# Moduli space of logarithmic states in AdS3/LCFT2

Yannick Mvondo-She
Department of Physics, University of Pretoria
Private Bag X20, Hatfield 0028, South Africa
<EMAIL_ADDRESS>
###### Abstract

In this work, we extend results recently obtained concerning partition
functions in the AdS3/LCFT2 correspondence. Originally derived by Grumiller,
Gaberdiel and Vassilevich in the theory of topologically (and new) massive
gravity at the critical point, these partition functions were recently
reformulated in terms of Bell polynomials. The latter were shown to be
generated by a function that is identical to the plethystic exponential. We
exploit these reformulations to study the moduli space of the logarithmic
states, and show using algebro-geometric methods that it describes an
$S^{n}(\mathbb{C}^{2})$ orbifold. We also show a connection between this
quotient of a complex vector space by the symmetric group action and the
untwisted sector of symmetric orbifold CFT models, looking at their generating
functions. Furthermore, the action of differential operators on symmetric
orbifolds is presented.

###### Contents

  1. 1 Introduction
  2. 2 Partition functions of critical massive gravities
  3. 3 Symmetric product and orbifold in AdS3/LCFT2
    1. 3.1 Symmetry and the cycle index
    2. 3.2 Moduli space of logarithmic states
      1. 3.2.1 Hilbert schemes of points on surfaces and symmetric products
      2. 3.2.2 The $n$-th symmetric product of $\mathbb{C}^{2}$
    3. 3.3 $Z_{log}(\nu,q,\bar{q};\mathbb{C}^{2})$ as an index of the untwisted sector of a symmetric orbifold model
  4. 4 Differential operators on symmetric orbifolds
  5. 5 Conjectures
  6. 6 Conclusion and outlook
  7. A Derivations of plethystic exponential and Bell polynomial forms of $Z_{log}(\nu,q,\bar{q};\mathbb{C}^{2})$
  8. B Cycle index of the symmetric group
  9. C Invariant theory
    1. C.1 Ring of polynomials
    2. C.2 Invariant rings of the symmetric group
    3. C.3 Counting the number of invariants
    4. C.4 Rings of differential operator

##  1 Introduction

Gravity in three dimensions has for some time now been an interesting model to
test theories of -classical and quantum- gravity, and a consistent non-trivial
theory would bring the prospect of clarifying many intricate aspects of
gravity. A fundamental breakthrough was made in the study of the asymptotics,
revealing the emergence of a Virasoro algebra at the boundary [1]. One can
thus expect a dual 2d CFT description, and this discovery can be thought of as
a precursor of the AdS/CFT correspondence. However, pure Einstein gravity in
three dimensions is locally trivial at the classical level and does not
exhibit propagating degrees of freedom. Hence there was a need to modify it.

One way of deforming pure Einstein gravity is by introducing a negative
cosmological constant, leading to a theory with black hole solutions [2].
Another possibility of deformation is to add gravitational Chern-Simons term.
In that case the theory is called topologically massive gravity (TMG), and
contains a massive graviton [3] [4]. When both cosmological and Chern-Simons
terms are included in a theory, it yields cosmological topologically massive
gravity (CTMG). Such theory features both gravitons and black holes.

Following Witten’s proposal in 2007 to find a CFT dual to Einstein gravity
[5], the graviton 1-loop partition function was calculated in [6]. However,
discrepancies were found in the results. In particular, the left- and right-
contributions did not factorize, therefore clashing with the proposal of [5].

Soon after, a non-trivial slightly modified version of Witten’s construct was
proposed by Li, Song and Strominger [7]. Their theory, in which Einstein
gravity was replaced by chiral gravity can be viewed as a special case of
topologically massive gravity [3] [4], at a specific tuning of the couplings,
and is asymptotically defined with AdS3 boundary conditions, according to
Fefferman-Graham-Brown-Henneaux [1]. A particular feature of the theory was
that one of the two central charges vanishes, whilst the other one can have a
non-zero value. This gave an indication that the partition function could
factorize.

Shortly after the proposal of [7], Grumiller et al. noticed that relaxing the
Brown-Henneaux boundary conditions allowed for the presence of a massive mode
that forms a Jordan cell with the massless graviton, leading to a degeneracy
at the critical point [8]. In addition, it was observed that the presence of
the massive mode spoils the chirality of the theory, as well as its unitarity.
Based on these results, the dual CFT of critical cosmological topologically
massive gravity (cTMG) was conjectured to be logarithmic, and the massive mode
was called the logarithmic partner of the graviton. Indeed, Jordan cell
structures are a noticeable feature of logarithmic CFTs, that are non unitary
theories (see [9] as well as the very nice introductory notes [10] and [11]).
The correspondence distinguishes itself on the conjectured dual LCFT side by a
left-moving energy-momentum tensor $T$ that has a logarithmic partner state
$t$ with identical conformal weight, forming the following Jordan cell

$\displaystyle L_{0}\begin{pmatrix}\ket{T}\\\
\ket{t}\end{pmatrix}=\begin{pmatrix}2&0\\\
1&2\end{pmatrix}\begin{pmatrix}\ket{T}\\\ \ket{t}\end{pmatrix}.$ (1)

A major achievement towards the formulation of the correspondence was the
calculation of correlation functions in TMG [12, 13], which confirmed the
existence of logarithmic correlators of the type $\langle
T(x)t(y)\rangle=b_{L}/(x-y)^{4}$ that arise in LCFT, with $b_{L}$ commonly
referred as the logarithmic anomaly. Subsequently, the 1-loop graviton
partition function of cTMG on the thermal AdS3 background was calculated in
[14], resulting in the following expression

$\displaystyle{Z_{\rm{cTMG}}}(q,\bar{q})=\prod_{n=2}^{\infty}\frac{1}{|1-q^{n}|^{2}}\prod_{m=2}^{\infty}\prod_{\bar{m}=0}^{\infty}\frac{1}{1-q^{m}\bar{q}^{\bar{m}}},$
(2)

where the first product can be identified as the three-dimensional gravity
partition function $Z_{0,1}$ in [6], and is therefore not modular invariant.
The double product is the partition function of the logarithmic single and
multi particle logarithmic states, and will be the central object of this
work.

The corresponding expression of the partition function on the CFT side was
derived in [14] and given the form

${\rm{Z_{LCFT}}}(q,\bar{q})={\rm{Z^{0}_{LCFT}}}(q,\bar{q})+\sum_{h,\bar{h}}N_{{h,\bar{h}}}q^{h}\bar{q}^{\bar{h}}\prod_{n=1}^{\infty}\frac{1}{|1-q^{n}|^{2}},$
(3)

with

${\rm{Z^{0}_{LCFT}}(q,\bar{q})=Z_{\Omega}+Z_{t}}=\prod_{n=2}^{\infty}\frac{1}{|1-q^{n}|^{2}}\left(1+\frac{q^{2}}{|1-q|^{2}}\right),$
(4)

where $\Omega$ is the vacuum of the holomorphic sector, and $t$ denotes the
logarithmic partner of the energy momentum tensor $T$.

As pointed out in [15], a better understanding of the partition function from
the CFT side is desirable, in particular how to precisely match the
combinatorics of multi particle logarithmic states on the gravity side to
states on the CFT side. Motivated by that, the partition function was
reformulated in [16], and recast in terms of Bell polynomials. Furthermore, it
was shown that the partition function could be rewritten using the more usual
language of Hilbert series, leading to an identity between the generating
function of Bell polynomials and the celebrated plethystic exponential. In
addition, the Bell polynomials formalism showed an interesting use in
revealing hidden symmetry actions on the $n$-particle terms of the partition
function (this point will be given an interpretation in this work).

Despite the aforementioned achievements made since the conjecture of the
AdS3/LCFT2 in 2008, it is fair to say that very little is known about the
nature of the logarithmic states. In this paper, we try to deal with this
issue by exploiting the results of [16] to study the moduli space of the
logarithmic sector. The concept of moduli space originates from algebraic
geometry. The behaviour of certain geometric objects such as collections of
$n$ distinct ordered points on a given topological space can be understood by
finding a space $X$ which parametrizes these objects, i.e a space whose points
are in bijection with equivalence classes of these objects. A space $X$ with
such a correspondence is called a moduli space, and it parametrizes the types
of objects of interest, which in our case will be the logarithmic states. The
geometry of moduli spaces can be encoded in their generating functions. We
take advantage of that fact to give a symmetric group interpretation of the
results obtained in [16], and to show that the moduli space of the logarithmic
states is an $S^{n}(\mathbb{C}^{2})$ orbifold.

This paper is organized as follows. In section 2, we give a brief description
of partition functions in critical massive gravities. In particular, we recall
how some infinite products can usefully be rewritten as generating functions
of Bell polynomials. This is an interesting application of Bell polynomials in
theoretical physics. In section 3, we discuss the symmetric product structure
that appears from the sub-partition function of the logarithmic sector. We
start by showing that the latter, expressed in terms of Bell polynomials is in
fact the cycle index of the symmetric group, $i.e$ a polynomial in several
variables that counts objects that are invariant under the action of the
symmetric group. Then, in the spirit of Hilbert schemes of point on surfaces,
and from their connection to symmetric products, we show that the moduli space
of the logarithmic states is the $n$-th symmetric product of $\mathbb{C}^{2}$.
This is followed by showing that the partition function of the logarithmic
states is an index of the untwisted sector of symmetric orbifold CFT models.
In section 4, we revisit results from [16] through the construction of
differential operators acting on the generating function of
$S^{n}(\mathbb{C})$ orbifolds. Based on the results obtained, we make a
conjecture on the nature of the logarithmic states in section 5. Finally, a
conclusion and a discussion on future work are given in section 6.

##  2 Partition functions of critical massive gravities

The graviton 1-loop partition function of cosmological topologically massive
gravity and new massive gravity both at the critical point were calculated in
[14]. In the case of topologically massive gravity, the computation was given
the form

$\displaystyle{Z_{\rm{cTMG}}}(q,\bar{q})=\prod_{n=2}^{\infty}\frac{1}{|1-q^{n}|^{2}}\prod_{m=2}^{\infty}\prod_{\bar{m}=0}^{\infty}\frac{1}{1-q^{m}\bar{q}^{\bar{m}}},$
(5)

and for new massive gravity, the partition function was derived as

$\displaystyle{Z_{\rm{cNMG}}}(q,\bar{q})=\prod_{n=2}^{\infty}\frac{1}{|1-q^{n}|^{2}}\prod_{m=2}^{\infty}\prod_{\bar{m}=0}^{\infty}\frac{1}{1-q^{m}\bar{q}^{\bar{m}}}\prod_{l=2}^{\infty}\prod_{\bar{l}=0}^{\infty}\frac{1}{1-q^{l}\bar{q}^{\bar{l}}}.$
(6)

Shortly after these results, topologically massive gravity was generalized to
higher spins in [17], and the 1-loop partition function for topologically
massive higher spin gravity (cTMHSG) for arbitrary spin was calculated in
[18]. A special attention was given to spin-3 case for which the partition
function was expressed as

$\displaystyle{\rm{Z^{(3)}_{cTMHSG}}}(q,\bar{q})$ $\displaystyle=$
$\displaystyle\prod_{n=2}^{\infty}\frac{1}{|1-q^{n}|^{2}}\prod_{m=2}^{\infty}\prod_{\bar{m}=0}^{\infty}\frac{1}{1-q^{m}\bar{q}^{\bar{m}}}$
$\displaystyle\times$
$\displaystyle\left[\prod_{n=3}^{\infty}\frac{1}{|1-q^{n}|^{2}}\prod_{m=3}^{\infty}\prod_{\bar{m}=0}^{\infty}\frac{1}{1-q^{m}\bar{q}^{\bar{m}}}\prod_{m=4}^{\infty}\prod_{\bar{m}=3}^{\infty}\frac{1}{1-q^{m}\bar{q}^{\bar{m}}}\right].$

Recently, motivated by the desire for a better understanding of the
combinatorics of the logarithmic excitations in the partition function of
these critical massive gravities, and with the eventual goal of having a more
concrete grasp of their conjectured holographic (L)CFT duals, the partition
functions were shown to be usefully expressed in terms of Bell polynomials
[16].

Bell polynomials are very useful in many areas of mathematics and have enjoyed
many applications in physics as well. For instance, recently expressions of
canonical and grand canonical partition functions of interacting quantum gases
of Statistical Mechanics systems were rederived in terms of Bell Polynomials
by the authors of [19], in which the Bell polynomials are the Mayer (cluster)
expansion. Also, in [20] [21] [22] [23] and references therein, the use of
Bell polynomials is discussed for partition functions, suggesting some
interaction of particles in the theories concerned.

Bell polynomials have also appeared in the study of partition functions of BPS
bosonic operators. Following [24], if we consider such partition functions at
finite $N$, and denote them $Z_{k}(\vec{\beta};N)$ where
$\vec{\beta}=(\beta_{1},\ldots,\beta_{k})$ is a set of $k$ chemical potentials
conjugate to $n_{i}$, the quantum numbers of the various conserved charges in
the superconformal theory in question, and $N$ the rank of the gauge group,
then these partition functions typically take the following expression

$\displaystyle\mathcal{Z}_{k}(\vec{\beta},p)=\prod_{n_{1},n_{2},\ldots,n_{k}\geq
0}^{\infty}\frac{1}{1-p\exp{(-\vec{\beta}\cdot\vec{n})}},$ (8)

where the infinite product converges if $|p|<1$ and $\Re{(\beta_{i})}>0$.
Furthermore, the infinite product is the generating function for
$Z_{k}(\vec{\beta};N)$ expressed as

$\displaystyle\mathcal{Z}_{k}(\vec{\beta},p)=\sum_{N=0}^{\infty}Z_{k}(\vec{\beta};N)p^{N}.$
(9)

Eq. (9) is the grand canonical partition function for bosons in a
$k$-dimensional harmonic oscillator potential with $p$ as the fugacity,
defined as the chemical potential that keeps track of particle number $N$.
These partition functions correspond for instance to $\frac{1}{2}$-BPS or
$\frac{1}{4}$-BPS states in $\mathcal{N}=4$ SYM when $k=1,2$ respectively, or
to $\frac{1}{8}$-BPS states in the M2-brane world-volume for $k=4$ and to
$\frac{1}{4}$-BPS states in the M5-brane world-volume for $k=2$ (the $(2,0)$
SCFT in six dimensions) [24].

In [16], it was shown that the partition functions of critical gravities can
be expressed in terms of Bell polynomials. In the specific case of critical
cosmological TMG, writing Eq. (2) as

$\displaystyle{Z_{\rm{cTMG}}}(q,\bar{q})={Z_{\rm{gravity}}}(q,\bar{q})\cdot{Z_{\rm{log}}}(q,\bar{q}),$
(10)

where

$\displaystyle{Z_{\rm{gravity}}}(q,\bar{q})=\prod_{n=2}^{\infty}\frac{1}{|1-q^{n}|^{2}},\hskip
14.22636pt\mbox{and}\hskip
14.22636pt{Z_{\rm{log}}}(q,\bar{q})=\prod_{m=2}^{\infty}\prod_{\bar{m}=0}^{\infty}\frac{1}{1-q^{m}\bar{q}^{\bar{m}}},$
(11)

it was shown that ${Z_{\rm{log}}}(q,\bar{q})$ is the generating function of
Bell polynomials

$\displaystyle{Z_{\rm{log}}}(q,\bar{q})=\sum_{n=0}^{\infty}\frac{Y_{n}}{n!}\left(q^{2}\right)^{n}.$
(12)

In Eq. (12), $Y_{n}$ is the (complete) Bell polynomial with variables
$g_{1},g_{2},\ldots,g_{n}$ such that

$\displaystyle Y_{n}(g_{1},g_{2},\ldots,g_{n})=\sum_{\vec{k}\vdash
n}\frac{n!}{k_{1}!\cdots
k_{n}!}\prod_{j=1}^{n}\left(\frac{g_{j}}{j!}\right)^{k_{j}},$ (13)

with

$\displaystyle\vec{k}\vdash
n=\left\\{\vec{k}=\left(k_{1},k_{2},\ldots,k_{n}\right)\hskip 2.84544pt|\hskip
2.84544ptk_{1}+2k_{2}+3k_{3}+\cdots+nk_{n}=n\right\\},$ (14)

and

$\displaystyle g_{n}=(n-1)!\sum_{m\geq 0,\bar{m}\geq
0}q^{nm}\bar{q}^{n\bar{m}}=(n-1)!\frac{1}{|1-q^{n}|^{2}}.$ (15)

Similarly, Eq. (6) takes the form

$\displaystyle{\rm{Z_{cNMG}}}(q,\bar{q})$
$\displaystyle={Z_{\rm{gravity}}}(q,\bar{q})\cdot{Z_{\rm{log}}}(q,\bar{q})\cdot{\bar{Z}_{\rm{log}}}(q,\bar{q})$
(16a)
$\displaystyle=\prod_{n=2}^{\infty}\frac{1}{|1-q^{n}|^{2}}\left(\sum^{\infty}_{m=0}\frac{Y_{m}}{m!}\left(q^{2}\right)^{m}\right)\left(\sum^{\infty}_{l=0}\frac{Y_{l}}{l!}\left(\bar{q}^{2}\right)^{l}\right),$
(16b)

while Eq. (2) can be written as

$\displaystyle{\rm{Z^{(3)}_{cTMHSG}}}(q,\bar{q})$ $\displaystyle=$
$\displaystyle\chi_{0}(\mathcal{W}_{3})\times\bar{\chi}_{0}(\mathcal{W}_{3})\left(\sum^{\infty}_{m=0}\frac{Y_{m}}{m!}\left(q^{2}\right)^{m}\right)$
(17) $\displaystyle\times$
$\displaystyle\left(\sum^{\infty}_{l=0}\frac{Y_{l}}{l!}\left(q^{3}\right)^{l}\right)\left(\sum^{\infty}_{k=0}\frac{Y_{k}}{k!}\left(q^{4}\bar{q}^{3}\right)^{k}\right),$

with $\chi_{0}(\mathcal{W}_{3})$ and $\bar{\chi}_{0}(\mathcal{W}_{3})$ as the
holomorphic and antiholomorphic $W_{3}$ vacuum characters.

The logarithmic partition function can therefore be given a general form that
reads

$\displaystyle
Z_{\rm{log}}(\nu;q,\bar{q})=\prod_{m=0}^{\infty}\prod_{\bar{m}=0}^{\infty}\frac{1}{1-\nu
q^{m}\bar{q}^{\bar{m}}}=\sum_{n=0}^{\infty}\frac{Y_{n}}{n!}\nu^{n},$ (18)

where the variable $\nu$ represents a monomial in (highest) weight that can be
holomorphic denoted by $q^{h}$ or antiholomorphic denoted by
$\bar{q}^{\bar{h}}$, with $h$ and $\bar{h}$ as the conformal weights of
holomorphic and antiholomorphic logarithmic partner states.

Since the conjecture of critical topologically massive gravity as the dual of
a logarithmic conformal field theory, it is fair to say that little work has
been done in the description of the logarithmic states. In the next section,
we would like to make a few steps in that direction by using results recently
obtained from the partition function of critical massive gravities to extract
information about the moduli space of the logarithmic states.

##  3 Symmetric product and orbifold in AdS3/LCFT2

Moduli spaces of objects associated with a given space $\rm{X}$ encode
interesting and surprising information about the geometry and topology of
$\rm{X}$. In the present case, the objects under consideration are the
logarithmic partner states in critical massive gravities, associated with a
quotient of the space $\rm{X}=\mathbb{C}^{2}$.

This section deals with the the description of the moduli space of the
logarithmic sector appearing in critical massive gravities. In order to study
the moduli space, we use the partition function calculated in [14], and
exploit the fact that it was rederived in two different ways in [16]. We start
this section by showing that the space associated with logarithmic states is
invariant under the action of the symmetric group.

###  3.1 Symmetry and the cycle index

In combinatorics, only few formulae can be applied systematically in all cases
of a given problem. Pólya theory is one such example, as it enables to count
the number of items under specific constraints, such as number of colors or
more generally symmetry.

From a group theory perspective, counting objects such as states "up to
symmetry" means counting the orbits of some group of symmetries on the set of
states that are being counted. A standard procedure to solve this problem is
to use the orbit-counting (Burnside’s) lemma [25]. Alternatively, the counting
can be made systematic by using a multivariate polynomial associated with a
permutation group, called the cycle index.

Consider a finite set denoted by $\mathbb{S}$. A permutation of $\mathbb{S}$
is a one-to-one mapping of $\mathbb{S}$ onto itself. Given a permutation
$\pi$, $\mathbb{S}$ can be split into cycles, which are subsets of
$\mathbb{S}$ subject to cyclic permutation by $\pi$. Denoting the length of
the cycle by $l$, and any element of the cycle by $c$, the cycle can then be
expressed as

$\displaystyle c,\pi c,\pi^{2}c,\ldots,\pi^{l-1}c,$ (19)

with $\pi^{2}=\pi\pi$, $\pi^{3}=\pi\pi\pi$, etc…

######  Definition 3.1.

Let $\mathbb{S}=\left\\{1,\ldots,r\right\\}$ be a finite set. If $\mathbb{S}$
splits into $c_{1}$ cycles of length 1, $c_{2}$ cycles of length 2, etc…, then
$\pi$ is of type $\left\\{c_{1},c_{2},c_{3},\ldots\right\\}$.

######  Definition 3.2.

Let $G$ be the group whose elements $g$ are the permutations of $\mathbb{S}$,
and let $Z_{G}(x_{1},x_{2},\ldots,x_{l})$ be the polynomial in $l$ variables
$x_{1},x_{2},\ldots,x_{l}$ such that for each $g\in G$, the type of $g$ is
given by the product
$z_{g}(x_{1},x_{2},\ldots,x_{l})=x_{1}^{c_{1}}x_{2}^{c_{2}}\cdots
x_{l}^{c_{l}}$ as the partition of $r\left[1^{c_{1}}2^{c_{2}}\ldots
l^{c_{l}}\right]$, with $r=c_{1}+2c_{2}+\ldots+lc_{l}$. Then, the polynomial

$\displaystyle Z_{G}(x_{1},x_{2},\ldots,x_{l})$
$\displaystyle=\frac{1}{|G|}\sum_{g\in G}z_{g}(x_{1},x_{2},\ldots,x_{l})$
(20a)
$\displaystyle=\frac{1}{|G|}\sum_{f}f(c_{1},\ldots,c_{l})x_{1}^{c_{1}}x_{2}^{c_{2}}\cdots
x_{l}^{c_{l}}$ (20b)

is defined as the cycle index of $G$, with $f(c_{1},\ldots,c_{l})$ as the
number of permutations of type $\left[1^{c_{1}}2^{c_{2}}\ldots
l^{c_{l}}\right]$.

The formula above is reminiscent of Burnside’s lemma, except that here, one
distinguishes the cycles of different lengths, and specifies the number of
cycles there are.

In the present case, we are interested in the cycle index of the symmetric
group of degree $n$ denoted by $S_{n}$. It is defined by the formula [25]

$\displaystyle\mathcal{Z}\left(S_{n}\right)=\sum_{c_{1}+2c_{2}+\ldots+lc_{l}=n}\frac{1}{\prod_{k=1}^{n}k^{c_{k}}c_{k}!}\prod_{k=1}^{n}x_{k}^{c_{k}}.$
(21)

It is well known that the cycle index of the symmetric group $S_{n}$ can be
expressed in terms of (complete) Bell polynomials as follows

$\displaystyle Z(S_{n})=\frac{Y_{n}(0!a_{1},1!a_{2},\ldots,(n-1)!a_{n})}{n!}.$
(22)

Then, setting $a_{n}=\frac{1}{|1-q^{n}|^{2}}$, we can immediately identify the
arguments in the above equation with the term
$g_{n}=(n-1)!\frac{1}{|1-q^{n}|^{2}}$. $Z_{log}\left(q,\bar{q}\right)$ can
therefore be rewritten as

$\displaystyle
Z_{\rm{log}}\left(\nu;q,\bar{q}\right)=\sum_{n=0}Z\left(S_{n}\right)\left(\nu\right)^{n}.$
(23)

This means that $Z_{\rm{log}}\left(\nu;q,\bar{q}\right)$ counts geometric
objects that are invariant under the action of the symmetric group. This will
be made more precise in the next section as we discuss the moduli space of the
states that are being counted by $Z_{\rm{log}}\left(\nu;q,\bar{q}\right)$.

###  3.2 Moduli space of logarithmic states

In section 3.1, we made use of the expression of
$Z_{\rm{log}}\left(\nu;q,\bar{q}\right)$ in terms of Bell polynomials to show
how it encodes information about spaces invariant under the action of the
symmetric group. From a more formal perspective, the Bell polynomial
formulation will also be useful (section 4) in the combinatorial description
of a Fock space created by the action of an algebra. In this section, we use
the fact that $Z_{\rm{log}}\left(\nu;q,\bar{q}\right)$ can also be expressed
as the generating function of Hilbert series to show the symmetric product
orbifold structure of the moduli space of the logarithmic states.

Also called Molien or Poincaré function, the Hilbert series is a generating
function familiar in algebraic geometry for counting the dimension of graded
components of the coordinate ring (see Appendix C). Its approach has been
developed and extensively used in theoretical physics with for instance the
work of [26], and notably with several applications under the so-called
Plethystic Program initiated in [27, 28]. In connection with the Plethystic
Program, the Hilbert series have been the essential instrument of a systematic
method that yields the generating function of multi-trace operators in gauge
theory from the generating function of single-trace operators at large N. This
formalism was shown to hold in the present setting of a function generating
multi particle states from single particle ones. Indeed, we recall from [16]
that the multivariate Hilbert series

$\displaystyle\mathcal{G}_{1}(q,\bar{q})=\sum^{\infty}_{m\geq 0,\bar{m}\geq
0}q^{m}\bar{q}^{\bar{m}}=\frac{1}{|1-q|^{2}},$ (24)

that counts single particles of the logarithmic states, can be taken by the
bosonic plethystic exponential $PE^{\mathcal{B}}$ to generate new partition
functions such that

$\displaystyle
Z_{\rm{log}}\left(\nu;q,\bar{q}\right)=PE^{\mathcal{B}}\left[\mathcal{G}_{1}(q,\bar{q})\right]=\exp\left(\sum^{\infty}_{n=1}\frac{\left(\nu\right)^{n}}{n}\mathcal{G}_{1}\left(q^{n},\bar{q}^{n}\right)\right),$
(25)

with

$\displaystyle\mathcal{G}_{1}\left(q^{n},\bar{q}^{n}\right)=\frac{1}{|1-q^{n}|^{2}}=\frac{1}{(1-q^{n})(1-\bar{q}^{n})}.$
(26)

The connection between the plethystic exponential in Eq. (25) and the cycle
index discussion of section 3.1 in terms of Bell polynomials can be made clear
by noting that $a_{1}$ in Eq. (22) can be identified with the function
$\mathcal{G}_{1}\left(q,\bar{q}\right)$, and accordingly, $a_{n}\equiv
a_{n}\left(q,\bar{q}\right)=\mathcal{G}_{1}\left(q^{n},\bar{q}^{n}\right)$. In
analogy with the aforementioned applications, this shows that the Hilbert
series $\mathcal{G}_{1}{\left(q,\bar{q}\right)}$ counts single particle
states, while the plethystic exponential
$PE^{\mathcal{B}}\left[\mathcal{G}_{1}{\left(q,\bar{q}\right)}\right]$ counts
multi-particle states.

The formalism of Hilbert series acted upon by the plethystic exponential is
well known for its use in describing algebraic and geometric aspects of moduli
space. We will draw from that knowledge to study the configuration space of
logarithmic states. Essential to this will be a discussion of symmetric
products in the spirit of Hilbert schemes of points on surfaces [29].

####  3.2.1 Hilbert schemes of points on surfaces and symmetric products

The Hilbert scheme $X^{[n]}$ of points on a surface is a simple example of a
moduli space. It consists in the description of the configuration space of $n$
points on $X$, $i.e$ the space of unordered $n$-tuples of points of $X$ [29].

Formally, the Hilbert scheme of points can be defined as

$\displaystyle X^{[n]}:=\left\\{\mbox{{\it{I}} | {\it{I}} is an ideal of $X[x_{1},\ldots,x_{n}]$ with dim($X[x_{1},\ldots,x_{n}]/I$)={\it{n}}}\right\\},$ (27)

where $X[x_{1},\ldots,x_{n}]$ is the coordinate ring of $X$. In the above
definition, $X^{[n]}$ is considered as a set. It can be defined in a more
geometric flavor as

$\displaystyle X^{[n]}:=\left\\{\mbox{ {$Q_{Z}$} | {\it{$Q_{Z}$}} is a quotient ring of {\it{X}} with dim({\it{$Q_{Z}$}})={\it{n}} }\right\\}.$ (28)

The algebra-geometry correspondence is expressed as

$\displaystyle 0\rightarrow I\rightarrow X\rightarrow Q_{Z}\rightarrow 0,$
(29)

where $Z$ is the 0-dimensional subscheme of $X$, and $Q_{Z}$ is the coordinate
ring of $Z$, and allows for flexibility of terminology between schemes and
ideals.

The construction of a moduli space such as the Hilbert scheme can be
accomplished by taking the quotient of $X$ by a group acting on it. In that
endeavor, it is sensible to consider the quotient by the action of the
symmetric group, since we do not distinguish between points. This gives the
symmetric product

$\displaystyle S^{n}X=\underbrace{X\times X\times\cdots\times X}_{n\hskip
2.84544pt\mbox{times}}/S_{n}.$ (30)

However, the symmetric product $S^{n}X$ (also denoted $X^{(n)}$) is singular.
Indeed, if for instance we consider the case $n=2$, the group action is not
free along the diagonal $\mathcal{D}\subset X\times X$, which yields a
singular locus along the diagonal in $S^{2}X$. More precisely, approaching the
diagonal corresponds in $X^{2}$ to the two points approaching each other, and
eventually overlapping. At that stage, the system has lost one degree of
freedom. A possible resolution of the problem would be to keep track of the
direction the two points approach each other along. That is in fact the
difference between $X^{[n]}$ and $S^{n}X$: The Hilbert scheme $X^{[n]}$ is a
resolution of singularities of the symmetric product $S^{n}X$. When there
exist $n$ distinct points $p_{1},\ldots,p_{n}$ in $X$, each point defines both
a point in $X^{[n]}$ and a point in $S^{n}X$ [30], and setting the ideal of
Eq. (27) to

$\displaystyle I:=\left\\{\mbox{$f\in X[x_{1},\ldots,x_{n}]\hskip
5.69046pt|\hskip 5.69046ptf(p_{1})=\cdots=f(p_{n})=0$}\right\\},$ (31)

$I$ is indeed an ideal with dim($X[x_{1},\ldots,x_{n}]/I$)=n. This is the case
when $\rm{dim}X=1$ ($i.e$ $n=1$): the Hilbert scheme $X^{[n]}$ is isomorphic
to the $n$-th symmetric product $S^{n}X$ and we have

$\displaystyle X^{[n]}\simeq S^{n}X.$ (32)

A different situation is when some points collide. Looking at the case $n=2$,
two types of ideals must be taken into account in $X^{[2]}$. One can either
consider an ideal given by two distinct points $p_{1}$ and $p_{2}$, or the
ideal

$\displaystyle I=\left\\{f\hskip 5.69046pt|\hskip 5.69046ptf(p)=0,\hskip
5.69046ptdf_{P}(v)=0\right\\},$ (33)

where $p$ is a point of $X$ and $v$ is a vector in the tangent space $T_{p}X$.
The information of the direction in which $p_{1}$ approaches $p_{2}$ is
remembered in this ideal. In the symmetric product, this information is lost
and one just has $2p$. When $n>2$, more complicated ideals appear.

####  3.2.2 The $n$-th symmetric product of $\mathbb{C}^{2}$

In the spirit of the Hilbert scheme of points on surfaces briefly discussed
above, we consider the case when $X=\mathbb{C}^{2}$. More precisely, we
consider the family

$\displaystyle
S^{n}(\mathbb{C}^{2})\simeq\mathbb{C}\left[x_{1},y_{1};x_{2},y_{2};\ldots;x_{n},y_{n}\right]/S^{n},$
(34)

where $(x,y)$ are the coordinates of $\mathbb{C}^{2}$ and $S_{n}$ permutes the
$n$-tuple of variables $(x_{i},y_{i})$. The computation of the Hilbert series
of the invariant ring $S^{n}(\mathbb{C}^{2})$ then amounts to extending
Molien’s Theorem to the bi-graded case. Such extension has already been
studied [31, 32]. We now show that in our case, $Z_{log}(\nu;q,\bar{q})$ is a
$\nu$-inserted bi-graded Molien series of the symmetric group and can be
expressed as

$\displaystyle Z_{log}(\nu;q,\bar{q})\equiv
Z_{log}(\nu,q,\bar{q};\mathbb{C}^{2})=\sum_{n=0}\left(\nu\right)^{n}Z\left(S^{n}(\mathbb{C}^{2})\right),$
(35)

where $Z\left(S^{n}(\mathbb{C}^{2})\right)$ is defined in terms of Bell
polynomials as

$\displaystyle
Z\left(S^{n}(\mathbb{C}^{2})\right)=\frac{Y_{n}(g_{1},\ldots,g_{n})}{n!},$
(36)

with

$\displaystyle g_{n}=(n-1)!\frac{1}{(1-q)(1-\bar{q})}.$ (37)

For that we will look at the cases when the symmetric group acts on sets of
two and three objects.

The symmetric group on two objects can be presented as $S_{2}=<e,\sigma>$,
where $e$ is the identity element and $\sigma$ can be expressed in cycle
notation as $\sigma=(12)$. If we consider the action of $S_{2}$ on $2\times 2$
matrices by permuting coordinates $q$, we have

$\displaystyle\begin{pmatrix}q&0\\\
0&q\end{pmatrix}\xrightarrow[]{\pi(e)}\begin{pmatrix}q&0\\\
0&q\end{pmatrix}\hskip 14.22636pt\mbox{and}\hskip
14.22636pt\begin{pmatrix}q&0\\\
0&q\end{pmatrix}\xrightarrow[]{\pi(\sigma)}\begin{pmatrix}0&q\\\
q&0\end{pmatrix}.$ (38)

Then,

$\displaystyle\det\left(I-q\pi(e)\right)$
$\displaystyle=\det\left[\begin{pmatrix}1&0\\\
0&1\end{pmatrix}-\begin{pmatrix}q&0\\\ 0&q\end{pmatrix}\right]$ (39a)
$\displaystyle=\det\begin{pmatrix}1-q&0\\\ 0&1-q\end{pmatrix}$ (39b)
$\displaystyle=\left(1-q\right)^{2},$ (39c)

and

$\displaystyle\det\left(I-q\pi(\sigma)\right)$
$\displaystyle=\det\left[\begin{pmatrix}1&0\\\
0&1\end{pmatrix}-\begin{pmatrix}0&q\\\ q&0\end{pmatrix}\right]$ (40a)
$\displaystyle=\det\begin{pmatrix}1&-q\\\ -q&1\end{pmatrix}$ (40b)
$\displaystyle=\left(1-q^{2}\right).$ (40c)

In the same way, the action of $S_{2}$ on $2\times 2$ matrices by permuting
coordinates $\bar{q}$ allow us to write
$\det\left(I-\bar{q}\pi(e)\right)=\left(1-\bar{q}\right)^{2}$ and
$\det\left(I-\bar{q}\pi(\sigma)\right)=\left(1-\bar{q}^{2}\right)$. From
there, we have

$\displaystyle Z\left(S^{2}(\mathbb{C}^{2})\right)$
$\displaystyle=\frac{Y_{n}(g_{1},g_{2})}{2!}=\frac{1}{2!}\left(g_{1}^{2}+g_{2}\right)$
(41a)
$\displaystyle=\frac{1}{2!}\left[\frac{1}{(1-q)^{2}(1-\bar{q})^{2}}+\frac{1}{(1-q^{2})(1-\bar{q}^{2})}\right]$
(41b)
$\displaystyle=\frac{1}{2!}\left[\frac{1}{\det\left(I-qe\right)\det\left(I-\bar{q}e\right)}+\frac{1}{\det\left(I-q\sigma\right)\det\left(I-\bar{q}\sigma\right)}\right].$
(41c)

Next we consider the symmetric group $S_{3}=<e,\sigma,\tau>$, where the
elements correspond respectively to the identity, the three-element conjugacy
class that consists of swaping two coordinates, and the two-element conjugacy
class of cyclic permutations. Using cycle notations,
$\sigma=\left\\{(12),(13),(23)\right\\}$ and
$\tau=\left\\{(123),(132)\right\\}$. The action of $S_{3}$ on $3\times 3$
matrices by permuting coordinates $q$ would yield the following. Starting from
the identity

$\displaystyle\det\begin{pmatrix}1-q&0&0\\\ 0&1-q&0\\\
0&0&1-q\end{pmatrix}=(1-q)^{3}.$ (42)

Then, taking one of the three terms consisting of swaps of two coordinates,
say $\sigma=(12)$ identically results into

$\displaystyle\det\begin{pmatrix}1&-q&0\\\ -q&1&0\\\
0&0&1-q\end{pmatrix}=(1-q)(1-q^{2}).$ (43)

Finally, taking one of the two cyclic permutation terms, say $\tau=(132)$,
identically yields

$\displaystyle\det\begin{pmatrix}1&-q&0\\\ 0&1&-q\\\
-q&0&1\end{pmatrix}=(1-q^{3}).$ (44)

Acting in the same way on coordinates $\bar{q}$ allows us to eventually write

$\displaystyle
Z\left(S^{3}(\mathbb{C}^{2})\right)=\frac{Y_{n}(g_{1},g_{2},g_{3})}{3!}=\frac{1}{3!}g_{1}^{3}+3g_{1}g_{2}+g_{3}$
(45a)
$\displaystyle=\frac{1}{3!}\left[\frac{1}{(1-q)^{3}(1-\bar{q})^{3}}+\frac{3}{(1-q)(1-q^{2})(1-\bar{q})(1-\bar{q}^{2})}+\frac{2}{(1-q^{3})(1-\bar{q}^{3})}\right]$
(45b)
$\displaystyle=\frac{1}{3!}\left[\frac{1}{\det\left(I-q\pi(e)\right)\det\left(I-\bar{q}\pi(e)\right)}+\frac{3}{\det\left(I-q\pi(\sigma)\right)\det\left(I-\bar{q}\pi(\sigma)\right)}+\frac{2}{\det\left(I-q\pi(\tau)\right)\det\left(I-\bar{q}\pi(\tau)\right)}\right].$
(45c)

Repeating the same procedure for all permutations $\pi$ of the elements $g\in
S_{n}$, we can generalize the above analysis to

$\displaystyle Z_{log}(\nu;q,\bar{q})\equiv
Z_{log}(\nu,q,\bar{q};\mathbb{C}^{2})=\frac{1}{\left|S_{n}\right|}\sum_{g\in
S_{n}}\frac{(\nu)^{n}}{\det\left(I-q\pi(g)\right)\det\left(I-\bar{q}\pi(g)\right)}.$
(46)

$Z_{log}(\nu,q,\bar{q};\mathbb{C}^{2})$ is therefore a ($\nu$-inserted) bi-
graded Molien series of $S_{n}$, and a Hilbert series of the ring of
invariants $S^{n}(\mathbb{C}^{2})$.

Closer to our previous discussion on Hilbert schemes of points on surfaces,
generating functions taking the exponential form of
$Z_{log}(\nu,q,\bar{q};\mathbb{C}^{2})$ were considered in [33] and in [28].
Using our notation, we can then write

$\displaystyle
Z_{log}(\nu,q,\bar{q};\mathbb{C}^{2})=PE^{\mathcal{B}}\left[\frac{1}{(1-q)(1-\bar{q})}\right]=\exp\left(\sum^{\infty}_{n=1}\frac{\left(\nu\right)^{n}}{n(1-q^{n})(1-\bar{q}^{n})}\right).$
(47)

In the simplest case of CCTMG for instance, $i.e$ when $\nu=q^{2}$, one can
write

$\displaystyle
Z_{log}(q^{2};q,\bar{q};\mathbb{C}^{2})=PE^{\mathcal{B}}\left[\frac{1}{(1-q)(1-\bar{q})}\right]=\exp\left(\sum^{\infty}_{n=1}\frac{\left(q^{2}\right)^{n}}{n(1-q^{n})(1-\bar{q}^{n})}\right).$
(48)

$S^{n}(\mathbb{C}^{2})$ is an orbifold locally isomorphic to an open set of
the Euclidean space quotiented by the action of the symmetric group [30, 33].
The above analysis therefore brings forth the orbifold structure of the moduli
space of logarithmic partners states in critical massive gravities.

###  3.3 $Z_{log}(\nu,q,\bar{q};\mathbb{C}^{2})$ as an index of the untwisted
sector of a symmetric orbifold model

In this section we show an interesting connection between the previous results
in terms of the $S^{n}(\mathbb{C}^{2})$ orbifold and the symmetric product
orbifolds in CFT.

The $S_{n}$ orbifold partition function was first derived in [34] using
automorphic properties of elliptic genus. It was subsequently rederived in
[35] using the group theoretic properties of the symmetric products as
permutation orbifolds of the full symmetric group.

From [16], we recall that $Z_{log}(\nu,q,\bar{q};\mathbb{C}^{2})$ can be
obtained starting from writing its logarithmic expression as follows

$\displaystyle\log\left[Z_{log}(\nu,q,\bar{q};\mathbb{C}^{2})\right]$
$\displaystyle=-\sum_{h\geq 0,\bar{h}\geq 0}\log(1-\nu q^{h}q^{\bar{h}})$
(49a) $\displaystyle=-\sum_{h\geq 0,\bar{h}\geq
0}\left(-\sum_{n=1}^{\infty}\frac{(\nu)^{n}}{n}q^{nh}\bar{q}^{n\bar{h}}\right)$
(49b) $\displaystyle=\sum_{h\geq 0,\bar{h}\geq
0}\sum_{n=1}^{\infty}\frac{\nu^{n}}{n}q^{nh}\bar{q}^{n\bar{h}}$ (49c)
$\displaystyle=\sum_{n=1}^{\infty}\frac{\nu^{n}}{n}\left(\sum_{h\geq
0,\bar{h}\geq 0}q^{nh}\bar{q}^{n\bar{h}}\right).$ (49d)

Then, we simply write

$\displaystyle Z_{log}(\nu,q,\bar{q};\mathbb{C}^{2})$
$\displaystyle=\exp\left[\sum_{n=1}^{\infty}\frac{\nu^{n}}{n}\left(\sum_{h\geq
0,\bar{h}\geq 0}q^{nh}\bar{q}^{n\bar{h}}\right)\right]$ (50a)
$\displaystyle=\exp\left[\sum_{n=1}^{\infty}\frac{\nu^{n}}{n}Z\left(n\tau,n\bar{\tau}\right)\right].$
(50b)

From [36], $Z_{log}(\nu,q,\bar{q};\mathbb{C}^{2})$ can immediately be
recognized as the partition function of the untwisted sector of a totally
symmetric orbifold. Specifically, if one considers the "seed" partition
function of the untwisted sector of the "totally" symmetric orbifold model to
be

$\displaystyle Z(\tau,\bar{\tau})=\sum_{h,\bar{h}}q^{h}\bar{q}^{\bar{h}},$
(51)

then the generating function of the untwisted partition function can be
expressed as

$\displaystyle Z_{\rm{untwisted}}=\prod_{h,\bar{h}}^{\infty}\frac{1}{1-\nu
q^{h}\bar{q}^{\bar{h}}},$ (52)

which is the original double product derived in [14]. Hence, in our specific
case

$\displaystyle Z(\tau,\bar{\tau})=Y_{1}(g_{1}),$ (53)

and

$\displaystyle Z_{log}(\nu,q,\bar{q};\mathbb{C}^{2})=Z_{\rm{untwisted}}.$ (54)

This is in agreement with (Eqs. (4.25) to (4.28) of) [36], with the difference
that in our case, $t$ is specialized to $\nu$, and $\rho(h,\bar{h})=1$. The
reason for the setting of $\rho(h,\bar{h})$ is because the partition function
counts descendant states constructed by the action on the highest weight state
of only one (Virasoro) generator ($L_{-1}$). This agrees with the fact that
the $sl(2)$ algebra is composed of only three generators which are expressed
in terms of the Virasoro generators $\left\\{L_{-1},L_{0},L_{1}\right\\}$, and
for which $L_{-1}$ creates descendant states. It also agrees with results from
[8] where the logarithmic states are created by multiple action of $L_{-1}$
and $\bar{L}_{-1}$.

To make contact with the terminology of [36], we start by writing

$\displaystyle\chi(n\tau)=\frac{1}{\left|1-q^{n}\right|^{2}}.$ (55)

This gives the following correspondence between the work of [36] and our Bell
polynomials $Y_{n}$

$\displaystyle P_{2}\left(1,0,\leavevmode\hbox to12.3pt{\vbox
to6.3pt{\pgfpicture\makeatletter\hbox{\hskip 0.15pt\lower-0.15pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{{}}{{}}{{{}}{{}}{}{{}}{}
{{}{}}{{}}{}{}{}{}{{}}{}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{0.3pt}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}\pgfsys@fill@opacity{1}\pgfsys@invoke{
}\pgfsys@fill@opacity{1}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{0.0pt}{6.0pt}\pgfsys@lineto{12.0pt}{6.0pt}\pgfsys@lineto{12.0pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{12.0pt}{6.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{}}{{}}{}{{}}{}{{}{}}{
{}{}{}{}{ }{ }{ }{ }{ }{}}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{0.3pt}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{12.0pt}{0.0pt}\pgfsys@moveto{0.0pt}{5.99pt}\pgfsys@lineto{12.0pt}{5.99pt}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{0.0pt}{6.0pt}\pgfsys@moveto{6.0pt}{0.0pt}\pgfsys@lineto{6.0pt}{6.0pt}\pgfsys@moveto{11.99pt}{0.0pt}\pgfsys@lineto{11.99pt}{6.0pt}\pgfsys@moveto{12.0pt}{6.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\right)=\frac{1}{2!}\left[\chi(\tau)+\chi(2\tau)\right]=\frac{1}{2!}Y_{2}$
(56) $\displaystyle P_{3}\left(1,0,\leavevmode\hbox to18.3pt{\vbox
to6.3pt{\pgfpicture\makeatletter\hbox{\hskip 0.15pt\lower-0.15pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{{}}{{}}{{{}}{{}}{}{{}}{}
{{}{}}{{}}{}{}{}{}{{}}{}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{0.3pt}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}\pgfsys@fill@opacity{1}\pgfsys@invoke{
}\pgfsys@fill@opacity{1}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{0.0pt}{6.0pt}\pgfsys@lineto{18.0pt}{6.0pt}\pgfsys@lineto{18.0pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{18.0pt}{6.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{}}{{}}{}{{}}{}{{}{}}{
{}{}{}{}{ }{ }{ }{ }{ }{ }{}}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{0.3pt}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{18.0pt}{0.0pt}\pgfsys@moveto{0.0pt}{5.99pt}\pgfsys@lineto{18.0pt}{5.99pt}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{0.0pt}{6.0pt}\pgfsys@moveto{6.0pt}{0.0pt}\pgfsys@lineto{6.0pt}{6.0pt}\pgfsys@moveto{12.0pt}{0.0pt}\pgfsys@lineto{12.0pt}{6.0pt}\pgfsys@moveto{17.99pt}{0.0pt}\pgfsys@lineto{17.99pt}{6.0pt}\pgfsys@moveto{18.0pt}{6.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\right)=\frac{1}{3!}\left[\left(\chi(\tau)\right)^{3}+3\chi(\tau)\chi(2\tau)+2\chi(3\tau)\right]=\frac{1}{3!}Y_{3},$
(57)

and similarly for $n>0$. Finally, one gets

$\displaystyle
Z_{log}(\nu,q,\bar{q};\mathbb{C}^{2})=\sum_{n}\left|P_{n}(1,0,\overbrace{\leavevmode\hbox
to12.3pt{\vbox to6.3pt{\pgfpicture\makeatletter\hbox{\hskip
0.15pt\lower-0.15pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{{}}{{}}{{{}}{{}}{}{{}}{}
{{}{}}{{}}{}{}{}{}{{}}{}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{0.3pt}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}\pgfsys@fill@opacity{1}\pgfsys@invoke{
}\pgfsys@fill@opacity{1}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{0.0pt}{6.0pt}\pgfsys@lineto{12.0pt}{6.0pt}\pgfsys@lineto{12.0pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{12.0pt}{6.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{}}{{}}{}{{}}{}{{}{}}{
{}{}{}{}{ }{ }{ }{ }{ }{}}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{0.3pt}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{12.0pt}{0.0pt}\pgfsys@moveto{0.0pt}{5.99pt}\pgfsys@lineto{12.0pt}{5.99pt}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{0.0pt}{6.0pt}\pgfsys@moveto{6.0pt}{0.0pt}\pgfsys@lineto{6.0pt}{6.0pt}\pgfsys@moveto{11.99pt}{0.0pt}\pgfsys@lineto{11.99pt}{6.0pt}\pgfsys@moveto{12.0pt}{6.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\cdots\leavevmode\hbox
to12.3pt{\vbox to6.3pt{\pgfpicture\makeatletter\hbox{\hskip
0.15pt\lower-0.15pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{{}}{{}}{{{}}{{}}{}{{}}{}
{{}{}}{{}}{}{}{}{}{{}}{}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{0.3pt}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}\pgfsys@fill@opacity{1}\pgfsys@invoke{
}\pgfsys@fill@opacity{1}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{0.0pt}{6.0pt}\pgfsys@lineto{12.0pt}{6.0pt}\pgfsys@lineto{12.0pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{12.0pt}{6.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{}}{{}}{}{{}}{}{{}{}}{
{}{}{}{}{ }{ }{ }{ }{ }{}}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{0.3pt}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{12.0pt}{0.0pt}\pgfsys@moveto{0.0pt}{5.99pt}\pgfsys@lineto{12.0pt}{5.99pt}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{0.0pt}{6.0pt}\pgfsys@moveto{6.0pt}{0.0pt}\pgfsys@lineto{6.0pt}{6.0pt}\pgfsys@moveto{11.99pt}{0.0pt}\pgfsys@lineto{11.99pt}{6.0pt}\pgfsys@moveto{12.0pt}{6.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}^{n})\right|^{2}\left(\nu\right)^{n},$
(58)

which is the application of Eq. (4.26) of [36] to the present case. The
specific choice of representation is due to the fact that in a bosonic theory,
only the totally symmetric representation of the $S_{n}$ characters would be
considered.

##  4 Differential operators on symmetric orbifolds

In this section, using an invariant theoretic language, we revisit some of the
work done in [16], and give a more formal interpretation to the hidden
structures found in the study of $Z_{log}(\nu;q,\bar{q};\mathbb{C}^{2})$.

Fock spaces were designed as an algebraic framework to construct many-particle
states in quantum mechanics. They typically represent the state space of an
indefinite number of identical particles (an electron gas, photons, etc…).
These particles can be classified in two types, bosons and fermions, and their
Fock spaces look quite different. The reason why a Fock space are of great
interest is that several important algebras can naturally act on it. Fermionic
Fock spaces are naturally representations of a Clifford algebra, where the
generators correspond to adding or removing a particle in a given energy
state. In a similar way, bosonic Fock space is naturally a representation of a
Weyl algebra.

Proceeding with our discussion on Hilbert schemes, for the non-compact space
$\mathbb{C}^{2}$, a connection between the theory of Hilbert schemes of points
on surfaces and the infinite dimensional Heisenberg algebra was made through
the construction of a representation of the Heisenberg algebra on the homology
group of the Hilbert scheme, turning the homology group into a Fock space
[37]. The construction showed that the Fock space representation on the
polynomial ring of infinitely many variables is an important representation of
the Heisenberg algebra. In the present case, with an interest on the bosonic
Fock space, we construct a combinatorial model of creation and annihilation
operators that are generators of a Heisenberg-Weyl algebra and that act on the
bosonic Fock space.

In general, a Fock space is considered on a Hilbert space, but in the simplest
case and for the purpose of our discussion, the bosonic vector space is
obtained by considering a complex vector space $\mathbb{C}$. Then, the bosonic
Fock space as a vector space is essentially a space of polynomials of
infinitely many variables. A typical basis can be constructed using Schur
symmetric functions. In our case, we consider a space of Bell polynomials
$Y_{n}$ of infinitely many variables $g_{1},g_{2},\cdots,g_{n}$.

Given the $n$-dimensional polynomial ring $\mathbb{C}[x_{1},\ldots,x_{n}]$, a
subring of invariant polynomials denoted
$\mathbb{C}[x_{1},\ldots,x_{n}]^{S_{n}}$ can be constructed from invariants
$g_{1},\ldots,g_{n}\in\mathbb{C}[x_{1},\ldots,x_{n}]$. The polynomials of the
invariant ring $\mathbb{C}[x_{1},\ldots,x_{n}]^{S_{n}}$ can take the form of
Bell polynomials $Y_{n}$ with coordinates $g_{1},\ldots,g_{n}$ such that
$\mathbb{C}[x_{1},\ldots,x_{n}]^{S_{n}}\simeq\mathbb{C}[g_{1},\ldots,g_{n}]$.
From the ring of differential operators

$\displaystyle
D\left(\mathbb{C}[g_{1},\ldots,g_{n}]\right)=\mathbb{C}<g_{1},\ldots,g_{n},\partial_{g_{1}},\ldots,\partial_{g_{n}}>,$
(59)

we construct the multiplication operator
$\hat{X}_{n}=g_{1}+\sum^{n}_{k=1}g_{k+1}\frac{\partial}{\partial g_{k}}$ and
the derivative operator $\hat{D}=\frac{\partial}{\partial g_{1}}$ such that
they satisfy the Heisenberg-Weyl algebra $\left[\hat{X}_{n},\hat{D}\right]=1$.
Then, defining the function

$\displaystyle Z\left(\mathbb{C}[x_{1},\ldots,x_{n}]^{S_{n}}\right)\equiv
Z\left(S^{n}(\mathbb{C})\right)=\frac{Y_{n}(g_{1},\ldots,g_{n})}{n!},$ (60)

these operators act as ladder operators on the function
$Z\left(\mathbb{C}[x_{1},\ldots,x_{n}]^{S_{n}}\right)$ at each $n$ level in
the following way.

######  Proposition 4.1.

Let $Z\left(S^{n}(\mathbb{C})\right)$ be defined in terms of Bell polynomials
$Y_{n}$ as

$\displaystyle
Z\left(S^{n}(\mathbb{C})\right)=\frac{Y_{n}(g_{1},\ldots,g_{n})}{n!}.$ (61)

The set of operators

$\displaystyle\hat{X}_{n}=g_{1}+\sum^{n}_{k=1}g_{k+1}\frac{\partial}{\partial
g_{k}};\hskip 28.45274pt\hat{D}=\frac{\partial}{\partial g_{1}},$

generating the Heisenberg-Weyl algebra $[\hat{D},\hat{X}_{n}]=1$,
$[\hat{D},1]=[\hat{X}_{n},1]=0$ acts on $Z\left(S^{n}(\mathbb{C})\right)$ as

$\displaystyle\hat{X}_{n}Z\left(S^{n}(\mathbb{C})\right)$
$\displaystyle=(n+1)Z\left(S^{n+1}(\mathbb{C})\right)$ (62a)
$\displaystyle\hat{D}Z\left(S^{n}(\mathbb{C})\right)$
$\displaystyle=Z\left(S^{n-1}(\mathbb{C})\right)$ (62b)
$\displaystyle\hat{X}_{n}\hat{D}Z\left(S^{n}(\mathbb{C})\right)$
$\displaystyle=nZ\left(S^{n}(\mathbb{C})\right).$ (62c)

###### Proof.

From [16], it is known that

$\displaystyle\hat{X}_{n}Y_{n}$ $\displaystyle=Y_{n+1},$ (63a)
$\displaystyle\hat{D}Y_{n}$ $\displaystyle=nY_{n-1},$ (63b)
$\displaystyle\hat{X}_{n}\hat{D}Y_{n}$ $\displaystyle=nY_{n}.$ (63c)

Hence we can write

$\displaystyle\hat{X}_{n}\left[n!Z\left(S^{n}(\mathbb{C})\right)\right]$
$\displaystyle=(n+1)!Z\left(S^{n+1}(\mathbb{C})\right)$ (64a)
$\displaystyle\Rightarrow\hskip
14.22636pt\hat{X}_{n}\left(Z\left(S^{n}(\mathbb{C})\right)\right)$
$\displaystyle=(n+1)Z\left(S^{n+1}(\mathbb{C})\right).$ (64b)

Similarly

$\displaystyle\hat{D}\left[n!Z\left(S^{n}(\mathbb{C})\right)\right]$
$\displaystyle=n\left[(n-1)!Z\left(S^{n-1}(\mathbb{C})\right)\right]$ (65a)
$\displaystyle\Rightarrow\hskip
14.22636pt\hat{D}\left(Z\left(S^{n}(\mathbb{C})\right)\right)$
$\displaystyle=Z\left(S^{n-1}(\mathbb{C})\right),$ (65b)

and

$\displaystyle\hat{X}_{n}\hat{D}\left[Z\left(S^{n}(\mathbb{C})\right)\right]$
$\displaystyle=n\left[Z\left(S^{n}(\mathbb{C})\right)\right]$ (66a)
$\displaystyle\Rightarrow\hskip
14.22636pt\hat{X}_{n}\hat{D}Z\left(S^{n}(\mathbb{C})\right)$
$\displaystyle=nZ\left(S^{n}(\mathbb{C})\right).$ (66b)

∎

##  5 Conjectures

In the discussions above, we delved quite intensively into algebro-geometric
and invariant theoretic issues that had not yet been addressed in the context
of critical massive gravities present in the AdS3/LCFT2 correspondence. In
this section, we come to the point where we would like to make some
conjectures about the nature of the logarithmic states.

Firstly, we would like to mention the following about the "characters"
generated by the log-partition $Z_{log}(\nu,q,\bar{q};\mathbb{C}^{2})$. From
the above analysis, these objects are linear combinations of the variables
$g_{1},g_{2},\ldots,g_{n}$ multiplied by a factor $\left(\nu\right)^{n}$.
However, $Z_{log}(\nu,q,\bar{q};\mathbb{C}^{2})$ also seem to describe the
symmetric tensor product of the characters of $sl(2,\mathbb{R})$ highest
weight representations

$\displaystyle\chi^{sl(2,\mathbb{R})}_{h}=\frac{q^{h}}{1-q},\hskip
28.45274pt\bar{\chi}^{sl(2,\mathbb{R})}_{\bar{h}}=\frac{\bar{q}^{\bar{h}}}{1-\bar{q}}.$
(67)

Taking $h=2,\bar{h}=0$ for instance, it is easy to see that
$\chi^{sl(2,\mathbb{R})}_{h=2}$ and
$\bar{\chi}^{sl(2,\mathbb{R})}_{\bar{h}=0}$ are respectively the single
particle holomorphic and antiholomorphic characters from which
$Z_{log}(q^{2},q,\bar{q};\mathbb{C}^{2})$ yields symmetric tensor product
(multiparticle) characters. Yet, if really there were a holomorphic and an
antiholomorphic character, the numerator of the antiholomorphic character of
CCTMG should not be constant since the antiholomorphic central charge is not
nul. In other words, there should systematically be a $(\bar{\nu})$ appear in
the logarithmic partition functions. This argument show that the logarithmic
highest weight states have descendants that are both holomorphic and
antiholomorphic, just as in the $c=0$ LCFT theory [38].

In the extension of holography to the present non-unitary case, critical
massive gravities fitting in the AdS3/LCFT2 correspondence are considered as
non-unitary AdS3 holographic duals of two dimensional non-unitary CFTs that
are known to exist. We would like to conjecture that in this setting, the
logarithmic states of the critical massive gravities are conical
singularities. This idea is not new. Indeed, it was already mentioned in [39]
that one could expect conical spaces to play a role in this non-unitary
context. As we have seen from the study of the moduli space of logarithmic
states, the latter are singular point particles in an orbifold space. It is
therefore quite natural to think of them as conical singularities. Of course
it is essential to validate this conjecture with further checks, and that will
be the object of future work.

##  6 Conclusion and outlook

In this work, we used the partition function derived in [14] and reformulated
in [16] to extract information about the moduli space of the logarithmic
sector in critical massive gravities. Using the relation between the cycle
index of the symmetric group and Bell polynomials, we first showed that the
partition function of the logarithmic states is the generating function of
polynomials invariant under the action of the symmetric group $S_{n}$. Then,
it was shown that the configuration space of the logarithmic states is the
quotient space $S^{n}(\mathbb{C}^{2})$, by looking at the partition function
of the logarithmic states $Z_{log}(\nu,q,\bar{q})$ from an algebraic point of
view, and showing that it can be written as a bi-graded Molien series. From a
geometric point of view, the quotient space $S^{n}(\mathbb{C}^{2})$ has the
structure of an orbifold, and $Z_{log}(\nu,q,\bar{q};\mathbb{C}^{2})$ is then
the generating function of an $S^{n}(\mathbb{C}^{2})$ orbifold. Besides, an
interesting connection between the $S^{n}(\mathbb{C}^{2})$ orbifold and the
untwisted sector of a symmetric orbifold CFT model was made, looking at their
respective partition functions. Then, the construction of differential
operators on symmetric orbifolds was discussed. The operators were shown to
generate a Heisenberg-Weyl algebra, and to act as raising and lowering
operators on the Bell polynomials. Finally, based on the arguments developed
in the text, a conjecture about the nature of these logarithmic states in
terms of conical singularities was made.

On its own right, the logarithmic sector of these critical massive gravities
looks like an interesting topic of study, and much more seemingly remains to
be unraveled from it. For example, it would be interesting to study the
modular properties of $Z_{log}(\nu,q,\bar{q};\mathbb{C}^{2})$, and how the
study contributes to the modular properties of $Z_{\rm{TMG}}$. It was also
observed that the counting in the multi particle sector of
$Z_{log}(\nu,q,\bar{q};\mathbb{C}^{2})$ can only be done correctly on the
account of quantum groupoid (Hopf algebroid) coproducts [40]. This will be
discussed in an upcoming publication.

Lastly, the author would like to mention that the realization that the
partition functions of critical massive gravities could be recast in terms of
Bell polynomials must be credited to [41, 42]. Therefore, inspired by ideas
from [41, 42, 43, 44] where partition functions expressed in terms of Bell
polynomials and recast into infinite products eventually lead to the
construction of quantum group, knots and link invariants, it would be
interesting to investigate how $Z_{log}(\nu,q,\bar{q};\mathbb{C}^{2})$ can be
useful in the construction of topological invariants.

##### Acknowledgements

Y. M-S is grateful to A. Hanany for communication regarding the plethystic
exponential, and to K. Zoubos for proofreading and commenting on earlier
versions of this work.

##  Appendix A Derivations of plethystic exponential and Bell polynomial forms
of $Z_{log}(\nu,q,\bar{q};\mathbb{C}^{2})$

In this appendix, we recall the derivation of
$Z_{log}(\nu,q,\bar{q};\mathbb{C}^{2})$ as a plethystic exponential and in a
Bell polynomial form [40].

Starting from

$\displaystyle
Z_{log}(\nu,q,\bar{q};\mathbb{C}^{2})=\prod_{m=0}^{\infty}\prod_{\bar{m}=0}^{\infty}\frac{1}{1-\nu
q^{m}\bar{q}^{\bar{m}}},$ (A.1)

we first compute the logarithmic function of
$Z_{log}(\nu,q,\bar{q};\mathbb{C}^{2})$ as

$\displaystyle\log\left[Z_{log}(\nu,q,\bar{q};\mathbb{C}^{2})\right]=-\sum_{m,\bar{m}\geq
0}\log(1-\nu q^{m}\bar{q}^{\bar{m}}).$ (A.2)

Using the well know Maclaurin series

$\displaystyle\log(1-x)=-\sum_{x=1}^{\infty}\frac{x^{n}}{n},$ (A.3)

we write

$\displaystyle\log\left[Z_{log}(\nu,q,\bar{q};\mathbb{C}^{2})\right]=\sum_{m,\bar{m}\geq
0}\sum_{n=1}^{\infty}\frac{\nu^{n}}{n}q^{nm}\bar{q}^{n\bar{m}}.$ (A.4)

Then, using the Maclaurin series of geometric series

$\displaystyle\frac{1}{1-x}=\sum_{x=0}^{\infty}x^{n},$ (A.5)

we write

$\displaystyle\log\left[Z_{log}(\nu,q,\bar{q};\mathbb{C}^{2})\right]$
$\displaystyle=\sum_{m,\bar{m}\geq
0}\sum_{n=1}^{\infty}\frac{\nu^{n}}{n}q^{nm}\bar{q}^{n\bar{m}}$ (A.6a)
$\displaystyle=\sum_{n=1}^{\infty}\frac{\nu^{n}}{n}\sum_{m,\bar{m}\geq
0}q^{nm}\bar{q}^{n\bar{m}}$ (A.6b)
$\displaystyle=\sum_{n=1}^{\infty}\frac{\nu^{n}}{n}\frac{1}{1-q^{n}}\frac{1}{1-\bar{q}^{n}}.$
(A.6c)

Finally, exponentiating the above equation yields the plethystic exponential
form of $Z_{log}(\nu,q,\bar{q};\mathbb{C}^{2})$

$\displaystyle
Z_{log}(\nu,q,\bar{q};\mathbb{C}^{2})=PE^{\mathcal{B}}\left[\frac{1}{(1-q)(1-\bar{q})}\right]=\exp\left(\sum^{\infty}_{n=1}\frac{\left(\nu\right)^{n}}{n(1-q^{n})(1-\bar{q}^{n})}\right).$
(A.7)

To obtain the Bell polynomial version of
$Z_{log}(\nu,q,\bar{q};\mathbb{C}^{2})$, it suffices to continue from Eq.
(A.6c) as follows

$\displaystyle\log\left[Z_{log}(\nu,q,\bar{q};\mathbb{C}^{2})\right]$
$\displaystyle=\sum_{n=1}^{\infty}\frac{\nu^{n}}{n}\frac{1}{1-q^{n}}\frac{1}{1-\bar{q}^{n}}$
(A.8a)
$\displaystyle=\sum_{n=1}^{\infty}\frac{\nu^{n}}{n}\frac{1}{\left|1-q^{n}\right|}$
(A.8b)
$\displaystyle=\sum_{n=1}^{\infty}\frac{\nu^{n}}{n!}(n-1)!\frac{1}{\left|1-q^{n}\right|}.$
(A.8c)

Introducing the function $g_{n}$ such that

$\displaystyle g_{n}=(n-1)!\frac{1}{\left|1-q^{n}\right|},$ (A.9)

we get

$\displaystyle\log\left[Z_{log}(\nu,q,\bar{q};\mathbb{C}^{2})\right]=\sum_{n=1}^{\infty}\frac{\nu^{n}}{n!}g_{n}.$
(A.10)

Finally, exponentiation the above logarithmic function gives

$\displaystyle Z_{log}(\nu,q,\bar{q};\mathbb{C}^{2})$
$\displaystyle=\exp\left(\sum_{n=1}^{\infty}\frac{\nu^{n}}{n!}g_{n}\right)$
(A.11a) $\displaystyle=\sum_{n=0}^{\infty}\frac{Y_{n}}{n!}\nu^{n},$ (A.11b)

where in Eq. (A.11b), $Z_{log}(\nu,q,\bar{q};\mathbb{C}^{2})$ is the
generating function of the Bell polynomials $Y_{n}$.

##  Appendix B Cycle index of the symmetric group

In this appendix, the notion of cycle index applied to the symmetric group is
reviewed.

######  Definition B.1.

The symmetric group $S_{n}$ defined over a finite set $X$ of $n$ objects, is
the group of bijective functions from $X$ to $X$ under the operation of
composition, which consists of permutations the $n$ objects.

The permutations of $S_{n}$ can be expressed in terms of cycles. For instance,
considering the set $X={1,2,3,4,5,6}$, the permutation $\pi=(124)(35)(6)$
tells us that $\pi$ maps 1 to 2, 2 to 4, 3 to 5, and 6 to itself. In this
case, $\pi$ consists of 3 disjoint cycles.

######  Definition B.2.

A k-cycle, or cycle of length k, is a cycle containing k elements.

Looking back at the example considered above, $\pi=(124)(35)(6)$ contains a
3-cycle, a 2-cycle, and a 1-cycle.

In group theory, the elements of any group may be partitioned into conjugacy
classes.

######  Definition B.3.

In any group G, the elements g and h are conjugates if

$\displaystyle g=khk^{-1}$

for $k\in G$. The set of all elements conjugate to a given g is called the
conjugacy class of g.

Hence, when $S_{n}$ acts on a set $X$, the cycle decomposition of each $\pi\in
S_{n}$ as product of disjoint cycles is associated to the partitions of the
objects in the set. For example, if one considers $S_{4}$, the partitions of 4
and the corresponding conjugacy classes are

$\displaystyle(1,1,1,1)$ $\displaystyle\rightarrow\left\\{(e)\right\\}$ (B.1a)
$\displaystyle(2,1,1)$
$\displaystyle\rightarrow\left\\{(12),(13),(14),(23),(24),(34)\right\\}$
(B.1b) $\displaystyle(2,2)$
$\displaystyle\rightarrow\left\\{(12)(34),(13)(24),(14)(23)\right\\}$ (B.1c)
$\displaystyle(3,1)$
$\displaystyle\rightarrow\left\\{(123),(132),(124),(142),(134),(143),(234),(243)\right\\}$
(B.1d) $\displaystyle(4)$
$\displaystyle\rightarrow\left\\{(1234),(1432),(1423),(1324),(1342),(1243)\right\\}$
(B.1e)

To keep track of the cycle decomposition of the elements of $S_{4}$, one can
use the cycle index polynomial. Representing each object of the set by a
coordinate, $i.e$ 1 by $g_{1}$, 2 by $g_{2}$, 3 by $g_{3}$ and 4 by $g_{4}$,
the cycle index of $S_{4}$ reads

$\displaystyle
Z(S_{4})=\frac{1}{24!}\left(g_{1}^{4}+6g_{1}^{2}g_{2}+3g_{2}^{2}+8g_{1}g_{3}+6g^{4}\right).$
(B.2)

The coefficients before the monomials (the products of coordinates) count the
number of elements in a given conjugacy class, the powers on the monomials
indicate the number of times the object appears in a given partition, and the
denominator $24!$ is the order of $S_{4}$, $i.e$ the total number of elements
in $S_{4}$. As such, the cycle index is simply the average of the number of
elements in $X$, that are left invariant by the action of $\pi\in S_{4}$.

##  Appendix C Invariant theory

Invariant theory appears in the description of moduli spaces whose points
parametrize spaces of interest, and is also useful in the construction of
Hilbert schemes with associated Hilbert polynomials. In this appendix, we
review some basic concepts of invariant theory (a good reference on this topic
can be found in [45]).

###  C.1 Ring of polynomials

######  Definition C.1.

Let $V$ be a complex vector space, and denote the dual vector space by V*=
$\left\\{f:V\rightarrow\mathbb{C}\right\\}$. The coordinate ring
$\mathcal{R}(V)$ of V is the algebra of functions F: $V\rightarrow\mathbb{C}$
generated by the elements of V*. The elements of $\mathcal{R}(V)$ are called
polynomial functions on V.

For a fixed basis $e_{1},e_{2},\ldots,e_{n}$ of $V$, a dual basis of $V*$ can
be expressed by the coordinates $x_{1},x_{2},\ldots,x_{n}$ such that
$x_{i}(c_{1}e_{1}+\cdots+c_{n}e_{n})=c_{i}$. The coordinate ring
$\mathcal{R}(V)$ obtained is
$\mathbb{C}\left[x_{1},x_{2},\ldots,x_{n}\right]$, the ring of polynomials in
$n$ variables $f(x_{1},x_{2},\cdots,x_{n})$ with complex coefficients.

###  C.2 Invariant rings of the symmetric group

The fundamental question at the heart of invariant theory is to ask whether
the orbits of a group $G$ that acts on a space $V$ can form a space in their
own right. In what follows we will consider the case where $G=S_{n}$.

Let the symmetry group $S_{n}$ act on the $n$-dimensional complex vector space
$V$. The action of $S_{n}$ on $V$ translates into an action of $S_{n}$ on the
polynomial ring $\mathbb{C}[x_{1},\ldots,x_{n}]:=\mathbb{C}[\bm{x}]$. The
objective is then to describe the subring of invariant polynomials, which in
the present case is denoted $\mathbb{C}[{\bm{x}}]^{S_{n}}$. According to a
theorem of Hilbert, $\mathbb{C}[{\bm{x}}]^{S_{n}}$ is finitely generated as an
algebra. This means that there exist invariants
$I_{1},\ldots,I_{n}\in\mathbb{C}[\bf{x}]$ such that
$\mathbb{C}[{\bm{x}}]^{S_{n}}$ consists exactly of polynomials of the
invariant ring $\mathbb{C}[{\bm{g}}]=\mathbb{C}[I_{1},\ldots,I_{n}]$.

We can summarize the results of section 3.1 in the following way. The
polynomials invariant under action of $S_{n}$ are precisely the Bell
polynomials $Y$ with coordinates ${\bm{g}}=\left(g_{1},\ldots,g_{n}\right)$.
In particular, $Y({\bm{g}})\in\mathbb{C}[{\bm{x}}]^{S_{n}}$ are uniquely
written as polynomial in the $g_{1},\ldots,g_{n}$ such that we have the
isomorphism

$\displaystyle\mathbb{C}[\bm{x}]^{S_{n}}\sim\mathbb{C}[\bm{g}].$ (C.1)

###  C.3 Counting the number of invariants

In this subappendix, we are interested in counting the polynomials that remain
invariant under the action of the symmetric group. The treatment of this
enumerative problem can be made systematic by keeping track of the degrees in
which these invariants occur.

Let $\mathbb{C}[\bm{x}]^{S_{n}}_{d}$ be the set of all homogeneous invariants
of degree $d$. The invariant ring
$\mathbb{C}[\bm{x}]^{S_{n}}=\bigoplus_{d=0}^{\infty}\mathbb{C}[\bm{x}]^{S_{n}}_{d}$
is the direct sum of the finite dimensional $\mathbb{C}$-vector spaces
$\mathbb{C}[\bm{x}]^{S_{n}}_{d}$. The Hilbert series (or Poincaré series) of
the graded algebra $\mathbb{C}[\bm{x}]^{S_{n}}$ is the formal power series in
$t$ defined by

$\displaystyle
H\left(\mathbb{C}[\bm{x}]^{S_{n}},t\right)=\sum_{d=0}^{\infty}\mbox{dim}\left(\mathbb{C}[\bm{x}]^{S_{n}}_{d}\right)t^{d},$
(C.2)

which encodes in a convenient way the dimensions of the
$\mathbb{C}[\bm{x}]^{S_{n}}_{d}$-vector space of degree $d$.

In 1897, Molien proved that for any group finite group $G$ acting on
$\mathbb{C}[\bm{x}]^{G}$, it is possible to compute
$H\left(\mathbb{C}[\bm{x}]^{G},t\right)$ without first computing
$\mathbb{C}[\bm{x}]^{G}$. This is captured in the beautiful theorem below

######  Theorem C.1.

(Molien’s theorem). Let $\rho:G\rightarrow\rm{GL(V)}$ be a representation of a
finite group G of order $\left|G\right|$. If G acts on
$\mathbb{C}[V]=\mathbb{C}[\bm{x}]$, then the Hilbert series of the invariant
ring $\mathbb{C}[\bm{x}]^{G}$ can be expressed as

$\displaystyle H\left(\mathbb{C}[\bm{x}]^{G},t\right)=\frac{1}{|G|}\sum_{g\in
G}\frac{1}{\det\left(I-\rho(g)t\right)}.$ (C.3)

We refer the reader to [45] for a very readable proof. In the case of the
symmetric group, one simply writes the Hilbert series (C.2) as

$\displaystyle
H\left(\mathbb{C}[\bm{x}]^{S_{n}},t\right)=\frac{1}{|S_{n}|}\sum_{g\in
S_{n}}\frac{1}{\det\left(I-\rho(g)t\right)}.$ (C.4)

###  C.4 Rings of differential operator

As the algebra of differential operators on affine $n$-spaces, the Weyl
algebra is perhaps the most important ring of differential operator. It is
denoted

$\displaystyle
D\left(\mathbb{C}[x_{1},\ldots,x_{n}]\right)=\mathbb{C}<x_{1},\ldots,x_{n},\partial_{x_{1}},\ldots,\partial_{x_{n}}>,$
(C.5)

where the variables $x_{i}$ commute with each other, the variables
$\partial_{x_{j}}=\frac{\partial}{\partial x_{j}}$ commute with each other,
and the two sets of variables interact via the product rule
$\partial_{j}x_{i}=x_{i}\partial_{j}+\delta_{ij}$ [32].

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