,9$ |  $64(Z_{A}+Z_{D}-Z_{E}^{(10)})$
|  $4,8$ |  $32(3Z_{A}-3Z_{D}+2Z_{E}^{(10)})$
$(11,\frac{33}{13},52)$ |  $1,\dots,12$ |  $84\,Z_{A}$
$(12,\frac{18}{7},56)$ |  odd $,\neq 7$ |  $8\,(8Z_{A}+5Z_{D})$
|  even |  $64\,(2Z_{A}-Z_{D})$
|  $7$ |  $144\,Z_{D}$
$(13,\frac{13}{5},60)$ |  $1,2,4,7,8,11,13,14$ |  $8\,(6Z_{A}+Z_{new}^{(13)})$
|  $3,6,9,12$ |  $72\,(2Z_{A}-Z_{new}^{(13)})$
|  $5,10$ |  $96\,Z_{new}^{(13)}$
$(14,\frac{21}{8},64)$ |  $1,3,5,7,8,9,11,13,15$ |  $64\,(Z_{A}+Z_{D})$
|  $2,4,6,10,12,14$ |  $32\,(5Z_{A}-3Z_{D})$
$(15,\frac{45}{17},68)$ |  $1,\dots,16$ |  $108\,Z_{A}$
$(16,\frac{8}{3},72)$ |  $1,5,7,11,13,17$ |  $72\,(Z_{A}+Z_{E})$
|  $2,4,6,8,10,12,14,16$ |  $72\,(2Z_{A}-Z_{D})$
|  $3,15$ |  $24\,(3Z_{A}+7Z_{D}-7Z_{E})$
|  $9$ |  $48\,(5Z_{D}-2Z_{E})$
$(17,\frac{51}{19},76)$ |  $1,\dots,18$ |  $120\,Z_{A}$
$(18,\frac{27}{10},80)$ |  $1,3,7,9,11,13,17,19$ |  $32\,(2Z_{A}+2Z_{D}+Z_{new}^{(18)})$
|  $2,6,14,18$ |  $16\,(13Z_{A}-5Z_{D}-4Z_{new}^{(18)})$
|  $4,8,12,16$ |  $48\,(3Z_{A}-3Z_{D}+2Z_{new}^{(18)})$
|  $5,10,15$ |  $128\,(Z_{A}+Z_{D}-Z_{new}^{(18)})$
$(19,\frac{19}{7},84)$ |  $\neq 0\mbox{ mod }3,7$ |  $24\,(4Z_{A}+Z_{new}^{(19)})$
|  $=0\mbox{ mod }3$ |  $96\,(2Z_{A}-Z_{new}^{(19)})$
|  $=0\mbox{ mod }7$ |  $144\,Z_{new}^{(19)}$
$(20,\frac{30}{11},88)$ |  odd $,\neq 11$ |  $24\,(4Z_{A}+3Z_{D})$
|  even |  $96\,(2Z_{A}-Z_{D})$
|  $11$ |  $240\,Z_{D}$
$(21,\frac{63}{23},92)$ |  $1,\dots,22$ |  $144\,Z_{A}$
$(22,\frac{11}{4},96)$ |  $1,5,7,11,13,17,19,23$ |  $32\,(2Z_{A}+2Z_{D}+Z_{new1}^{(22)})$
|  $2,10,14,22$ |  $16\,(10Z_{A}-6Z_{D}+Z_{new1}^{(22)}+Z_{new2}^{(22)})$
|  $3,9,12,15,21$ |  $64\,(2Z_{A}+2Z_{D}-Z_{new1}^{(22)})$
|  $4,20$ |  $32\,(10Z_{A}-6Z_{D}-Z_{new1}^{(22)}-2Z_{new2}^{(2)})$
|  $6,18$ |  $32\,(2Z_{A}+2Z_{D}-Z_{new1}^{(22)}+3Z_{new2}^{(2)})$
|  $8,16$ |  $64\,(4Z_{A}-4Z_{D}+Z_{new1}^{(22)}-Z_{new2}^{(2)})$
$(23,\frac{69}{25},100)$ |  $1,\dots,24$ |  $150\,Z_{A}$
$(24,\frac{36}{13},104)$ |  odd $,\neq 13$ |  $8\,(14Z_{A}+11Z_{D})$
|  even |  $112\,(2Z_{A}-Z_{D})$
|  $13$ |  $288\,Z_{D}$
$(25,\frac{25}{9},108)$ |  $\neq 0\mbox{ mod }3$ |  $54\,(3Z_{A}-Z_{new}^{(25)})$
|  $=0\mbox{ mod }3$ |  $36\,(3Z_{A}+5Z_{new}^{(25)})$
$(26,\frac{39}{14},112)$ |  $\neq 0\mbox{ mod }2,7$ |  $32\,(3Z_{A}+3Z_{D}+Z_{new}^{(26)})$
|  $=2\mbox{ mod }4,\neq 14$ |  $96\,(3Z_{A}-Z_{D}-Z_{new}^{(26)})$
|  $=0\mbox{ mod }4$ |  $64\,(3Z_{A}-3Z_{D}+2Z_{new}^{(26)})$
|  $=0\mbox{ mod }7$ |  $192\,(Z_{A}+Z_{D}-Z_{new}^{(26)})$
$(27,\frac{81}{29},116)$ |  $1,\dots,28$ |  $180\,Z_{A}$
$(28,\frac{14}{5},120)$ |  $1,7,11,13,17,19,23,29$ |  $8\,(12Z_{A}+8Z_{D}+11Z_{E}+Z_{new}^{(28)})$
|  $2,4,8,14,16,22,26,28$ |  $16\,(12Z_{A}-4Z_{D}-Z_{E}+Z_{new}^{(28)})$
|  $3,9,21,27$ |  $48\,(4Z_{A}-Z_{E}-Z_{new}^{(28)})$
|  $5,25$ |  $32\,(10Z_{D}-5Z_{E}+2Z_{new}^{(28)})$
|  $6,12,18,24$ |  $96\,(4Z_{A}-4Z_{D}+Z_{E}-Z_{new}^{(28)})$
|  $10,20$ |  $128\,(2Z_{D}-Z_{E}+Z_{new}^{(28)})$
|  $15$ |  $192\,(2Z_{D}-Z_{E})$
$(29,\frac{87}{31},124)$ |  $1,\dots,30$ |  $192\,Z_{A}$
$(30,\frac{45}{16},128)$ |  odd, $16$ |  $128\,(Z_{A}+Z_{D})$
|  even, $\neq 4,12,16,20,28$ |  $64\,(5Z_{A}-3Z_{D}-2Z_{new}^{(30)})$
|  $4,12,20,28$ |  $64\,(3Z_{A}-Z_{D}+5Z_{new}^{(30)})$
$(31,\frac{31}{11},132)$ |  $\neq 0\mbox{ mod }3,11$ |  $48\,(3Z_{A}+Z_{new}^{(31)})$
|  $=0\mbox{ mod }3$ |  $144\,(2Z_{A}-Z_{new}^{(31)})$
|  $=0\mbox{ mod }11$ |  $240\,Z_{new}^{(31)}$
$(32,\frac{48}{17},136)$ |  odd $,\neq 17$ |  $24\,(6Z_{A}+5Z_{D})$
|  even |  $144\,(2Z_{A}-Z_{D})$
|  $17$ |  $384\,Z_{D}$
$(33,\frac{99}{35},140)$ |  $\neq 0\mbox{ mod }5,7$ |  $12\,(16Z_{A}+Z_{new}^{(33)})$
|  $=0\mbox{ mod }5$ |  $192\,(2Z_{A}-Z_{new}^{(33)})$
|  $=0\mbox{ mod }7$ |  $216\,Z_{new}^{(33)}$
$(34,\frac{17}{6},144)$ |  odd, $\neq 3,9,15,21,27,33$ |  $48\,(3Z_{A}+3Z_{D}-2Z_{new1}^{(34)})$