DESY 23-170

#  Generalized Narain Theories $\mathfrak{Decoded}$:
$\mathfrak{D}$iscussions on $\mathfrak{E}$isenstein series,
$\mathfrak{C}$haracteristics, $\mathfrak{O}$rbifolds,
$\mathfrak{D}$iscriminants & $\mathfrak{E}$nsembles in any
$\mathfrak{D}$imension

Meer AshwinkumarN,a, Abhiram Kidambir,a,
Jacob M. Leedomi, Masahito Yamazakin,a

N Albert Einstein Center for Fundamental Physics, Institute for Theoretical
Physics
University of Bern, Sidlerstrasse 5, CH-3012 Bern, Switzerland
a Kavli IPMU, Uni. Tokyo, Kashiwanoha 5-1-5, 277-8583 Kashiwa, Chiba, Japan
r Erwin Schrödinger Institute, Boltzmanngasse 1A, 1090 Vienna, Austria
i Deutsches Elektronen-Synchrotron DESY, Notkestr. 85, 22607 Hamburg, Germany
n Trans-Scale Quantum Science Institute, The University of Tokyo, Tokyo
113-0033, Japan
MA<EMAIL_ADDRESS>AK<EMAIL_ADDRESS>
JML<EMAIL_ADDRESS>MY<EMAIL_ADDRESS>

Abstract

We study a class of newly-introduced CFTs associated with even quadratic forms
of general signature, which we call generalized Narain theories. We first
summarize the properties of these theories. We then consider orbifolds of
these theories, thereby obtaining a large class of non-supersymmetric CFTs
with exactly marginal deformations. We then discuss ensemble averages of such
theories over their moduli space, and obtain a modular form associated with
the quadratic form and an element of the discriminant group. The modular form
can be written as a Poincaré series, which contains novel invariants of lens
spaces and suggests the interpretation of the holographic bulk as a theory of
anyons.

###### Contents

  1. 1 Introduction
  2. 2 Generalized Narain Theories
    1. 2.1 Review of Standard Narain CFTs
    2. 2.2 Defining Data for Generalized Narain Theories
    3. 2.3 Torus Partition Functions
    4. 2.4 Operators and Moduli Space
  3. 3 Orbifolded Narain CFTs
    1. 3.1 Orbifold Actions
    2. 3.2 Torus Partition Functions
  4. 4 Ensemble Average
    1. 4.1 Unorbifolded Case
    2. 4.2 Orbifolded Case
    3. 4.3 Flavored Case
  5. 5 Comments on the Holographic Bulk
  6. 6 Summary and Discussion
  7. A Conventions and Notations
  8. B T-dualities
  9. C Orbifold Jacobi Theta Functions
    1. C.1 Modular Transformations
    2. C.2 Asymptotics at Cusps
  10. D Congruence Subgroup
  11. E Transformation of Eta and Theta Functions

##  1\. Introduction

In physics one often makes progress by studying simple models which capture
the essence of complicated physical phenomena. This comment may well be true
in the theories of holography: instead of complicated full-fledged string
theory setups for holography, one can hope to make progress by first studying
simpler solvable setups in detail and then try to embed the resulting setups
back into string theory.

Ensemble averages of Narain Conformal Field Theories (CFTs) as recently
discussed in [1, 2] (see also [3, 4, 5, 6, 7, 8, 9, 10, 11]) can be regarded
as perfect examples of such simplified models, especially when we wish to
explore ensemble averages in holography [12, 13]. The ensemble averages of 2d
CFTs can be computed exactly over the CFT moduli space, and the result in the
bulk generates an exotic theory of quantum gravity, where we have a sum over
geometries and the contribution from each geometry is captured by an Abelian
Chern-Simons theory. It was moreover suggested that these holographic
dualities can be embedded into string theory [14]111See e.g. [10, 15, 16, 17]
for a sample of papers which discuss the embedding of ensemble averages into
string theory, albeit in different setups., based on an earlier claim of [18].

In our previous paper [19] we analyzed ensemble averages of generalized Narain
CFTs associated with a general even integer lattice. Since the lattice is in
general not unimodular, the torus partition function is not modular invariant,
and this is probably one of the reasons why such theories have rarely been
discussed in string theory literature.222In the literature, modular invariance
is sometimes included in the definition of the two-dimensional CFT, and most
of our theories are not CFTs under this definition. We find it useful,
however, to be more flexible in the definition of the CFTs, at least for the
purposes of this paper. Note that even under the strict definition requiring
modular invariance, our theories can be used as building blocks for modular-
invariant theories, as discussed in Section 2. The generalized CFTs, however,
are otherwise well-defined theories, and we can average over their CFT moduli
spaces, to obtain an exotic theory of gravity in the bulk described by a
three-dimensional Abelian Chern-Simons theory. Here the non-modular-invariance
of the theory is accounted for by the existence of the Chern-Simons term in
the bulk. Moreover, the non-modular-invariance implies that we have a non-
trivial set of anyons in the holographic bulk, and this leads to global
symmetries in the bulk, some of which are emergent only after the ensemble
averaging [14].

In this paper we discuss ensemble averages of orbifolds of the Narain CFT [20,
21] associated with a general quadratic form, generalizing the previous
analysis [19] of the un-orbifolded case and [8, 22] for even self-dual
quadratic forms.333See also [11] for another paper on ensemble averages for
orbifolded Narain theories for special choices of unimodular quadratic forms.
We will also discuss generalizations of the aforementioned ensemble averages
where chemical potentials for flavor symmetries are turned on in the Narain
CFT partition function.

There are several motivations for studying orbifolds of generalized Narain
CFTs. Firstly, by studying orbifolds we can construct a new class of
irrational (and rational) CFTs. Secondly, it serves to understand the less-
studied CFTs associated to general (indefinite) even quadratic forms. Thirdly,
it is rare to find non-supersymmetric CFTs with exactly marginal deformations,
and orbifolded Narain CFTs will provide further concrete examples to test the
ideas of ensemble averages in holography. Fourthly, orbifolding gauges part of
the T-duality symmetry, and this can be regarded as gauging of the “ensemble
symmetries” in holography, as articulated in [14] (see also [23]). Note that
global symmetries are present only in exotic holographies involving non-
Einsteinian gravity, such as the ones discussed for generalized Narain
theories. Lastly, our analysis generates automorphic forms which are
generalizations of the Siegel-Eisenstein series, and could be of independent
mathematical interest.

The rest of this paper is organized as follows. In Section 2 we first
summarize generalized Narain theories of [19]. In Section 3 we discuss
orbifolds of the Narain theories. In Section 4 we then discuss ensemble
averages of the torus partition functions over the CFT moduli space. We will
also extend the discussion of ensemble averages by including chemical
potentials for the flavor symmetries, and will encounter new Siegel-Jacobi
forms associated with quadratic forms. In Section 5 we briefly comment on the
holographic-dual theories. We conclude with summaries and discussions in
Section 6. The appendices contain technical materials needed for the
understanding of the main text.

##  2\. Generalized Narain Theories

In this section, we describe the generalized Narain theories, which contains
the original Narain CFTs (toroidal CFTs) [20, 21] as special examples. While
we mostly follow our previous paper [19], we expand the discussion to clarify
the construction. Our conventions and notations are summarized in Appendix A.

###  2.1. Review of Standard Narain CFTs

We begin with a recasting of a familiar story—Narain CFTs. Narain CFTs can be
constructed from toroidal compactifications of superstring theories, in
particular the heterotic string444For a nice discussion whose conventions we
follow, see [24]. [20, 21]. These theories are defined via Narain lattices
$\mathcal{P}_{\rm Narain}$ that are even and self-dual with respect to the
inner product. For now, we focus on Narain CFTs with equal left- and right-
moving central charges $c_{L}=c_{R}=D$ so that the signature of
$\mathcal{P}_{\rm Narain}$ is $(D+,D-)$. The momenta $(p_{L},p_{R})$ depend on
the moduli of the compactification: the background metric $G$, and the two-
form $B$. These moduli parameterize the marginal deformations of the CFT and
take values in the typical Narain moduli space $\mathcal{M}_{\rm Narain}$
defined below in 2.5. At a point $m\in\mathcal{M}_{\rm Narain}$, the
components of a lattice vector $(p_{L},p_{R})\in\mathcal{P}_{\rm Narain}$ can
be written555For $c_{L}\neq c_{R}$, there are also Wilson line moduli that
will further modify these expressions.

$\displaystyle\begin{split}p_{L,i}&=n_{i}+\frac{1}{2}(G_{ij}-B_{ij})w^{j}\;,\\\
p_{R,i}&=n_{i}-\frac{1}{2}(G_{ij}+B_{ij})w^{j}\;,\end{split}$ (2.1)

with $n_{i},w^{j}\in\mathbb{Z}$ ($i,j=1,\cdots D$). The operator content of
the Narain CFT consists of currents $J^{M}=\partial X^{M}$ and
$\bar{J}^{M}=\bar{\partial}X^{M}$ and vertex operators
$\mathcal{V}=\;:\exp\bigg{(}ip_{L}\cdot X_{L}+ip_{R}\cdot X_{R}\bigg{)}:$,
where the $X^{M}=X^{M}_{L}+X^{M}_{R}$ ($M=1,\dots,D$) are the compact bosons
of the CFT.

The above can be reformulated in a more suggestive manner. The Narain lattice
consists of discrete data in the form of lattice points and continuous data
from the moduli. The discrete and continuous data can be separated using the
Narain vielbein $\mathcal{E}$. Re-expressing an element of $\mathcal{P}_{\rm
Narain}$ as $(p_{L},p_{R})=\mathcal{E}\ell$, where $\ell\in\mathbb{Z}^{2D}$,
we see that $p_{L}^{2}-p_{R}^{2}=\ell^{T}Q_{\rm Narain}\ell$, where $Q_{\rm
Narain}=\mathcal{E}^{T}\mathbbm{1}_{D,D}\mathcal{E}$ is a quadratic form
defining an inner product on the lattice

$\Lambda_{\rm Narain}:=\left\\{\ell\in\mathbb{Z}^{2D}\,\Big{|}\,\ell^{T}Q_{\rm
Narain}\ell\in 2\mathbb{Z}\right\\}\;.$ (2.2)

This lattice is the discrete data defining the Narain CFT. The moduli, which
constitute the continuous data, define $\mathcal{E}$ and can packaged in a
secondary quadratic form $H=\mathcal{E}^{T}\mathcal{E}$ on the lattice, which
acts as $\ell^{T}H\ell=p_{L}^{2}+p_{R}^{2}$. To make this more concrete, we
can consider the torus partition function of Narain CFTs,

$\lvert\eta(\tau)\rvert^{2D}Z_{\rm
Narain}(\tau;m):=\sum_{(p_{L},p_{R})\in\;\mathcal{P}_{\rm
Narain}}q^{p_{L}^{2}/2}\bar{q}^{p_{R}^{2}/2}=\sum_{\ell\in\mathbb{Z}^{2D}}\exp\bigg{[}i\pi\tau_{1}Q(\ell)-\pi\tau_{2}H(\ell)\bigg{]}~{},$
(2.3)

where $\eta(\tau)$ is the Dedekind eta function, $m$ collectively denotes the
CFT moduli and $q=e^{2\pi i\tau}$, with $\tau=\tau_{1}+i\tau_{2},\ \tau_{2}>0$
the modular parameter of the torus. Thus we see that the discrete and
continuous data can be directly used to write the partition function of the
Narain CFT. They can also be used to construct the vertex operators in the
canonical manner.

Finally, we come to the Narain moduli space itself, $\mathcal{M}_{\rm
Narain}$. An arbitrary Narain lattice can be obtained by the left action of an
element of $O(D,D;\mathbb{R})$ on a reference Narain lattice. These lattices
define unique Narain CFTs only up to independent $O(D)$ rotations of the left-
and right-moving momenta and outer automorphisms of the lattice:

$O(D,D;\mathbb{Z}):=\left\\{\Sigma\in\mathrm{GL}(2D,\mathbb{Z})\,\Big{|}\,\Sigma^{T}Q_{\rm
Narain}\Sigma=Q_{\rm Narain}\right\\}~{},$ (2.4)

which is precisely the T-duality group of the Narain CFT. Then the Narain
moduli space is given by the usual double quotient space

$\mathcal{M}_{\rm Narain}:=O(D,D;\mathbb{Z})\backslash
O(D,D;\mathbb{R})/O(D;\mathbb{R})\times O(D;\mathbb{R})~{}.$ (2.5)

Finally, we note that T-duality manifests as an invariance of the partition
function under elements of $O(D,D;\mathbb{Z})$, i.e.

$\displaystyle Z_{\rm Narain}(\tau;g\cdot m)$ $\displaystyle=Z_{\rm
Narain}(\tau;m)\;,$ (2.6)

for all $\Sigma\in O(D,D;\mathbb{Z})$.

###  2.2. Defining Data for Generalized Narain Theories

We now formalize the concepts of the previous subsection and utilize them to
construct generalizations of the Narain CFTs. More specifically, we first
define the discrete and continuous data (moduli) for a class of theories and
use this data to define partition functions and operators of the CFTs. In
analogy with 2.2, we first consider a lattice associated with an even,
integral, quadratic form $Q$ of signature $(p+,q-)$

$Q(\ell)=\sum_{i,j=1}^{p+q}Q_{ij}\ell^{i}\ell^{j}\in
2\mathbb{Z},~{}~{}Q_{ij}\in\text{Mat}_{(p+q)\times(p+q)}(\mathbb{Z}),~{}\ell\in\mathbb{Z}^{p+q}.$
(2.7)

We shall henceforth refer to an even, integral quadratic form simply as a
quadratic form, unless specified otherwise. A quadratic form corresponds to an
even integer lattice, which we denote by $\Lambda$:666An integer lattice is
simply a freely generated $\mathbb{Z}-$module.

$\Lambda:=\left\\{\ell\in\mathbb{R}^{p+q}\,\Big{|}\,Q(\ell)\in
2\mathbb{Z}\right\\}\;.$ (2.8)

Owing to the relation between quadratic forms and integer lattices, one may
think of an even quadratic form as a norm function/inner product for vectors
in an integer lattice. In other words, an even quadratic form is bilinear
$\displaystyle Q:\Lambda\times\Lambda\rightarrow 2\mathbb{Z}$ defined as

$Q(\ell,\ell^{\prime}):=\frac{Q(\ell+\ell^{\prime})-Q(\ell)-Q(\ell^{\prime})}{2}=\sum_{i,j=1}^{p+q}Q_{ij}\ell^{i}\ell^{\prime
j}\in 2\mathbb{Z}\;.$ (2.9)

Unlike the previously mentioned self-dual lattices $\Lambda_{\rm Narain}$ of
typical Narain CFTs, we will not restrict $\Lambda$ to be self-dual. The dual
lattice

$\Lambda^{*}:=\left\\{x\in\mathbb{R}^{p+q}\,\Big{|}\,Q(x,\ell)\in\mathbb{Z}\quad(\forall\
\ell\in\Lambda)\right\\}\;,\ \text{with}\ \Lambda\subset\Lambda^{*}~{}.$
(2.10)

This implies that $\Lambda$ has a non-trivial discriminant group

$\mathscr{D}_{\Lambda}:=\Lambda^{*}/\Lambda\;.$ (2.11)

The elements of $\mathscr{D}_{\Lambda}$ are equivalence classes $[x]$ such
that if $[x]=[y]$ for $x,y\in\Lambda^{*}$, then $x-y\in\Lambda$. We resort to
a mild abuse of notation and use $\alpha$ to refer to both elements of
$\mathscr{D}_{\Lambda}$ as well as representatives of the equivalence classes.
We will specify the distinction in situations where it is important. We will
also let $[0]$ denote the equivalence class of the zero element of
$\Lambda^{*}$ so that all representatives of $[0]$ are elements of $\Lambda$.

For the continuous data, we must define a second quadratic form acting on the
elements of $\Lambda$. To define sensible theta functions that we will use to
construct partition functions, we must look at elements of the representation
space $\mathfrak{h}_{Q}$ of the quadratic form $Q$ defining $\Lambda$ [25]:

$\mathfrak{h}_{Q}:=\left\\{H\in\mathrm{GL}(p+q,\mathbb{R})\,\Big{|}\,HQ^{-1}H=Q\right\\}\;.$
(2.12)

One could solve the condition on $H$ in 2.12 and parameterize the solutions —
the resulting expression for $H$ will depend on $pq$ number of parameters,
giving dim${}_{\mathbb{R}}\mathfrak{h}_{Q}=pq$. This is equivalent to
diagonalizing the quadratic form $Q$ via the Narain vielbein777where “viel”
$=p+q$. $\mathcal{E}$ as $Q=\mathcal{E}^{T}\mathbbm{1}_{p,q}\mathcal{E}$888The
signature of the diagonalized quadratic form remains the same due to
Sylvester’s law of inertia [26]. and defining $H=\mathcal{E}^{T}\mathcal{E}$,
as described in 2.1. Clearly, two distinct vielbeins $\mathcal{E}$ and
$\mathcal{E}^{\prime}$ are related by an element $\mathfrak{R}\in
O(p,q;\mathbb{R})$ as $\mathcal{E}^{\prime}=\mathfrak{R}\mathcal{E}$ and so
there exists $H,H^{\prime}\in\mathfrak{h}_{Q}$ such that
$H^{\prime}=\mathfrak{R}^{T}H\mathfrak{R}$.

To properly define the continuous data of the generalized Narain CFTs, we must
describe the moduli space $\mathcal{M}_{\Lambda}$ that defines distinct
quadratic forms $H$ and thereby distinct CFTs. Due to the existence of a non-
trivial discriminant group, this is a subtle point we return to below. For
now, we simply assume the existence of this moduli space and state that
distinct Narain CFTs correspond to points $m\in\mathcal{M}_{\Lambda}$.

With the discrete and continuous data $\Lambda$ and $\mathfrak{h}_{Q}$ in
hand, we can now define the generalized Narain lattice $\mathcal{P}_{\Lambda}$
associated to $\Lambda$ (and hence $Q$) as

$\mathcal{P}_{\Lambda}:=\left\\{\mathfrak{p}=(p_{L},p_{R})=\mathcal{E}\ell\,\Big{|}\,\ell\in\Lambda\;\;\&\;\;H=\mathcal{E}^{T}\mathcal{E}\in\mathfrak{h}_{Q}\right\\}\;.$
(2.13)

The inner product on $\mathcal{P}_{\Lambda}$ is induced from that on $Q(\ell)$
as

$\displaystyle\mathbb{I}(\mathfrak{p})$
$\displaystyle=\mathfrak{p}^{T}\mathbbm{1}_{p,q}\mathfrak{p}=p_{L}^{2}-p_{R}^{2}=\ell^{T}Q\ell\in
2\mathbb{Z}$ (2.14)

for all $\mathfrak{p}=\mathcal{E}\ell\in\mathcal{P}_{\Lambda}$. Thus we see
clearly that the introduction of the Narain vielbien allows a decomposition
into left- and right-movers in analogy with standard Narain CFTs. Indeed we
also see that $H(\ell)=p_{L}^{2}+p_{R}^{2}$. A convenient way to encapsulate
the decomposition of the momenta is via the quantities

$Q_{L}(\ell):=\frac{Q}{2}+\frac{H}{2}\;,\quad
Q_{R}(\ell):=\frac{H}{2}-\frac{Q}{2}\;,$ (2.15)

such that $Q_{L}(\ell)=p_{L}^{2}$ and $Q_{R}(\ell)=p_{R}^{2}$.

We also define the dual Narain lattice

$\mathcal{P}_{\Lambda}^{*}:=\left\\{\rho=(\rho_{L},\rho_{R})=\mathcal{E}\alpha\,\Big{|}\,\alpha\in\Lambda^{*}\;\;\&\;\;H:=\mathcal{E}^{T}\mathcal{E}\in\mathfrak{h}_{Q}\right\\}\;,$
(2.16)

whose elements satisfy the expected property
$\mathbb{I}(\rho,\mathfrak{p})=\rho^{T}\mathbbm{1}_{p,q}\mathfrak{p}=x^{T}Q\ell\in\mathbb{Z}$
for all $\rho=\mathcal{E}\alpha\in\mathcal{P}_{\Lambda}^{*}$ and
$\mathfrak{p}=\mathcal{E}\ell\in\mathcal{P}_{\Lambda}$. Let us also define a
subset of $\mathcal{P}_{\Lambda}^{*}$ associated with a particular element
$[\alpha]$ of $\mathscr{D}_{\Lambda}$:

$\mathcal{P}^{*}_{\Lambda,[\alpha]}:=\left\\{\rho=(\rho_{L},\rho_{R})=\mathcal{E}\beta\,\Big{|}\,\beta\in\Lambda^{*}\;\;\&\;\;[\alpha]=[\beta]\in\mathscr{D}_{\Lambda}\;\;\&\;\;H:=\mathcal{E}^{T}\mathcal{E}\in\mathfrak{h}_{Q}\right\\}\;,$
(2.17)

Note that this is not a lattice in general, and satisfies
$\mathcal{P}_{\Lambda^{*},[\alpha]}\cdot\mathcal{P}_{\Lambda^{*},[\beta]}\subset\mathcal{P}_{\Lambda^{*},[\alpha+\beta]}$.

Discrete Data:
${\color[rgb]{0.9453125,0.5625,0.12109375}\definecolor[named]{pgfstrokecolor}{rgb}{0.9453125,0.5625,0.12109375}\Lambda}\Rightarrow{\color[rgb]{0.6015625,0,0.0703125}\definecolor[named]{pgfstrokecolor}{rgb}{0.6015625,0,0.0703125}\mathscr{D}_{\Lambda}}$

Continuous Data: $\mathcal{M}_{\Lambda}$$m$Vertex
Operators$\mathcal{V}^{[\alpha]}_{(k_{L},k_{R})}$Hilbert
Space$\mathcal{H}_{\Lambda}$Partition Functions$Z^{\Lambda}_{\alpha}$

Generalized Lattices $\mathcal{P}_{\Lambda}$ and $\mathcal{P}_{\Lambda}^{*}$

Figure 1. Logical flow of the construction of the generalized CFTs described
in this section. A choice of integral lattice
${\color[rgb]{0.9453125,0.5625,0.12109375}\definecolor[named]{pgfstrokecolor}{rgb}{0.9453125,0.5625,0.12109375}\Lambda}$
(2.8) defines a discriminant group
${\color[rgb]{0.6015625,0,0.0703125}\definecolor[named]{pgfstrokecolor}{rgb}{0.6015625,0,0.0703125}\mathscr{D}_{\Lambda}}$
(2.11) as well as the moduli space
${\color[rgb]{0,0.625,0.875}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.625,0.875}\mathcal{M}_{\Lambda}}$
(2.51) of the CFT. This data, with a choice of
$m\in{\color[rgb]{0,0.625,0.875}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.625,0.875}\mathcal{M}_{\Lambda}}$,
can then be used to define the generalized Narain lattice
$\mathcal{P}_{\Lambda}$ (2.13) and its dual $\mathcal{P}_{\Lambda}^{*}$
(2.16), vertex operators ( 2.20), the Hilbert space $\mathcal{H}_{\Lambda}$,
and partition functions $Z^{\Lambda}_{\alpha}(\tau,\overline{\tau};m)$ (
2.27).

Similar to standard Narain CFTs, the primary operators of our generalized
theories are built from compact chiral bosons with periodicity conditions

$\begin{pmatrix}X^{a}_{L}\\\
X^{\overline{m}}_{R}\end{pmatrix}\sim\begin{pmatrix}X^{a}_{L}\\\
X^{\overline{m}}_{R}\end{pmatrix}+\mathcal{E}L$ (2.18)

with $L\in\mathbb{Z}^{p+q}$, $a=1,2,\dots,p$ and $\overline{m}=1,\dots,q$. The
operators consist of holomorphic and anti-holomorphic currents

$\displaystyle J^{a}(z)$ $\displaystyle:=i\partial X^{a}_{L}(z)\;,$ (2.19)
$\displaystyle\overline{J}^{\overline{m}}(\overline{z})$
$\displaystyle:=-i\bar{\partial}X^{\overline{m}}_{R}(\bar{z})\;,$

and vertex operators

$\mathcal{V}^{[\alpha]}_{(k_{L},k_{R})}(z,\overline{z}):=\;\;:\exp\bigg{(}ik_{L}\cdot
X_{L}(z)+ik_{R}\cdot X_{R}(\overline{z})\bigg{)}:\;,$ (2.20)

where $:\,:$ denotes the normal ordering,
$(k_{L}^{a},k_{R}^{\overline{m}})\in\mathcal{P}^{*}_{\Lambda,[\alpha]}$, and
the dot products are simply $k_{L}\cdot
X_{L}=\sum_{a=1}^{p}k_{L}^{a}X^{a}_{L}$ and $k_{R}\cdot
X_{R}=\sum_{m=1}^{q}k^{m}_{R}X_{R}^{\overline{m}}$. We also can define
holomorphic and anti-holomorphic stress tensors

$\displaystyle T(z)$ $\displaystyle:=\;\;\sum_{a}:J^{a}(z)J^{a}(z):\;,$ (2.21)
$\displaystyle\bar{T}(\overline{z})$
$\displaystyle:=\;\;\sum_{\overline{m}}:\overline{J}^{\overline{m}}(\overline{z})\overline{J}^{\overline{m}}(\overline{z}):\;,$

The currents have the standard operator product expansions (OPEs)

$J^{a}(z)J^{b}(0)=\frac{\delta^{ab}}{2z^{2}}+\cdots\quad\overline{J}^{\overline{m}}(\overline{z})\overline{J}^{\bar{n}}(0)=\frac{\delta^{\overline{m}\bar{n}}}{2\overline{z}^{2}}+\cdots\;,$
(2.22)

which give the vertex operator OPE

$\mathcal{V}^{[\alpha_{1}]}_{(k_{L},k_{R})}(z,\overline{z})\mathcal{V}^{[\alpha_{2}]}_{(k^{\prime}_{L},k^{\prime}_{R})}(0,0)=z^{k_{L}\cdot
k_{L}^{\prime}}\overline{z}^{k_{R}\cdot
k^{\prime}_{R}}\mathcal{V}^{[\alpha_{1}+\alpha_{2}]}_{(k_{L}+k^{\prime}_{L},k_{R}+k^{\prime}_{R})}(0,0)+\cdots\;.$
(2.23)

The existence of $\mathcal{P}_{\Lambda}$ guarantees the closure of this OPE.
Note while the vertex operators $\mathcal{V}^{[\alpha]}_{(k_{L},k_{R})}$ are
mutually local for $\alpha=0$, general vertex operators
$\mathcal{V}^{[\alpha]}_{(k_{L},k_{R})}$ with $\alpha\neq 0$ are not
necessarily mutually local.

The Virasoro algebra can be realized from the mode expansion of the stress-
energy tensor in the usual way, and we denote the operators by $L_{h}$ and
$\widetilde{L}_{h}$.

We can define states in the theory using the typical state-operator
correspondence. Apart from oscillator modes, we have momentum modes

$|\alpha;k_{L},k_{R}\rangle=\lim_{z,\overline{z}\rightarrow
0}\mathcal{V}^{[\alpha]}_{(k_{L},k_{R})}(z,\overline{z})|0\rangle\;,$ (2.24)

which satisfy the usual relations

$L_{0}|\alpha;k_{L},k_{R}\rangle=\frac{k_{L}^{2}}{4}|\alpha;k_{L},k_{R}\rangle\;,\qquad\widetilde{L}_{0}|\alpha;k_{L},k_{R}\rangle=\frac{k_{R}^{2}}{4}|\alpha;k_{L},k_{R}\rangle\;.$
(2.25)

One can also define higher oscillator states in the usual way, and the
combination of momenta and oscillator states defines the Hilbert space
$\mathcal{H}_{\Lambda}$ of the theory.

From the above, we have obtained definitions for primary operators, the
stress-energy tensor, sensible and closed OPEs, and states in the theory. It
is also clear that the above is a unitary CFT with the typical inner product
between the momenta states defined in 2.24. One may also wonder about the
modular invariance of the partition function. We define the partition
functions in the next section and return to this question there.

Finally, one might be interested in the question of formulating the existence
of generalized CFTs in a mathematically rigorous fashion. For this purpose,
one needs to choose a mathematical definition of CFT999Discussions on toroidal
CFTs can also be found in [27, 28, 29, 30]. 101010While the Vertex Operator
Algebras (VOAs) have often been used in the mathematical formulations of CFT,
this applies only to the chiral (anti-chiral) part of the CFTs, and is
insufficient for our purposes here.. One attempt in this direction is [31],
which defined “full VOAs” for CFTs on the plane and showed that generalized
Narain CFTs associated with a lattice satisfy the axioms therein. The
mathematical existence of generalized Narain CFTs is proven in this sense.

###  2.3. Torus Partition Functions

The torus partition function is given by the trace over the Hilbert space
$\mathcal{H}_{\Lambda}$ of the theory on the spatial $S^{1}$:

$\displaystyle
Z^{\Lambda}(\tau,\overline{\tau};m)=\textrm{Tr}_{\mathcal{H}_{\Lambda}}\left(q^{L_{0}-\frac{c}{24}}\overline{q}^{\overline{L}_{0}-\frac{\overline{c}}{24}}\right)=\frac{1}{\eta(\tau)^{p}\overline{\eta}(\overline{\tau})^{q}}\sum_{\ell\in\Lambda}e^{i\pi\tau
Q_{L}(\ell)-i\pi\overline{\tau}Q_{R}(\ell)}\;,$ (2.26)

where $q:=\exp(2\pi i\tau)$ and $\overline{q}:=\exp(-2\pi i\overline{\tau})$,
$Q_{L}$ and $Q_{R}$ are defined in terms of the Narain data as in 2.15, and we
have explicitly shown the dependence on the moduli
$m\in\mathcal{M}_{\Lambda}$.

The modular transformations of the partition function $Z^{\Lambda}$ is better
described in terms of building blocks

$\displaystyle
Z^{\Lambda}_{\alpha}(\tau,\overline{\tau};m)=\frac{\vartheta^{\Lambda}_{\alpha}(\tau,\overline{\tau};m)}{\eta(\tau)^{p}\overline{\eta}(\overline{\tau})^{q}}\;,$
(2.27)

which serve as the basis for more general partition functions.

Here the theta function $\vartheta^{\Lambda}_{\alpha}$ is given by

$\displaystyle\vartheta^{\Lambda}_{\alpha}(\tau,\overline{\tau};m)$
$\displaystyle:=\sum_{\ell\in{\Lambda+\alpha}}e^{i\pi\tau
Q_{L}(\ell)-i\pi\overline{\tau}Q_{R}(\ell)}=\sum_{\ell\in{\Lambda+\alpha}}e^{i\pi\tau_{1}Q(\ell)-\pi\tau_{2}H(\ell)}\;.$
(2.28)

Here $\alpha$ is a representative of $[\alpha]\in\mathscr{D}_{\Lambda}$. There
are $\lvert\mathscr{D}_{\Lambda}\rvert$ theta functions, which are defined
only by the equivalence class and not by the particular representatives
chosen, as can be seen by re-defining the sum over $\Lambda$. The partition
function 2.27 reduces to 2.26 when $\alpha=0\in\mathscr{D}_{\Lambda}$. By
relabeling the sum in 2.27 as $\ell\to-\ell$, we obtain

$\displaystyle\vartheta^{\Lambda}_{\alpha}(\tau,\overline{\tau};m)=\vartheta^{\Lambda}_{-\alpha}(\tau,\overline{\tau};m)\;.$
(2.29)

The generators of $\mathrm{SL}(2,\mathbb{Z})$ are given by the matrices $T$
and $S$:

$\displaystyle T=\left(\begin{array}[]{cc}1&1\\\ 0&1\end{array}\right)\;,\quad
S=\left(\begin{array}[]{cc}0&-1\\\ 1&0\end{array}\right)\;.$ (2.34)

We have relations $(ST)^{3}=1,S^{2}=-1$. The corresponding modular
transformations are111111Strictly speaking we should rather consider the
double cover of $\mathrm{SL}(2,\mathbb{Z})$, the metaplectic group
$\textrm{Mp}(2,\mathbb{Z})$, and the representation is in 2.35 is the Weil
representation of $\textrm{Mp}(2,\mathbb{Z})$ [32].

$\begin{split}&T:\quad\vartheta^{\Lambda}_{\alpha}(\tau+1;m)=e^{i\pi
Q(\alpha)}\vartheta^{\Lambda}_{\alpha}(\tau;m)\;,\\\
&S:\quad\vartheta^{\Lambda}_{\alpha}\left(-\frac{1}{\tau};m\right)=\frac{1}{\sqrt{|\textrm{det}\,Q|}}\,e^{-\frac{\pi
i}{4}(p-q)}\tau^{\frac{p}{2}}\overline{\tau}^{\frac{q}{2}}\sum_{\gamma\in\mathscr{D}_{\Lambda}}e^{-2\pi
iQ(\alpha,\gamma)}\vartheta^{\Lambda}_{\gamma}(\tau;m)\;,\end{split}$ (2.35)

where here and in the following we often drop the $\overline{\tau}$ dependence
from the notation.

Let us consider a modular transformation

$\displaystyle\tau\to\tau_{[M]}:=\frac{a\tau+b}{c\tau+d}\;,\quad[M]=\pm\left(\begin{array}[]{cc}a&b\\\
c&d\end{array}\right)\in\mathrm{PSL}(2,\mathbb{Z})\quad(c\neq 0)\;,$ (2.38)

where in the following we will denote an element of
$\mathrm{PSL}(2,\mathbb{Z})$ as $[M]=\pm M$, with $M$ being an
$\mathrm{SL}(2,\mathbb{Z})$ representative. Under the modular transformation
2.38 the theta functions mix among themselves:

$\displaystyle\vartheta^{\Lambda}_{\alpha}(\tau_{[M]};m)$
$\displaystyle=\sum_{\beta\in\mathscr{D}_{\Lambda}}\mathcal{U}^{\Lambda}_{\alpha,\beta}(M,\tau)\,\vartheta^{\Lambda}_{\beta}(\tau;m)\;.$
(2.39)

Here the transformation matrix $\mathcal{U}$ is given by

$\displaystyle\mathcal{U}^{\Lambda}_{\alpha,\beta}(M,\tau):=\begin{cases}\displaystyle(c\tau+d)^{\frac{p}{2}}(c\overline{\tau}+d)^{\frac{q}{2}}\lambda^{\Lambda}_{\alpha,\beta}(M)&(c\neq
0)\;,\\\ \displaystyle\mathcal{U}^{\Lambda}_{\alpha,\beta}(T,\tau):=e^{i\pi
Q(\alpha)}\delta_{\alpha,\beta}&(c=0)\;,\end{cases}$ (2.40)

and a version of the quadratic Gauss sum by

$\displaystyle\lambda^{\Lambda}_{\alpha,\beta}(M):=\frac{1}{\sqrt{|\textrm{det}\,Q|}}\,e^{-\frac{\pi
i}{4}(p-q)}c^{-\frac{p+q}{2}}\sum_{\ell_{c}\in\Lambda/(c\Lambda)}e^{\frac{\pi
i}{c}\left(aQ[\ell_{c}+\alpha]-2Q[\ell_{c}+\alpha,\beta]+dQ[\beta]\right)}\;.$
(2.41)

It is easy to verify that $\lambda_{\alpha,\beta}$ is preserved when we shift
$\alpha,\beta$ by elements of $\Lambda$, so that we can regard $\alpha,\beta$
to be elements of the discriminant $\mathscr{D}_{\Lambda}$, as implied by the
notation.

Note that $\mathcal{U}^{\Lambda}_{\alpha,\beta}(M,\tau)$ and
$\lambda^{\Lambda}_{\alpha,\beta}(M)$ both depend on the choice of the
$\mathrm{SL}(2,\mathbb{Z})$ representative $M$. We can also verify, however,
that the relation 2.39 is not affected by this choice; this is verified by the
relations 2.29,
$\mathcal{U}_{\alpha,\beta}(M)=\mathcal{U}_{\alpha,-\beta}(-M)$ and then a
relabeling of the sum as $\beta\to-\beta$. In the following we will denote
both $[M]$ and $M$ simply by $M$, to avoid clutter in notations.

The modular transformation formula 2.39 was known long ago to C. Siegel [25,
§3]. The matrix $\mathcal{U}$ satisfies

$\displaystyle\mathcal{U}^{\Lambda}_{\alpha,\beta}(M\cdot
M^{\prime},\tau)=\sum_{\gamma\in\mathscr{D}_{\Lambda}}\mathcal{U}^{\Lambda}_{\alpha,\gamma}(M,\tau_{M^{\prime}})\,\mathcal{U}^{\Lambda}_{\gamma,\beta}(M^{\prime},\tau)\;,$
(2.42)

which ensures the consistency of the modular transformation 2.39 under the
composition of $\mathrm{SL}(2,\mathbb{Z})$ elements. Note that $\lambda$
themselves satisfy $\tau$-independent relations

$\displaystyle\lambda^{\Lambda}_{\alpha,\beta}(M\cdot
M^{\prime})=\sum_{\gamma\in\mathscr{D}_{\Lambda}}\lambda^{\Lambda}_{\alpha,\gamma}(M)\,\lambda^{\Lambda}_{\gamma,\beta}(M^{\prime})\;.$
(2.43)

The modular transformations of the partition functions 2.27 are given by

$\begin{split}&T:\quad Z^{\Lambda}_{\alpha}(\tau+1;m)=e^{-\frac{2\pi
i(p-q)}{24}}e^{i\pi Q(\alpha)}Z^{\Lambda}_{\alpha}(\tau;m)\;,\\\ &S:\quad
Z^{\Lambda}_{\alpha}\left(-\frac{1}{\tau};m\right)=\frac{1}{\sqrt{|\textrm{det}\,Q}|}\sum_{\gamma\in\mathscr{D}_{\Lambda}}e^{-2\pi
iQ(\alpha,\gamma)}Z^{\Lambda}_{\gamma}(\tau;m)\;,\end{split}$ (2.44)

We comment here on the role of modular invariance and the generalized CFTs
defined above. The consistency conditions for a $2d$ CFT on arbitrary $2d$
surfaces include crossing symmetry of the sphere four-point function and
modular invariance of the torus partition function and torus one-point
function [33]. From the transformation properties in 2.44, it is clear that a
given torus partition function $Z^{\Lambda}_{\alpha}$ is not invariant under
$\mathrm{PSL}(2,\mathbb{Z})$ since the elements of the discriminant group
transform into one another under the $S$ transformation. (The exception to
this statement is if the quadratic form $Q$ corresponds to a self-dual lattice
and thus the discriminant group is trivial.) One possible viewpoint is that
our CFTs are limited and not defined on arbitrary $2d$ surfaces.
Alternatively, we can consider the CFTs defined above as being a subsector of
a CFT with expanded content such that the total partition function is modular
invariant.121212For example, a modular invariant theory could be constructed
by combining a generalized Narain CFT defined by quadratic form $Q$ with a
“conjugate” CFT that is defined by $-Q$. Such a sector would furnish a
projective representation of $\mathrm{PSL}(2,\mathbb{Z})$ similar to that
defined in 2.44, although the theory in itself need not be identical to the
generalized Narain CFT sector. There is some indication that such an expansion
is necessary. For example, the bulk Chern-Simons dual of the generalized
Narain CFTs contains a global one-form symmetry that should not exist within
the context of quantum gravity. Thus at a bare minimum the content of the
expanded CFT is such that this global symmetry is eliminated. While [18]
discusses a stringy origin of the bulk Chern-Simons theories, we expect that
the discussion there should be supplemented with additional ingredients based
on the arguments here.

###  2.4. Operators and Moduli Space

Similar to standard Narain CFTs, the primary operators of our generalized
theories are holomorphic and anti-holomorphic currents formed from compact
bosons

$\displaystyle J^{a}(z)$ $\displaystyle:=\partial X^{a}\;,$ (2.45)
$\displaystyle\bar{J}^{\overline{m}}(\bar{z})$
$\displaystyle:=\bar{\partial}X^{\overline{m}}\;,$

and vertex operators

$\mathcal{V}:=\;\;:\exp\bigg{(}ip_{L}\cdot X_{L}+ip_{R}\cdot
X_{R}\bigg{)}:\;,$ (2.46)

where $a=1,\dots,p$ and $\overline{m}=1,\dots,q$, and $:\,:$ denotes the
normal ordering. The existence of $\mathcal{P}_{\Lambda}$ guarantees the
closure of the associated Operator Product Expansion (OPE).

We now return to the issue of the moduli space of the generalized CFTs.
Naively, one may draw an analogy with the T-duality group of Narain CFTs in
2.4 and define the T-duality group for the generalized theories by simply
collecting integral transformations that preserve the quadratic form $Q$:

$\mathrm{O}_{Q}(p,q;\mathbb{Z}):=\left\\{\Sigma\in\mathrm{GL}(p,q;\mathbb{Z})\,\Big{|}\,\Sigma^{T}Q\Sigma=Q\right\\}\;.$
(2.47)

This is essentially correct, but there are several subtleties that must be
addressed. First, the generalized partition functions in 2.27 are not
necessarily invariant under the elements of 2.47, and we instead have [14]

$\vartheta^{\Lambda}_{g\cdot\alpha}(\tau,\bar{\tau};g\cdot
m)=\vartheta^{\Lambda}_{\alpha}(\tau,\bar{\tau};m)\;,$ (2.48)

where $g\cdot\alpha:=\Sigma\alpha$ and $g\cdot m$ implies the replacement
$H\rightarrow\Sigma^{T}H\Sigma$ for $\Sigma\in O_{Q}(p,q;\mathbb{Z})$.

One is thus led to define T-duality groups that not only preserve $Q$ but also
an element $\alpha$ the discriminant group:

$\mathrm{O}_{Q,\alpha}(p,q;\mathbb{Z}):=\left\\{\Sigma\in\mathrm{GL}(p+q;\mathbb{Z})\,\Big{|}\,\Sigma^{T}Q\Sigma=Q\;\&\;[\Sigma\alpha]=[\alpha]\in\mathscr{D}_{\Lambda}\right\\}\;.$
(2.49)

Note that $\displaystyle O_{Q,\alpha}(p,q;\mathbb{Z})\subseteq
O_{Q}(p,q;\mathbb{Z})$. Then we can define a moduli space for each element of
the discriminant group as

$\mathcal{M}_{Q,\alpha}:=\mathrm{O}_{Q,\alpha}(p,q;\mathbb{Z})\big{\backslash}\widetilde{\mathcal{M}}_{Q}\;,\quad\widetilde{\mathcal{M}}_{Q}:=(O(p,q;\mathbb{R})\big{/}(O(p;\mathbb{R})\times
O(q;\mathbb{R}))\,.$ (2.50)

which we sometimes denote as $\mathcal{M}_{\Lambda,\alpha}$. However, these
spaces are not the moduli space of the complete theory, which includes all of
the elements of $\mathscr{D}_{\Lambda}$. To determine this, we utilize several
more facts about the T-duality group and $\mathscr{D}_{\Lambda}$ (see Appendix
B for proofs).

The elements of $\mathrm{O}_{Q}(p,q;\mathbb{Z})$ define bijective maps from
$\mathscr{D}_{\Lambda}$ to itself that leave $[0]$ invariant. Thus the
T-duality group associated to $[0]$ is
$\mathrm{O}_{Q,[0]}(p,q;\mathbb{Z})=\mathrm{O}_{Q}(p,q;\mathbb{Z})$. For the
other equivalence classes in $\mathscr{D}_{\Lambda}$, the elements of
$\mathrm{O}_{Q}(p,q;\mathbb{Z})$ can either i) leave the class invariant, as
for the case of $[0]$ or ii) map different equivalence classes into each
other. Classes that satisfy the former property again have T-duality groups
given by 2.47. For classes that satisfy the latter property, we associate with
them the more limited groups of 2.49. However, so long as we include all the
different equivalence classes in the definition of the theory, the T-duality
group of the total theory is the full one presented in 2.47. These points are
illustrated in Figure 2.

Thus in the full theory that contains all of the elements of
$\mathscr{D}_{\Lambda}$, the true moduli space is

$\mathcal{M}_{Q}:=\mathrm{O}_{Q}(p,q;\mathbb{Z})\big{\backslash}\widetilde{\mathcal{M}}_{Q}\,.$
(2.51)

which we sometimes denote as $\mathcal{M}_{\Lambda}$.

The moduli spaces 2.50 are endowed with a metric, known as the Zamolodchikov
metric whose measure we denote by $[dm]$. This measure is the Haar measure of
an orthogonal symmetric space with a canonical normalization. The dimension of
the moduli space 2.50 is
$\textrm{dim}_{\mathbb{R}}\,\mathcal{M}_{Q,\alpha}=pq$.

$[\alpha_{2}]$$[\alpha_{3}]$$\mathcal{M}_{\Lambda}$$[\alpha_{1}]$$[\alpha_{1}]$$\mathcal{M}_{\Lambda,\alpha_{2}}$
Figure 2. Illustration of T-duality on elements of $\mathscr{D}_{\Lambda}$ in
the space of moduli values of the generalized CFT. The blue region is the
moduli space of the generalized CFT, $\mathcal{M}_{\Lambda}$, the orange
region is the moduli space of $[\alpha_{2}]\in\mathscr{D}_{\Lambda}$,
$\mathcal{M}_{\Lambda,\alpha_{2}}$, and the white space corresponds to moduli
values outside the fundamental domain. The two possibilities outlined in the
text are i) an element of $[\alpha_{1}]\in\mathscr{D}_{\Lambda}$ is preserved
by all elements of $\mathrm{O}_{Q}(p,q;\mathbb{Z})$ and so is unchanged when
mapping into $\mathcal{M}_{\Lambda}$ or ii) an element
$[\alpha_{1}]\in\mathscr{D}_{\Lambda}$ is mapped to
$[\alpha_{2}]\neq[\alpha_{1}]$ under $\mathrm{O}_{Q}(p,q;\mathbb{Z})$.

We can illustrate the points above by a simple example

$Q=\begin{pmatrix}0&2\\\ 2&0\end{pmatrix}\,.$ (2.52)

Then $|\text{det}(Q)|=\lvert\mathscr{D}_{\Lambda}\rvert=4$ and we can choose
representatives of the elements of $\mathscr{D}_{\Lambda}$ as

$\displaystyle\alpha_{0}$ $\displaystyle=\left(0,0\right)\;,\quad\alpha_{1}$
$\displaystyle=\left(\frac{1}{2},\frac{1}{2}\right)\;,\quad\alpha_{2}$
$\displaystyle=\left(\frac{1}{2},0\right)\;,\quad\alpha_{3}$
$\displaystyle=\left(0,\frac{1}{2}\right)\;.$ (2.53)

The vielbein is parameterized by elements of $\mathrm{O}(1,1;\mathbb{R})$ as

$\mathcal{E}=\begin{pmatrix}\cosh\phi&\sinh\phi\\\
\sinh\phi&\cosh\phi\end{pmatrix}\begin{pmatrix}1&1\\\ -1&1\end{pmatrix}\;,$
(2.54)

which gives a parameterization for the elements of $\mathfrak{h}_{Q}$:

$H=\begin{pmatrix}2(\cosh(2\phi)-\sinh(2\phi))&0\\\
0&2(\cosh\phi+\sinh\phi)^{2}\end{pmatrix}=:\begin{pmatrix}A&0\\\
0&4A^{-1}\end{pmatrix}\,.$ (2.55)

Finally, the T-duality group consists of a single element other than the
identity, up to an overall sign:

$\Sigma=\begin{pmatrix}0&1\\\ 1&0\end{pmatrix}\,.$ (2.56)

The action $H\rightarrow\Sigma^{T}H\Sigma$ of $\Sigma$ is equivalent to the
replacement $A\rightarrow 4A^{-1}$. Then the naive moduli space is the
interval $[4,\infty)$ (or equivalently, the interval $(0,4]$). However, we
first note the action of $\Sigma$ on elements of $\mathscr{D}_{\Lambda}$.
Using the representatives in 2.53, we see that $[\alpha_{0}]$ and
$[\alpha_{1}]$ are invariant but

$\displaystyle\begin{split}[\Sigma\alpha_{2}]&=[\alpha_{3}]\;,\quad[\Sigma\alpha_{3}]=[\alpha_{2}]\end{split}$
(2.57)

Thus the moduli spaces of the sectors of the theory described by
$\alpha_{0,1}$ are identical to the naive one:
$\mathcal{M}_{\Lambda,\alpha_{0}}=\mathcal{M}_{\Lambda,\alpha_{1}}=[4,\infty)$.
For the remaining sectors, we have
$\mathcal{M}_{\Lambda,\alpha_{2}}=\mathcal{M}_{\Lambda,\alpha_{3}}=(0,\infty)$.
However, if we consider any point $m\in(0,4)$ for either of these latter two
classes, we see that it can be mapped to the interval $[4,\infty)$, provided
that we swap the sector simultaneously as
$\alpha_{2}\leftrightarrow\alpha_{3}$. Thus we see that the moduli space of
the full theory is indeed the naive one, in agreement with the discussion
above.

##  3\. Orbifolded Narain CFTs

Let us next further enlarge the class of theories, by considering orbifolds of
generalized Narain CFTs.

###  3.1. Orbifold Actions

We start with the un-orbifolded theory specified by the lattice $\Lambda$.
Given a group $G$, let us consider an action of $g\in G$ on the momentum
$p=(p_{L},p_{R})$:

$\displaystyle
g:(p_{L},p_{R})\mapsto(\theta_{L}(g)p_{L}+V_{L}(g),\theta_{R}(g)p_{R}+V_{R}(g))\;.$
(3.1)

This is defined by a pair of the rotation matrix
$\theta(g)=(\theta_{L}(g),\theta_{R}(g))$ and the “shift vector”
$V(g)=(V_{L}(g),V_{R}(g))$, where the latter is defined modulo the lattice
$\Lambda$. In general, there are multiple ways of expressing the same orbifold
action in terms of $\theta$ and $V$. We will see, however, that such
ambiguities will not affect the subsequent discussions. For consistency with
the identification of the un-orbifolded theory we need to impose the condition
that $G$ is an automorphism of the lattice $\Lambda$, namely
$g\cdot\Lambda=\Lambda$ for any $g\in G$.

The rotations $\theta(g)$ generate the point group, while the pair
$(\theta(g),V(g))$, together with the translations of the original lattice
$\Lambda$, generates the space group. Its elements satisfy the consistency
condition

$\displaystyle(\theta(g_{1}),V(g_{1}))\circ(\theta(g_{2}),V(g_{2}))=(\theta(g_{1}g_{2}),V(g_{2})+\theta(g_{2})V(g_{1}))\;.$
(3.2)

The toroidal orbifold is defined by the identification of the momentum under
the space group.

For simplicity of presentations, we choose $G=\mathbb{Z}_{N}$ for the rest of
this paper. We use the symbol $\omega$ to be a generator of $\mathbb{Z}_{N}$,
and sometimes denote the rotation $\theta(\omega)$ (shift vector $V(\omega)$)
simply as $\theta$ ($V$). In this case,
$\omega^{n}=(\theta,V)^{N}=(\theta^{N},NP_{\omega}V)=(1,0)$ with

$\displaystyle P_{\omega}:=(1+\theta+\theta^{2}+\dots+\theta^{N-1})/N$ (3.3)

being a projector onto $\theta$-invariant states. Let us define $I$ to be the
invariant lattice of $\Lambda$ under $\omega$ (hence under
$G=\mathbb{Z}_{N}$):

$\displaystyle I:=\left\\{\ell\in\Lambda|\,\omega\cdot\ell=\ell\right\\}\;.$
(3.4)

This lattice is moduli-independent for symmetric orbifolds
$\theta_{L}=\theta_{R}$, and we will in the following restrict our analysis to
symmetric orbifolds.131313For asymmetric orbifolds the invariant lattice $I$
is in itself moduli-dependent, and it becomes more challenging to discuss
ensemble averages.

We define $W$ to be the “projected shift vector”

$\displaystyle W:=P_{\omega}V\;.$ (3.5)

Since $(\theta,V)^{N}=1$, we need $\theta^{N}=1$ and $NW\in\Lambda$. The
former condition states that we can denote the eigenvalues of
$\theta_{L}(\omega)$ and $\theta_{R}(\omega)$ in terms of “twist vectors”
$l_{j},r_{j}\in\mathbb{Z}/N\mathbb{Z}$:

$\displaystyle\theta_{L}(g):\\{e^{2\pi
i\,l_{j}}\\}\;,\quad\theta_{R}(g):\\{e^{2\pi i\,r_{j}}\\}\;.$ (3.6)

The latter condition can be written as a condition on $W$:

$\displaystyle W\in I(N)/I\;,$ (3.7)

where we defined the $I(N)$ to be

$\displaystyle I(N):=\left\\{N\ell\in\Lambda|\,N\ell\in I\right\\}\;.$ (3.8)

Note that the ambiguity of the shift of $V$ by an element of $\Lambda$ does
not affect the element 3.7 modulo $I$. Moreover, the lattice $I$ and the
projected shift vector $W$ is not affected by the ambiguity of expressing
$\omega$ in term of the rotation $\theta$ and the shift $V$.

In the following we consider symmetric orbifolds with $\theta_{L}=\theta_{R}$
(and hence for example $l_{j}=r_{j}$), so that the orbifold action 3.1 can be
written in a moduli-independent manner:

$\displaystyle g:p\mapsto\theta_{L}(g)p+V(g)\;.$ (3.9)

The orbifold projects out the Narain moduli space 2.51 onto a smaller subspace

$\displaystyle\mathcal{M}_{I}:=\mathrm{O}_{I}(\mathbb{Z})\big{\backslash}(O(p_{I},q_{I};\mathbb{R})\big{/}(O(p_{I};\mathbb{R})\times
O(q_{I};\mathbb{R}))\subset\mathcal{M}_{Q}\;,$ (3.10)

where we denoted the signature of $I$ by $(p_{I},q_{I})$. In the following we
consider an ensemble average over this moduli space.

Notice that in general the lattice $I$ could be trivial—in the examples of
$\mathbb{Z}_{2}$-orbifolds of toroidally-compactified theories discussed in
[8], for example, all the twisted sectors have no moduli dependence, so that
the ensemble average over $\mathcal{M}_{I}$ is trivial. Nevertheless, since we
consider a rather general class of orbifolds of general lattices, we will in
general have non-trivial invariant lattices and hence the associated moduli
space.

###  3.2. Torus Partition Functions

Let us next discuss the torus partition functions of the theory. While such
partition functions have been extensively studied in the context of heterotic
string compactifications, we are not focusing on specific lattices and will
keep the lattice to be a general lattice associated with even quadratic forms.

The torus partition function of the $\mathbb{Z}_{N}$-orbifolded theory is
built out of the following $N^{2}$ partition functions:

$\displaystyle\textrm{untwisted
sector}:Z(1,\omega^{n})\quad(n=0,1,\dots,N-1)\;,$ (3.11)
$\displaystyle\textrm{twisted
sector}:Z(\omega^{m},\omega^{n})\quad(m=1,\dots,N-1;n=0,1,\dots,N-1)\;.$
(3.12)

Here $Z(g,h)$ denotes the torus partition function with twists $g\in G$ (and
$h\in G$) along spatial (and temporal) directions:

$\displaystyle
Z(g,h)=\begin{picture}(20.0,20.0)(0.0,0.0)\put(10.0,0.0){\framebox(15.0,15.0)[]{}}
\put(15.0,-7.0){\mbox{$g$}} \put(2.0,4.0){\mbox{$h$}} \end{picture}\quad\;.$
(3.13)

In general such twists require a compatibility condition $gh=hg$. This
condition is automatically satisfied for Abelian orbifolds.

The full torus partition function of the $G$-orbifolded theory is given by

$\displaystyle Z_{Q/G}=\frac{1}{N}\left[\overbrace{\sum_{h\in
G}Z(1,h)}^{\textrm{untwisted
sector}}+\overbrace{\sum_{\genfrac{}{}{0.0pt}{}{g,h\in G}{[g,h]=1;g\neq
1}}c(g,h)Z(g,h)}^{\textrm{twisted sector}}\right]\;,$ (3.14)

and in the literature the coefficients $c(g,h)$ are chosen appropriately for
the modular invariance of the full partition function. We are, however, not
imposing the modular invariance in this paper,141414In [19] the modular non-
invariance of the torus partition function on the boundary CFT matches with
the framing anomaly of the Chern-Simons theory in the bulk, and the modular
non-invariance was a crucial ingredient for the consistency of the discussion.
and for our paper it is enough to discuss individual blocks $Z(g,h)$
separately.

For the untwisted sectors, the orbifold action is along the temporal direction
and hence we can still use the same Hilbert space $\mathcal{H}_{Q}$ as in the
un-orbifolded case, as long as we insert an appropriate operator $U(g)$
representing the twist by $g\in G$:

$\displaystyle
Z(1,g)=\textrm{Tr}_{\mathcal{H}_{\Lambda}}\left(U(g)\,q^{L_{0}-\frac{c}{24}}\overline{q}^{\overline{L}_{0}-\frac{\overline{c}}{24}}\right)\;.$
(3.15)

For $g=1$, $Z(1,1)$ is nothing but the un-orbifolded partition function
$Z^{\Lambda}$ given in 2.26. For $Z(1,g)$, from the $g$-action given in 3.1,
we find that the only contribution comes from the subspace of the un-
orbifolded Hilbert space left invariant under the generator $\omega$; this is
described by the the $G$-invariant sublattice $I$ 3.4. This means that the
partition function 3.15 contains the contribution

$\displaystyle\sum_{\ell\in I}e^{i\pi\tau
Q_{L}(\ell)-i\pi\overline{\tau}Q_{R}(\ell)}e^{2\pi
i(p_{L}(\ell)V_{L}-p_{R}(\ell)V_{R})}=\sum_{\ell\in I}e^{i\pi\tau
Q_{L}(\ell)-i\pi\overline{\tau}Q_{R}(\ell)}e^{2\pi iQ(\ell,W)}\;,$ (3.16)

where in the second line we used $Q(\ell,V)=Q(P_{\omega}\ell,V)=Q(\ell,W)$ for
$\ell\in I$.

We also need to take into account contributions from oscillators. For
$(ml_{j},nl_{j})=(0,0)$ we have the familiar contribution $1/\eta(\tau)$ from
the full set of chiral oscillators. For $(ml_{j},nl_{j})\neq(0,0)$ the
contribution from chiral oscillators is
$\eta(\tau)/\vartheta\\!\left[\genfrac{}{}{0.0pt}{}{\frac{1}{2}+ml_{j}}{\frac{1}{2}+nl_{j}}\right]\\!(\tau)$,
where the theta function
$\vartheta\\!\left[\genfrac{}{}{0.0pt}{}{\theta}{\phi}\right]\\!(\tau)$ is
given by

$\displaystyle\begin{split}\vartheta\\!\left[\genfrac{}{}{0.0pt}{}{\theta}{\phi}\right]\\!(\tau)&:=\sum_{n\in\mathbb{Z}}q^{\frac{1}{2}(n+\theta)^{2}}e^{2\pi
i(n+\theta)\phi}\\\ &=\eta(\tau)\,e^{2\pi
i\theta\phi}q^{\frac{1}{2}\theta^{2}-\frac{1}{24}}\prod_{k=1}^{\infty}(1+q^{k+\theta-\frac{1}{2}}e^{2\pi
i\phi})(1+q^{k-\theta-\frac{1}{2}}e^{-2\pi i\phi})\;.\end{split}$ (3.17)

There are similar oscillator contributions for anti-chiral oscillators, with
$\tau$ replaced by $\overline{\tau}$.

By collecting all the ingredients, we obtain

$\displaystyle
Z(\omega^{m},\omega^{n})=\begin{cases}Z_{\Lambda}=\displaystyle\frac{\vartheta_{\Lambda}}{\eta^{p}{\overline{\eta}^{q}}}&(m,n)=(0,0)\;,\\\
Z^{I^{\perp}}(\omega^{m},\omega^{n})Z^{I}(\omega^{m},\omega^{n})&(m,n)\neq(0,0)\;,\end{cases}$
(3.18)

The building blocks, $Z_{I^{\perp}}$ for the oscillators and $Z_{I}$ for the
lattice, are given by

$\displaystyle
Z^{I^{\perp}}(\omega^{m},\omega^{n})=C(I,\vartheta)\prod_{j:\,l_{j}\neq
0}\frac{\eta(\tau)}{\vartheta\\!\left[\genfrac{}{}{0.0pt}{}{\frac{1}{2}+ml_{j}}{\frac{1}{2}+nl_{j}}\right]\\!(\tau)}\prod_{j:\,r_{j}\neq
0}\frac{\overline{\eta}(\overline{\tau})}{\vartheta\\!\left[\genfrac{}{}{0.0pt}{}{\frac{1}{2}+mr_{j}}{\frac{1}{2}+nr_{j}}\right]\\!(\overline{\tau})}\;,$
(3.19) $\displaystyle
Z^{I}(\omega^{m},\omega^{n})=\frac{\vartheta^{I}_{(mW,nW)}(\tau,\overline{\tau};m)}{\eta(\tau)^{p_{I}}\overline{\eta}(\overline{\tau})^{q_{I}}}\;,$
(3.20)

and $C(I,\vartheta)$ is a moduli-independent numerical factor which is not
important for our analysis.151515This includes an integer
$\sqrt{\det\\!{}^{\prime}(1-\theta)/|I^{*}/I|}$ counts the number of fixed
points under the $\theta$-rotation [34]. Here $I^{*}$ is defined in 3.24 and
the primed determinant $\det\\!{}^{\prime}$ is defined by disregarding the
zero eigenvalues.

In 3.20 we defined a generalized version of the theta function associated with
the invariant lattice $I$:

$\displaystyle\vartheta^{I}_{(\delta,\eta)}(\tau,\overline{\tau};m):=\sum_{\ell\in
I+\delta}e^{i\pi\tau Q_{L}(\ell)-i\pi\overline{\tau}Q_{R}(\ell)}e^{2i\pi
Q_{I}(\ell,\eta)}\;,\quad(\delta\in\mathbb{R}^{p_{I}+q_{I}}/I,\quad\eta\in\mathbb{R}^{p_{I}+q_{I}})\;,$
(3.21)

where we have shown the dependence on the moduli $m\in\mathcal{M}_{I}$, and
$Q_{I}$ denotes the quadratic form associated with the lattice $I$. Since
$Q_{I}$ is a restriction of $Q$, for notational simplify we mostly use the
same symbol $Q$ to denote the quadratic form $Q_{I}$; however, the distinction
matters when we discuss the determinants of the quadratic forms, for example.

We can verify from the definition that $\vartheta_{I,(\delta,\eta)}$ is indeed
preserved when we shift $\delta$ by an element of $I$, as suggested in 3.21.
By contrast, when we shift $\eta$ by an element of $\beta\in I^{*}$ we obtain
an extra phase factor

$\displaystyle\vartheta^{I}_{(\delta,\eta+\beta)}(\tau,\overline{\tau};m)=e^{2i\pi
Q(\delta,\beta)}\,\vartheta^{I}_{(\delta,\eta)}(\tau,\overline{\tau};m)\;,$
(3.22)

and the phase factor is in general non-trivial even when $\beta\in I$.

By relabeling the sum as $\ell\to-\ell$ we obtain

$\displaystyle\vartheta^{I}_{(\delta,\eta)}(\tau,\overline{\tau};m)=\vartheta^{I}_{(-\delta,-\eta)}(\tau,\overline{\tau};m)\;.$
(3.23)

Let us denote the signature of $Q_{I}$ by $(p_{I},q_{I})$, and define the dual
lattice $I^{*}$ of $I$ as a dual inside $\mathbb{R}^{p_{I}+q_{I}}$ (and not
$\mathbb{R}^{p+q}$):

$\displaystyle
I^{*}:=\left\\{x\in\mathbb{R}^{p_{I}+q_{I}}\,\Big{|}\,Q(x,\ell)\in\mathbb{Z}\quad(\forall\
\ell\in I)\right\\}\;.$ (3.24)

Similar to the case of $\Lambda$, we define the discriminant for $I$ to be

$\displaystyle\mathscr{D}_{I}=I^{*}/I\;.$ (3.25)

For $T$ and $S$ generators of $\mathrm{SL}(2,\mathbb{Z})$ we have the modular
transformation (see Appendix C for derivation):161616 Special cases of these
transformation rules appeared in string theoretic literature [35, section 3].

$\displaystyle T:\quad\vartheta^{I}_{(\delta,\eta)}(\tau+1;m)=e^{-\pi
iQ(\delta)}\,\vartheta^{I}_{(\delta,\eta+\delta)}(\tau;m)\;,$ (3.26)
$\displaystyle
S:\quad\vartheta^{I}_{(\delta,\eta)}\left(-\frac{1}{\tau};m\right)=\frac{e^{-\frac{i\pi(p_{I}-q_{I})}{4}}}{\sqrt{\textrm{det}\,Q_{I}}}\tau^{\frac{p_{I}}{2}}\overline{\tau}^{\frac{q_{I}}{2}}e^{2\pi
iQ(\delta,\eta)}\sum_{\gamma\in\mathscr{D}_{I}}\vartheta^{I}_{(\gamma+\eta,-\delta)}(\tau;m)\;.$
(3.27)

In terms of the ratio

$\displaystyle
Z^{I}_{(\delta,\eta)}(\tau;m):=\frac{\vartheta^{I}_{(\delta,\eta)}(\tau;m)}{\eta^{p_{I}}(\tau)\overline{\eta}^{q_{I}}(\overline{\tau})}\;,$
(3.28)

we have

$\displaystyle T:Z^{I}_{(\delta,\eta)}(\tau+1;m)=e^{-\frac{2\pi
i(p_{I}-q_{I})}{24}}e^{-\pi
iQ(\delta)}\,Z^{I}_{(\delta,\eta+\delta)}(\tau;m)\;,$ (3.29) $\displaystyle
S:Z^{I}_{(\delta,\eta)}\left(-\frac{1}{\tau};m\right)=\frac{1}{\sqrt{\textrm{det}\,Q_{I}}}\,e^{2\pi
iQ(\delta,\eta)}\sum_{\gamma\in\mathscr{D}_{I}}Z^{I}_{(\gamma+\eta,-\delta)}(\tau;m)\;.$
(3.30)

For our practical purposes, we do not need fully general choices of
$\delta,\eta$, and we only need those needed for the discussion of the modular
transformations of $Z^{I}_{(mW,nW)}$ 3.20. This picks up the choice
$\delta=\alpha+mW$ and $\eta=\beta+nW$ for $\alpha,\beta\in\mathscr{D}_{I}$
and $m,n\in\mathbb{Z}_{N}$. The only effect of $\beta$, however, is to change
the overall normalization of the partition function

$\displaystyle\vartheta^{I}_{(\alpha+mW,\beta+nW)}(\tau;m)=e^{2\pi
iQ(\alpha+mW,\beta)}\,\vartheta^{I}_{(\alpha+mW,nW)}(\tau;m)\;,$ (3.31)

and hence we can set $\beta=0$ when we are discussing basic building blocks up
to overall normalization factors. The modular transformations of the blocks
$Z^{I}_{(\alpha+mW,nW)}$ are given by, after using 3.31:

$\displaystyle T:Z^{I}_{(\alpha+mW,nW)}(\tau+1;m)=e^{-\frac{2\pi
i(p_{I}-q_{I})}{24}}e^{-\pi im^{2}Q(W)+\pi
iQ(\alpha)}\,Z^{I}_{(\alpha+mW,(m+n)W)}(\tau;m)\;,$ (3.32) $\displaystyle
S:Z^{I}_{(\alpha+mW,nW)}\left(-\frac{1}{\tau};m\right)=\frac{1}{\sqrt{\textrm{det}\,Q_{I}}}\,e^{2\pi
imnQ(W)}\sum_{\gamma\in\mathscr{D}_{I}}e^{-2\pi
iQ(\gamma,\alpha)}Z^{I}_{(\gamma+nW,-mW)}(\tau;m)\;.$ (3.33)

As we prove in Appendix D, the partition functions $Z_{I,(\alpha+mW,nW)}$ are
modular functions with respect to a congruence subgroup $\Gamma(N^{2}L)$ of
$\mathrm{PSL}(2,\mathbb{Z})$, where an integer $L$ is a integer multiple of
the level $L_{I}$ of the quadratic form $Q_{I}$, where the level is defined as
the smallest integer such that $L_{I}Q_{I}^{-1}$ is integral.171717The level
$L_{I}$ was denoted as $N$ in [19].

By comparing the modular transformations 3.32 with those of the un-orbifolded
case 2.44, one finds that the $(m,n)\neq 0$ sector we have the same
transformation as the un-orbifolded case (represented by $m=n=0$), the only
difference being that

  1. (i)

the lattice $\Lambda$ replaced by the $G$-invariant lattice $I$ and

  2. (ii)

the discriminant $\mathscr{D}=\Lambda^{*}/\Lambda$ is replaced by

$\displaystyle\hat{\mathscr{D}}:=\mathscr{D}_{I}\oplus\mathscr{D}_{N}\;.$
(3.34)

which is larger than $\mathscr{D}_{I}=I^{*}/I$ by a factor of
$\mathscr{D}_{N}:=\mathbb{Z}_{N}$, the choice of which is determined by the
projected shift vector $W$.

The braiding between the two elements of $\mathscr{D}_{N}$ is determined by
the quadratic form $Q$ and the projected shift vector $W$:

$\displaystyle B(m,n)=e^{2\pi iQ(mW,nW)}\;,\quad(m,n\in\mathbb{Z}_{N})\;.$
(3.35)

For the special case $N=2$ we have

$\displaystyle\vartheta^{I}_{(\alpha+mW,nW)}(\tau,\overline{\tau}):=\sum_{\ell\in
I+\alpha+mW}e^{i\pi\tau
Q_{L}(\ell)-i\pi\overline{\tau}Q_{R}(\ell)}(-1)^{nQ(\ell,2W)}\;,$ (3.36)

with $\alpha,\beta\in I^{*}$, $W\in I/2$ and $m,n=0,1$. This expression is
reminiscent of the theta function for spin CFTs in our previous paper [19],
and the projected shift vector $2W\in I$ plays a role analogous to the Wu
class [36, 37] for the spin CFT.

The modular transformation for the orbifolded lattice theta function 3.21 is
given by (see Appendix C for derivation)

$\displaystyle\vartheta^{I}_{(\delta,\eta)}(\tau_{M};m)$
$\displaystyle=\mu_{(\delta,\eta)\cdot
M}\sum_{\gamma\in\mathscr{D}_{I}}\mathcal{U}_{\alpha,\gamma}^{I}(M)\,\vartheta^{I}_{(\gamma,0)+(\delta,\eta)\cdot
M}(\tau;m)\;,$ (3.37)

Here we defined a phase $\mu_{(\delta,\eta)}$ by

$\displaystyle\mu_{(\delta,\eta)}:=e^{-i\pi Q(\delta,\eta)}$ (3.38)

so that $\mu_{(-\delta,-\eta)}=\mu_{(\delta,\eta)}$ and
$\mu_{(\delta,\eta)\cdot M}=\mu_{(a\delta+c\eta,b\delta+d\eta)}=e^{-i\pi
Q(a\delta+c\eta,b\delta+d\eta)}$. The matrix
$\mathcal{U}^{I}_{\alpha,\gamma}(M,\tau)$ (for $M$ given as in 2.38 with
entries $a,b,c,d$) is given by the essentially same expression as
$\mathcal{U}^{\Lambda}_{\alpha,\gamma}(M,\tau)$, with $\Lambda$ replaced by
$I$:

$\displaystyle\mathcal{U}^{I}_{\alpha,\gamma}(M,\tau):=\begin{cases}\displaystyle(c\tau+d)^{\frac{p}{2}}(c\overline{\tau}+d)^{\frac{q}{2}}\lambda^{I}_{\alpha,\gamma}(M)&(c\neq
0)\;,\\\ \displaystyle\mathcal{U}^{I}_{\alpha,\gamma}(T,\tau):=e^{i\pi
Q(\alpha)}\delta_{\alpha,\gamma}&(c=0)\;,\end{cases}$ (3.39)

and

$\displaystyle\lambda^{I}_{\alpha,\gamma}(M,\tau):=\frac{1}{\sqrt{\textrm{det}\,Q_{I}}}\,e^{-\frac{\pi
i}{4}(p_{I}-q_{I})}c^{-\frac{p_{I}+q_{I}}{2}}\sum_{\ell_{c}\in
I/(cI)}e^{\frac{\pi
i}{c}\left(aQ[\ell_{c}+\alpha]-2Q[\ell_{c}+\alpha,\gamma]+dQ[\gamma]\right)}\;.$
(3.40)

As in the un-orbifolded case 2.42, we have the consistency relations

$\displaystyle\mathcal{U}^{I}_{\alpha,\beta}(M\cdot
M^{\prime},\tau)=\sum_{\gamma\in\mathscr{D}_{\Lambda}}\mathcal{U}^{I}_{\alpha,\gamma}(M,\tau_{M^{\prime}})\,\mathcal{U}^{I}_{\gamma,\beta}(M^{\prime},\tau)\;,$
(3.41)

for two $\mathrm{SL}(2,\mathbb{Z})$ matrices $M,M^{\prime}$. We also have the
sign-flip relations

$\displaystyle\mathcal{U}^{I}_{\alpha,\gamma}(M,\tau)=\mathcal{U}^{I}_{\alpha,-\gamma}(-M,\tau)\;,$
(3.42)

which ensures that the right hand side of 3.37 does not depend on the choice
of the $\mathrm{SL}(2,\mathbb{Z})$ representative $M$.

For the special case $\delta=\alpha+mW,\eta=\beta+nW$, we have

$\displaystyle\vartheta^{I}_{(\alpha+mW,\beta+nW)}(\tau_{M};m)$
$\displaystyle=\mu_{(\alpha+mW,\beta+nW)\cdot
M}\sum_{\gamma\in\mathscr{D}_{I}}\mathcal{U}_{\alpha,\gamma}^{I}(M)\,\vartheta^{I}_{(\gamma,0)+(\alpha+mM,\beta+nW)\cdot
M}(\tau;m)\;,$ (3.43)

##  4\. Ensemble Average

###  4.1. Unorbifolded Case

The ensemble average of the theta function $\vartheta_{Q,\alpha}$ over the
moduli space 2.49181818We impose $p+q>4$ for the convergence of the integral.

$\displaystyle\langle\vartheta^{Q}_{\alpha}\rangle_{\mathcal{M}_{Q}}(\tau)$
$\displaystyle:=\frac{1}{\textrm{Vol}(\widetilde{\mathcal{M}}_{Q})}\int_{\tilde{\mathcal{M}}_{Q}}[dm]\,\vartheta^{Q}_{\alpha}(\tau;m)$
(4.1)

is given by the Siegel-Eisenstein series [38, 25, 2, 19] (henceforth referred
to simply as the Eisenstein series)

$\displaystyle\langle\vartheta^{Q}_{\alpha}(\tau)\rangle_{\mathcal{M}_{\Lambda}}=E^{Q}_{\alpha}(\tau):=\delta_{\alpha\in\Lambda}+\sum_{(c,d)=1,\,c>0}\,\frac{\gamma^{\Lambda}_{\alpha}(c,d)}{(c\tau+d)^{\frac{p}{2}}(c\overline{\tau}+d)^{\frac{q}{2}}}\;,$
(4.2)

associated with the lattice $\Lambda$, where $\delta_{\alpha}=1$ for
$\alpha\in\Lambda$, and $\delta_{\alpha}=0$ for $\alpha\notin\Lambda$. The
factor $\gamma^{\Lambda}_{\alpha}(c,d)$ is given by

$\displaystyle\gamma^{\Lambda}_{\alpha}(c,d):=\lambda_{\alpha,0}(M^{-1})=\frac{1}{\sqrt{\textrm{det}\,Q}}\,e^{-\frac{\pi
i(p-q)}{4}}(-c)^{-\frac{p+q}{2}}\sum_{\ell_{c}\in\Lambda/c\Lambda}\exp\left[-\pi
i\frac{d}{c}Q(\ell_{c}+\alpha)\right]\;.$ (4.3)

with $M$ given as in 2.38. Note that this depends only on the two entries
$c,d$ of the matrix $M$, as opposed to general $\lambda_{\alpha,\beta}(M)$
given in 2.41.

Since the Eisenstein series is an ensemble average of the theta function, and
since the modular transformation matrices $\mathcal{U}$ in 2.39 are
independent of the moduli, the modular transformation of the Eisenstein series
should be the same as in 2.39:

$\displaystyle E^{\Lambda}_{\alpha}(\tau_{M})$
$\displaystyle=\sum_{\beta\in\mathscr{D}_{\Lambda}}\mathcal{U}^{\Lambda}_{\alpha,\beta}(M,\tau)\,E^{\Lambda}_{\beta}(\tau)\;.$
(4.4)

We can verify this expression explicitly from the definition 4.2 with the help
of 4.3 and 2.42. Finally, we have the counterparts of 2.29 and 2.48:

$\displaystyle E^{\Lambda}_{\alpha}(\tau)=E^{\Lambda}_{-\alpha}(\tau)\;,\quad
E^{\Lambda}_{g\cdot\alpha}(\tau)=E^{\Lambda}_{\alpha}(\tau)\;.$ (4.5)

###  4.2. Orbifolded Case

Let us next discuss the ensemble average of the partition functions over the
projected Narain moduli space $\mathcal{M}_{I}$ 3.10.191919We impose $p+q>4$
for the convergence of the integral. We evaluate the ensemble average of the
theta function $\vartheta_{I,(\delta,\eta)}$ introduced in 3.21, which we
denote as $E^{I}_{(\delta,\eta)}$ (in anticipation of the fact this is a
certain generalization of the Siegel-Eisenstein series) :

$\displaystyle\langle\vartheta_{I,(\delta,\eta)}\rangle_{\mathcal{M}_{I}}(\tau,\overline{\tau})=\frac{1}{\textrm{Vol}(\mathcal{M}_{I})}\int_{\mathcal{M}_{I}}[dm]\,\vartheta_{I,(\delta,\eta)}(\tau,\overline{\tau};m)\;.$
(4.6)

To evaluate this average we need some generalization of the Siegel-Weil
formula, which does not seem to exist in the literature.202020The exception is
Ref. [22], where the authors considered the special cases of CFTs defined by
even self-dual lattices and vanishing chiral central charge ($p=q$ in our
notation). They considered the $\mathrm{SU}(N)$ WZW models at level $k$, which
describes the $\mathbb{Z}_{N}$-quotient of a product theory, i.e., Narain
theory times the parafermion theory. The theta functions in the two references
are related as $\vartheta^{\rm(here)}_{(\delta,\eta)}(\tau)=e^{\pi
i\delta\cdot\eta}\vartheta^{\rm(there)}_{(-\delta,\eta)}(\tau)$. Since the
difference is only an overall constant, the same trivially propagates into the
definition of the Eisenstein series after the ensemble averages. It should be
noticed, however, that the phase factors transform non-trivially under the
modular transformation and hence we obtain slightly different expressions for
the Poincaré sum.

Fortunately for us, the Siegel-Weil formula was re-derived in our previous
paper [19, section 2.3] and we can use the same logic to derive the formula
necessary for our present purposes, as explained below.

We find that the ensemble average
$\langle\vartheta_{I,(\delta,\eta)}\rangle_{\mathcal{M}_{I}}$ evaluates to

$\displaystyle E^{I}_{(\delta,\eta)}(\tau,\overline{\tau})$
$\displaystyle:=\delta_{\delta\in
I}+\frac{1}{\sqrt{\textrm{det}\,Q_{I}}}\,e^{-\frac{\pi
i}{4}(p_{I}-q_{I})}\sum_{(c,d)=1,c>0}c^{-\frac{p_{I}+q_{I}}{2}}(c\tau+d)^{-\frac{p_{I}}{2}}(c\bar{\tau}+d)^{-\frac{q_{I}}{2}}$
$\displaystyle\quad\times\mu_{(\delta,\eta)\cdot
M^{-1}}\lambda^{I}_{0,-d\delta+c\eta}(M^{-1})\;,$ (4.7)

where $M$ is a $\mathrm{PSL}(2,\mathbb{Z})$ matrix with the second row given
by $(c,d)$ as in 2.38, and we used previously-defined $\mu_{(\delta,\eta)}$ as
in 3.38, as well as

$\displaystyle\lambda^{I}_{0,-d\delta+c\eta}(M^{-1})$
$\displaystyle=\sum_{\ell_{c}\in I/(cI)}e^{-\frac{\pi
i}{c}\left(dQ[\ell_{c}]-2Q[\ell_{c},-d\delta+c\eta]+aQ[-d\delta+c\eta]\right)}$
(4.8)

as in 2.41 (previously defined for $\Lambda$). This is one of the main
technical results of this paper. We can slightly simplify the formula into

$\displaystyle E^{I}_{(\delta,\eta)}(\tau,\overline{\tau})$
$\displaystyle=\delta_{\delta\in
I}+\frac{1}{\sqrt{\textrm{det}\,Q_{I}}}\,e^{-\frac{\pi
i}{4}(p_{I}-q_{I})}\sum_{(c,d)=1,c>0}c^{-\frac{p_{I}+q_{I}}{2}}(c\tau+d)^{-\frac{p_{I}}{2}}(c\bar{\tau}+d)^{-\frac{q_{I}}{2}}$
$\displaystyle\quad\times\sum_{\ell_{c}\in I/(cI)+\delta}e^{-\frac{\pi
i}{c}\left(dQ[\ell_{c}]-2cQ[\ell_{c},\eta]+cQ[\delta,\eta]\right)}\;.$ (4.9)

In the latter form it is manifest that the expression reduces to the un-
orbifolded case 4.2 for $N=1$ (and hence $\delta=\eta=0$).

We call the function in 4.2 and 4.2 the “orbifold Eisenstein series,” since
this is an orbifold analogue of the non-holomorphic Eisenstein series. The
orbifolded Eisenstein series is a modular form with respect to a congruence
subgroup $\Gamma(N^{2}L_{I})$ with $L_{I}$ being directly proportional to the
level of $Q_{I}$; see also Appendix D. As far as we are aware, there is no
literature discussing this Eisenstein-like series in the mathematical
literature (apart from special cases). It would be interesting to further
study the properties of this function.

For application to orbifolds, we set $(\delta,\eta)=(\alpha+mW,\beta+nW)$ with
$\alpha,\beta\in\mathscr{D}_{I}$ and $m,n\in\mathbb{Z}_{N}$, leading to

$\displaystyle E^{I}_{(\alpha+mW,\beta+nW)}(\tau)=\delta_{\delta\in
I}+\sum_{(c,d)=1,c>0}\gamma^{I}_{(\alpha,\beta;m,n)}(c,d)(c\tau+d)^{-\frac{p_{I}}{2}}(c\bar{\tau}+d)^{-\frac{q_{I}}{2}}\;,$
(4.10)

with

$\displaystyle\gamma^{I}_{(\alpha,\beta;m,n)}(c,d)$
$\displaystyle=\frac{1}{\sqrt{\textrm{det}\,Q_{I}}}\,e^{-\frac{\pi
i}{4}(p_{I}-q_{I})}c^{-\frac{p_{I}+q_{I}}{2}}\mu_{(\alpha+mW,\beta+nW)\cdot
M^{-1}}\lambda^{I}_{0,-d(\alpha+mW)+c(\beta+nW)}(M^{-1})$ (4.11)
$\displaystyle=\frac{1}{\sqrt{\textrm{det}\,Q_{I}}}\,e^{-\frac{\pi
i}{4}(p_{I}-q_{I})}c^{-\frac{p_{I}+q_{I}}{2}}$
$\displaystyle\qquad\times\sum_{\ell_{c}\in I/(cI)+\alpha+mW}e^{-\frac{\pi
i}{c}\left(dQ[\ell_{c}]-2cQ[\ell_{c},\beta+nW]+cQ[\alpha+mW,\beta+nW]\right)}\;.$
(4.12)

The averaged partition functions are then

$\displaystyle\langle
Z^{I}_{(\alpha+mW,\beta+nW)}(\tau,\overline{\tau})\rangle$
$\displaystyle=\frac{E^{I}_{(\alpha+mW,\beta+nW)}(\tau,\overline{\tau})}{\eta(\tau)^{p_{I}}\overline{\eta}(\overline{\tau})^{q_{I}}}$
$\displaystyle=\frac{\delta_{mW\in
I}}{\eta(\tau)^{p_{I}}\overline{\eta}(\overline{\tau})^{q_{I}}}+\sum_{g\in\Gamma_{\infty}\backslash\mathrm{PSL}(2,\mathbb{Z}),g\neq
1}\epsilon(g)^{p_{I}-q_{I}}\frac{\gamma^{I}_{(\alpha,\beta;m,n)}(g)}{\eta(g\cdot\tau)^{p_{I}}\overline{\eta}(g\cdot\overline{\tau})^{q_{I}}}\;,$
(4.13)

where the phase $\epsilon(g)$ is the multiplier system for the Dedekind eta
function and is described in Appendix E. Their modular transformations are
given by

$\displaystyle T:\langle
Z^{I}_{(\alpha+mW,nW)}(\tau+1;m)\rangle=e^{-\frac{2\pi
i(p_{I}-q_{I})}{24}}e^{-\pi iQ(mW)}e^{\pi iQ(\alpha)}\,\langle
Z^{I}_{(\alpha+mW,(m+n)W)}(\tau;m)\rangle\;,$ (4.14) $\displaystyle S:\langle
Z^{I}_{(\alpha+mW,nW)}\left(-\frac{1}{\tau};m\right)\rangle=\frac{1}{\sqrt{\textrm{det}\,Q_{I}}}\,e^{2\pi
iQ(mW,nW)}\sum_{\gamma\in\mathscr{D}_{I}}e^{-2\pi iQ(\alpha,\gamma)}\langle
Z^{I}_{(\gamma-\alpha-mW,nW)}(\tau;m)\rangle\;.$ (4.15)

##### Modular Transformation

Since the orbifold Eisenstein series is defined by the ensemble average of the
orbifold theta function, we expect the two have the same modular
transformation properties

$\displaystyle E^{I}_{(\delta,\eta)}(\tau_{M^{\prime}})$
$\displaystyle=\mu_{(\delta,\eta)\cdot
M^{\prime}}\sum_{\gamma\in\mathscr{D}_{I}}\mathcal{U}_{0,\gamma}(M^{\prime})\,E^{I}_{(\gamma,0)+(\delta,\eta)\cdot
M^{\prime}}(\tau)\;,$ (4.16)

and in particular

$\displaystyle T:\quad E^{I}_{(\delta,\eta)}(\tau+1)=e^{-\pi
iQ(\delta,\eta)}\,E^{I}_{(\delta,\eta+\delta)}(\tau)\;,$ (4.17) $\displaystyle
S:\quad
E^{I}_{(\delta,\eta)}\left(-\frac{1}{\tau}\right)=\frac{e^{-\frac{i\pi(p_{I}-q_{I})}{4}}}{\sqrt{\textrm{det}\,Q_{I}}}\tau^{\frac{p_{I}}{2}}\overline{\tau}^{\frac{q_{I}}{2}}e^{-\pi
iQ(\delta,\eta)}\sum_{\gamma\in\mathscr{D}_{I}}E^{I}_{(\gamma-\eta,\delta)}(\tau)\;.$
(4.18)

To verify this, let us write
$M^{\prime}=\left(\begin{array}[]{cc}a^{\prime}&b^{\prime}\\\
c^{\prime}&d^{\prime}\end{array}\right)$ and

$\displaystyle E_{I,(\delta,\eta)}(\tau)$ $\displaystyle=\delta_{\delta\in
I}+\sum_{(c,d)=1,c>0}\left(\frac{-c\tau+a}{c\tau+d}\right)^{-\frac{p}{2}}\left(\frac{-c\overline{\tau}+a}{c\bar{\tau}+d}\right)^{-\frac{q}{2}}$
$\displaystyle\qquad\times\mu_{(\delta,\eta)\cdot
M^{-1}}\sum_{\gamma\in\mathscr{D}_{I}}\delta_{\gamma+d\delta-c\eta\in
I}\,\mathcal{U}_{0,\gamma}(M_{c,d}^{-1},\tau)\;.$ (4.19)

When evaluating $E^{I}_{(\delta,\eta)}(\tau_{M^{\prime}})$ we can use from
3.41

$\displaystyle\mathcal{U}^{I}_{0,\gamma}(M_{c,d}^{-1},\tau_{M^{\prime}})=\sum_{\beta\in\mathscr{D}_{I}}\mathcal{U}^{U}_{0,\beta}\left((M^{\prime-1}M_{c,d})^{-1},\tau\right)\,\mathcal{U}^{I}_{\beta,\gamma}(M^{\prime-1},\tau_{M^{\prime}})$
(4.20)

and

$\displaystyle\mathcal{U}_{\beta,\gamma}(M^{\prime-1},\tau)=\mathcal{U}_{\beta,\gamma}(M^{\prime-1},\tau_{M^{\prime}})\left(\frac{-c\tau+a}{c\tau+d}\right)^{-\frac{p}{2}}\left(\frac{-c\overline{\tau}+a}{c\bar{\tau}+d}\right)^{-\frac{q}{2}}\;.$
(4.21)

In the sum over $M$ we can replace $M$ by $M^{\prime}M$, to obtain

$\displaystyle E_{I,(\delta,\eta)}(\tau_{M^{\prime}})$
$\displaystyle=\delta_{\delta\in
I}+\sum_{\beta\in\mathscr{D}_{I}}\mathcal{U}_{\beta,\gamma}(M^{\prime-1},\tau_{M^{\prime}})\sum_{(c,d)=1,c>0}\mu_{(\delta,\eta)\cdot
M^{\prime}\cdot
M^{-1}}\sum_{\gamma\in\mathscr{D}_{I}}\delta_{\gamma+d\delta-c\eta\in
I}\,\mathcal{U}_{0,\beta}(M_{c,d}^{-1},\tau)\;.$ (4.22)

##### Proof

We can now prove 4.2. Let us consider the function
$F_{Q}:=E_{Q}-\langle\vartheta_{Q}\rangle$. This function satisfies the
following three properties:

  1. (i)

$F_{Q}$ is a modular form for a particular congruence subgroup, denoted
$\Gamma$, of $\mathrm{PSL}(2,\mathbb{Z})$; see Appendix D. This congruence
subgroup depends on the dimension of the lattice/quadratic form, specifically
if whether the lattice dimension is even or odd.

  2. (ii)

$F_{Q}$ has no singularities at the cusps of $\mathbb{H}/\Gamma$, and hence is
square-integrable. This is because $E_{Q}$ and $\vartheta_{Q}$ have the same
behaviour at the cusps; see Section C.2.

  3. (iii)

$\tau_{2}^{(p_{I}+q_{I})/4}F_{Q}$ is an eigenfunction of the weight $k$
Laplacian

$\displaystyle\square_{k}:=-\tau_{2}^{2}\left(\partial_{1}^{2}+\partial_{2}^{2}\right)+ik\tau_{2}\partial_{1},$
(4.23)

where $k=(p_{I}-q_{I})/2$, with eigenvalue
$-\frac{((p_{I}+q_{I})/4-1)(p_{I}+q_{I})}{4}$. This follows since

$\displaystyle\left(\tau_{2}(\partial_{1}^{2}+\partial_{2}^{2})+\frac{p_{I}+q_{I}}{2}\partial_{2}+\frac{i(q_{I}-p_{I})}{2}\partial_{1}\right)F_{Q}(\tau)=0\;.$
(4.24)

These facts are enough to conclude $F_{Q}=0$ as in [19, section 2.3]. We have
thus proven the identity 4.2.

##### Poincaré Sum

We can interpret the right hand side of 4.2 as a sum over geometries in the
bulk. Using the transformation law of the Dedekind eta function from Appendix
E, we have

$\displaystyle\langle
Z^{I}_{(\alpha+mW,\beta+nW)}(\tau,\overline{\tau})\rangle$
$\displaystyle=\sum_{g\in\Gamma_{\infty}\backslash\mathrm{PSL}(2,\mathbb{Z})}\epsilon(g)^{\sigma_{I}}\frac{\gamma^{I}_{(\alpha,\beta;m,n)}(g)}{\eta(g\cdot\tau)^{p_{I}}\overline{\eta}(g\cdot\overline{\tau})^{q_{I}}}\;.$
(4.25)

This suggests that the average has a holographic interpretation. However, 4.25
describes only one sector of the boundary CFT, and we must cast the entire
partition function as a Poincarè sum to justify a holographic interpretation.
This can be done by folding in the twisted sector partition functions
$Z^{I^{\perp}}(\omega^{m},\omega^{n})$ into the Poincarè sum in the same way
as the Dedekind eta functions above. Thus we write the averaged twisted sector
contribution to the partition function as

$\displaystyle\begin{split}\langle
Z^{I^{\perp}}(\omega^{m},\omega^{n})&Z^{I}(\omega^{m},\omega^{n})\rangle=\\\
&C(I,\vartheta)\sum_{g\in\Gamma_{\infty}\backslash\mathrm{PSL}(2,\mathbb{Z})}\left(\prod_{j:\,l_{j}\neq
0}\frac{\eta(g\cdot\tau)}{\vartheta\\!\left[\genfrac{}{}{0.0pt}{}{\frac{1}{2}+m^{\prime}l_{j}}{\frac{1}{2}+n^{\prime}l_{j}}\right]\\!(g\cdot\tau)}\prod_{j:\,r_{j}\neq
0}\frac{\overline{\eta}(g\cdot\overline{\tau})}{\vartheta\\!\left[\genfrac{}{}{0.0pt}{}{\frac{1}{2}+m^{\prime}r_{j}}{\frac{1}{2}+n^{\prime}r_{j}}\right]\\!(g\cdot\overline{\tau})}\right)\\\
&\hskip
113.81102pt\times\frac{\Psi_{(\alpha,\beta;m,n)}(g)}{\eta(g\cdot\tau)^{p_{I}}\overline{\eta}(g\cdot\overline{\tau})^{q_{I}}}\end{split}\;,$
(4.26)

where

$\begin{pmatrix}m^{\prime}\\\ n^{\prime}\end{pmatrix}=\begin{pmatrix}d&-c\\\
-b&a\end{pmatrix}\begin{pmatrix}m\\\ n\end{pmatrix}$ (4.27)

and the phase $\Psi(g)$ is a combination of phases from the twist and shift
sectors:

$\displaystyle\Psi_{(\alpha,\beta;m,n)}(g)$
$\displaystyle=\prod_{j:\,l_{j}\neq 0}\prod_{j:\,r_{j}\neq
0}\frac{\zeta(g,mr_{j},nr_{j})}{\zeta(g,ml_{j},nl_{j})}\epsilon(g)^{\sigma_{I}}\gamma^{I}_{(\alpha,\beta;m,n)}(g)\;.$
(4.28)

The phases $\zeta(g,\alpha,\beta)$ are defined as

$\zeta(M,\alpha,\beta):=e^{i\pi(\alpha(\beta+1)-\alpha^{\prime}(\beta^{\prime}+1))}\;.$
(4.29)

(see Appendix E for further details). The untwisted sector average can also be
written as a Poincarè sum [19], and so we have shown that the entire averaged
CFT partition function can be cast as a Poincarè sum. Note that the summand in
4.26 appears to explicitly depend on the upper two components of the
$\mathrm{SL}(2,\mathbb{Z})$ matrix $g$. However, this is illusory since this
dependence in the theta functions cancels out with a similar dependence in the
phase $\Psi(g)$.

###  4.3. Flavored Case

Let us next go back to the un-orbifolded case $N=1$.

In [7], ensemble averaging for Jacobi theta functions defined with respect to
even self-dual lattices was considered. Here, the average was taken over
$O(D,D;\mathbb{Z})\backslash O(D,D;\mathbb{R})$ instead of the usual Narain
moduli space.

We shall generalize this analysis to the case of Jacobi theta functions
associated with general even integer lattices which take the form

$\displaystyle\vartheta_{\alpha}(\tau,z)=\sum_{\ell}\exp\left(i\pi\left\\{\tau
Q_{L}(\ell+\alpha)-\bar{\tau}Q_{R}(\ell+\alpha)\right\\}\right)e^{2\pi
i[Q_{L}(z,\ell+\alpha)-Q_{R}(z,\ell+\alpha)]}\;,$ (4.30)

where the integration is now over $O_{Q}(p,q;\mathbb{Z})\backslash
O(p,q;\mathbb{R})$.212121As in previous cases, we impose $p+q>4$ for the
convergence of the integral. These functions have a well-defined
transformation under $\textrm{PSL}(2,\mathbb{Z})$, as shown in Appendix C.

Let us identify $O(p,q;\mathbb{R})$ as a set of linear transformations which
preserve the quadratic form $\eta_{AB}Z^{A}Z^{B}$, where $A,B=1,2,\ldots,p+q$,
and where $\eta_{AB}=\operatorname{diag}\left(1^{p},-1^{q}\right)$. The
generators of $O(p,q,\mathbb{R})$ then have the following representation as
differential operators :

$\displaystyle J^{AB}=\eta^{BC}Z^{A}\frac{\partial}{\partial
Z^{C}}-\eta^{AC}Z^{B}\frac{\partial}{\partial Z^{C}}\;,$ (4.31)

which can be shown to satisfy

$\displaystyle[J^{AB},J^{CD}]=\eta^{BC}J^{AD}-\eta^{DB}J^{AC}-\eta^{AC}J^{BD}+\eta^{AD}J^{BC}\;.$
(4.32)

The quadratic Casimir will be especially useful, since it is proportional to
the Laplacian in the present representation. This takes the form

$\displaystyle J^{2}=\eta_{AC}\eta_{BD}J^{AB}J^{CD}.$ (4.33)

It will be convenient to use the generalized vielbein $\mathcal{E}$, as
defined in 2.2, to define a basis for $\Lambda$ where $Q$ is diagonal. In this
basis, the Jacobi theta function takes the form

$\displaystyle\vartheta_{h}(\tau,z)=\sum_{\tilde{l}}\exp\left(i\pi\left\\{\tau\mathbb{I}_{L}(\tilde{l})-\bar{\tau}\mathbb{I}_{R}(\tilde{l})\right\\}\right)e^{2\pi
i[\mathbb{I}_{L}(\tilde{z},\tilde{l})-\mathbb{I}_{R}(\tilde{z},\tilde{l})]}\;,$
(4.34)

where ${\tilde{l}}=\mathcal{E}(\ell+h)$, $\tilde{z}=\mathcal{E}z$, and where
$\mathbb{I}_{L}$ and $\mathbb{I}_{R}$ are defined such that
$Q_{L}=\mathcal{E}^{T}\mathbb{I}_{L}\mathcal{E}$ and
$Q_{R}=\mathcal{E}^{T}\mathbb{I}_{R}\mathcal{E}$. From 2.15, we may deduce
that $\mathbb{I}_{L}=(\mathbbm{1}+\mathbbm{1}_{p,q})/2$ and
$\mathbb{I}_{R}=(\mathbbm{1}-\mathbbm{1}_{p,q})/2$. Thus, we may rewrite 4.34
as

$\displaystyle\vartheta_{h}(\tau,z)=\sum_{\tilde{l}}\exp\left(i\pi\left\\{\tau\delta_{ab}\tilde{l}^{a}_{L}\tilde{l}^{b}_{L}-\bar{\tau}\delta_{\overline{m}\overline{n}}\tilde{l}^{\overline{m}}_{R}\tilde{l}^{\overline{n}}_{R}\right\\}\right)e^{2\pi
i[\delta_{ab}\tilde{z}^{a}_{L}\tilde{l}^{b}_{L}-\delta_{\overline{m}\overline{n}}\tilde{z}^{\overline{m}}_{R}\tilde{z}^{\overline{n}}_{R}]}\;,$
(4.35)

where we denote the $p$\- and $q$-dimensional projections of $\tilde{l}$ and
$\tilde{z}$ using the subscripts $L$ and $R$ respectively, and employ the
indices $a,b=1,2,\ldots p,\overline{m},\overline{n}=1,2,\ldots q$ as before.

Let us first consider the case where all the chemical potentials are set to
zero. The vectors ${\tilde{l}}=(\tilde{l}_{L},\tilde{l}_{R})$ transform as
contravariant vectors under $O(p,q,\mathbb{R})$ rotations, and we can combine
each of them into an $O(p,q,\mathbb{R})$ vector
$\tilde{l}=(\tilde{l}^{A})=(\tilde{l}^{{a}}=\tilde{l}_{L}^{a},\tilde{l}^{\overline{m}}=\tilde{l}_{R}^{\overline{m}})$.
Let us define a second-order differential operator $J^{2}$ by

$\displaystyle
J^{2}:=L^{a}{}_{b}L_{a}{}^{b}+R^{\overline{m}}{}_{\overline{n}}R_{\overline{m}}{}^{\overline{n}}-2T^{a}{}_{\overline{n}}T_{a}{}^{\overline{n}}\;,$
(4.36)

with

$\displaystyle
L^{a}{}_{b}:=\tilde{l}_{L}^{a}\frac{\partial}{\partial\tilde{l}_{L}^{b}}-\tilde{l}_{Lb}\frac{\partial}{\partial\tilde{l}_{La}}\;,\quad
R^{\overline{m}}{}_{\overline{n}}:=\tilde{l}_{R}^{\overline{m}}\frac{\partial}{\partial\tilde{l}_{R}^{\overline{n}}}-\tilde{l}_{R\overline{n}}\frac{\partial}{\partial\tilde{l}_{R\overline{m}}}\;,\quad
T^{a}{}{}_{\overline{n}}:=\tilde{l}_{L}^{a}\frac{\partial}{\partial\tilde{l}_{R}^{\overline{n}}}+\tilde{l}_{R\overline{n}}\frac{\partial}{\partial\tilde{l}_{La}}\;.$
(4.37)

With the definition $\Delta_{\mathcal{M}_{Q}}:=-J^{2}/8$, we can show that,
when all the chemical potentials, $z$, are set to zero,

$\displaystyle\left(-\tau_{2}^{2}\left(\partial_{2}^{2}+\partial_{1}^{2}\right)-\frac{(p+q)\tau_{2}}{2}\partial_{2}-i\frac{(q-p)\tau_{2}}{2}\partial_{1}+\Delta_{\mathcal{M}_{Q}}\right)\vartheta_{Q,h}(\tau,\bar{\tau};m)=0\;.$
(4.38)

When we turn on chemical potentials, the definition of the $O(p,q;\mathbb{R})$
generators ought to be extended:

$\displaystyle\hat{L}^{a}{}_{b}=\tilde{l}_{L}^{a}\frac{\partial}{\partial\tilde{l}_{L}^{b}}-\tilde{l}_{Lb}\frac{\partial}{\partial\tilde{l}_{La}}+\tilde{z}_{L}^{a}\frac{\partial}{\partial\tilde{z}_{L}^{b}}-\tilde{z}_{Lb}\frac{\partial}{\partial\tilde{z}_{La}}\;,$
(4.39)
$\displaystyle\hat{R}^{\overline{m}}{}_{\overline{n}}=\tilde{l}_{R}^{\overline{m}}\frac{\partial}{\partial\tilde{l}_{R}^{\overline{n}}}-\tilde{l}_{R\overline{n}}\frac{\partial}{\partial\tilde{l}_{R\overline{m}}}+\tilde{z}_{R}^{\overline{m}}\frac{\partial}{\partial\tilde{z}_{R}^{\overline{n}}}-\tilde{z}_{R\overline{n}}\frac{\partial}{\partial\tilde{z}_{R\overline{m}}}\;,$
$\displaystyle\hat{T}^{a}{}_{\overline{n}}=\tilde{l}_{L}^{a}\frac{\partial}{\partial\tilde{l}_{R}^{\overline{n}}}+\tilde{l}_{R\overline{n}}\frac{\partial}{\partial\tilde{l}_{La}}+\tilde{z}_{L}^{a}\frac{\partial}{\partial\tilde{z}_{R}^{\overline{n}}}+\tilde{z}_{R\overline{n}}\frac{\partial}{\partial\tilde{z}_{La}}.$

These generators all annihilate the $O(p,q,\mathbb{R})$ invariant combination
$\delta_{ab}\tilde{z}_{L}^{a}\tilde{l}_{L}^{b}-\delta_{\overline{m}\overline{n}}\tilde{z}_{R}^{\overline{m}}\tilde{l}_{R}^{\overline{n}}$.
Defining the Casimir operator $\hat{J}$ for the generators given in
LABEL:hatgen by analogy with 4.36, we now have

$\displaystyle\left(-\tau_{2}^{2}\left(\partial_{2}^{2}+\partial_{1}^{2}\right)-\frac{(p+q)\tau_{2}}{2}\partial_{2}-i\frac{(q-p)\tau_{2}}{2}\partial_{1}+\Delta_{\mathcal{M}_{Q}}\right)\vartheta_{Q,h}(\tau,\bar{\tau},\tilde{z};m)=0\;,$
(4.40)

since the statement in the previous sentence means that the proof of 4.40
follows from that of 4.38.

Since the ensemble average is over $O_{Q}(p,q;\mathbb{Z})\backslash
O_{Q}(p,q;\mathbb{R})$, the result of the averaging procedure will depend only
on the chemical potentials via the $O(p;\mathbb{R})\times O(q;\mathbb{R})$
invariant combinations
$\tilde{z}_{L}^{2}=\delta_{ab}\tilde{z}_{L}^{a}\tilde{z}_{L}^{b}$ and
$\tilde{z}_{R}^{2}=\delta_{\overline{m}\overline{n}}\tilde{z}_{R}^{\overline{m}}\tilde{z}_{R}^{\overline{n}}$.
Notably, the terms that depend on $z$ in $\hat{L}$ and $\hat{R}$ annihilate
$\tilde{z}_{L}^{2}$ and $\tilde{z}_{R}^{2}$. As a result, after averaging,
only the potential-dependent parts of $\hat{T}^{a}{}_{\overline{n}}$ in
$\Delta_{\mathcal{M}}$ contribute to the differential equation satisfied by
the average. This allows us to obtain the average, denoted $f(\tau,z)$, by
solving

$\displaystyle\bigg{(}$
$\displaystyle-\tau_{2}^{2}\left(\partial_{2}^{2}+\partial_{1}^{2}\right)-\frac{(p+q)\tau_{2}}{2}\partial_{2}-i\frac{(q-p)\tau_{2}}{2}\partial_{1}$
(4.41)
$\displaystyle+\frac{1}{4}\left(\tilde{z}_{L}^{a}\frac{\partial}{\partial\tilde{z}_{R}^{\overline{m}}}+\tilde{z}_{R\overline{m}}\frac{\partial}{\partial\tilde{z}_{La}}\right)\left(\tilde{z}_{La}\frac{\partial}{\partial\tilde{z}_{R\overline{m}}}+\tilde{z}_{R}^{\overline{m}}\frac{\partial}{\partial\tilde{z}^{a}_{L}}\right)\bigg{)}f(\tau,\tilde{z})=0\;.$

To achieve this, we first note that

$\displaystyle\left(\tilde{z}_{L}^{a}\frac{\partial}{\partial\tilde{z}_{R}^{\overline{m}}}+\tilde{z}_{R\overline{m}}\frac{\partial}{\partial\tilde{z}_{La}}\right)\left(\tilde{z}_{La}\frac{\partial}{\partial\tilde{z}_{R\overline{m}}}+\tilde{z}_{R}^{\overline{m}}\frac{\partial}{\partial\tilde{z}^{a}_{L}}\right)e^{i\pi\left(\frac{c\tilde{z}_{L}^{2}}{c\tau+d}-\frac{c\tilde{z}_{R}^{2}}{c\bar{\tau}+d}\right)}$
(4.42) $\displaystyle\quad=\left(\frac{4\pi
c^{2}\tau_{2}}{(c\tau+d)(c\bar{\tau}+d)}\right)^{2}\tilde{z}_{R}^{2}\tilde{z}_{L}^{2}e^{i\pi\left(\frac{c\tilde{z}_{L}^{2}}{c\tau+d}-\frac{c\tilde{z}_{R}^{2}}{c\bar{\tau}+d}\right)}$
$\displaystyle\quad+\left(\frac{4\pi
c^{2}\tau_{2}}{(c\tau+d)(c\bar{\tau}+d)}\right)(\tilde{z}_{L}^{2}q+\tilde{z}_{R}^{2}p)e^{i\pi\left(\frac{c\tilde{z}_{L}^{2}}{c\tau+d}-\frac{c\tilde{z}_{R}^{2}}{c\bar{\tau}+d}\right)}.$

In addition, we find

$\displaystyle\left(-\tau_{2}^{2}\left(\partial_{2}^{2}+\partial_{1}^{2}\right)-\frac{(p+q)\tau_{2}}{2}\partial_{2}-i\frac{(q-p)\tau_{2}}{2}\partial_{1}\right)\frac{e^{i\pi\left(\frac{c\tilde{z}_{L}^{2}}{c\tau+d}-\frac{c\tilde{z}_{R}^{2}}{c\bar{\tau}+d}\right)}}{(c\tau+d)^{\frac{p}{2}}(c\bar{\tau}+d)^{\frac{p}{2}}}$
(4.43) $\displaystyle=$ $\displaystyle-\left[4\left(\frac{\pi
c^{2}\tau_{2}}{(c\tau+d)(c\bar{\tau}+d)}\right)^{2}\tilde{z}_{R}^{2}\tilde{z}_{L}^{2}+\left(\frac{\pi
c^{2}\tau_{2}}{(c\tau+d)(c\bar{\tau}+d)}\right)(\tilde{z}_{L}^{2}q+\tilde{z}_{R}^{2}p)\right]\frac{e^{i\pi\left(\frac{c\tilde{z}_{L}^{2}}{c\tau+d}-\frac{c\tilde{z}_{R}^{2}}{c\bar{\tau}+d}\right)}}{(c\tau+d)^{\frac{p}{2}}(c\bar{\tau}+d)^{\frac{p}{2}}}\;.$

This allows us to propose a solution for the ensemble average. Recall that
$\tilde{z}_{R}^{I}$ and $\tilde{z}_{L}^{I}$ are defined in the basis where $Q$
is diagonal, via the generalized vielbein $\mathcal{E}$. Returning to our
original basis using $\delta_{ab}\tilde{z}^{a}_{L}\tilde{z}^{b}_{L}=Q_{L}(z)$
and
$\delta_{\overline{m}\overline{n}}\tilde{z}^{\overline{m}}_{R}\tilde{z}^{\overline{n}}_{R}=Q_{R}(z)$,
we claim that the average can be described in terms of an Eisenstein-Jacobi
series that takes the form

$\displaystyle
E_{Q,h}(\tau,z):=\delta_{h\in\Lambda}+\sum_{(c,d)=1,c>0}\frac{\gamma_{Q,h}(c,d)\,e^{i\pi\left(\frac{c}{c\tau+d}Q_{L}(z)-\frac{c}{c\bar{\tau}+d}Q_{R}(z)\right)}}{(c\tau+d)^{\frac{p}{2}}(c\bar{\tau}+d)^{\frac{q}{2}}}\;.$
(4.44)

The justification for this claim is the uniqueness of this solution, which
follows by comparison with the case where all chemical potentials are set to
zero, as well as the fact that the Jacobi theta function (and its ensemble
average) obeys a pair of heat equations. In the basis where $Q$ is diagonal,
these equations take the form

$\displaystyle\frac{\partial}{\partial\tau}f(\tilde{z},\tau)=\frac{1}{4\pi
i}\nabla_{L}^{2}f(\tilde{z},\tau)\;,\quad\frac{\partial}{\partial\bar{\tau}}f(\tilde{z},\tau)=-\frac{1}{4\pi
i}\nabla_{R}^{2}f(\tilde{z},\tau)\;,$ (4.45)

where

$\displaystyle\nabla_{L}^{2}=\frac{\partial^{2}}{\partial\tilde{z}_{Li}\partial\tilde{z}_{L}^{i}}\;,\quad\nabla_{R}^{2}=\frac{\partial^{2}}{\partial\tilde{z}_{RI}\partial\tilde{z}_{R}^{I}}\;.$
(4.46)

Since the dependence of the average is only via
$\tilde{z}_{L}=\sqrt{\tilde{z}_{L}^{2}}$ and
$\tilde{z}_{R}=\sqrt{\tilde{z}_{R}^{2}}$, the Laplace operators can be written
in spherical coordinates, with the angular parts acting trivially :

$\displaystyle\frac{\partial f}{\partial\tau}=\frac{1}{4\pi
i}\left[\frac{\partial^{2}f}{\partial\tilde{z}_{L}^{2}}+\frac{p-1}{\tilde{z}_{L}}\frac{\partial
f}{\partial\tilde{z}_{L}}\right]\;,\quad\frac{\partial
f}{\partial\bar{\tau}}=-\frac{1}{4\pi
i}\left[\frac{\partial^{2}f}{\partial\tilde{z}_{R}^{2}}+\frac{q-1}{\tilde{z}_{R}}\frac{\partial
Y}{\partial\tilde{z}_{R}}\right]\;.$ (4.47)

Given that it is possible to expand the solution of these equations as
$\sum_{m,n=0}^{\infty}f_{m,n}(\tau,\bar{\tau})\tilde{z}_{L}^{2m}\tilde{z}_{R}^{2n}$,
and since

$\displaystyle
f(\tau,\tilde{z}=0)=\delta_{h\in\Lambda}+\sum_{(c,d)=1,c>0}\frac{\gamma_{Q,h}(c,d)}{(c\tau+d)^{\frac{p}{2}}(c\bar{\tau}+d)^{\frac{q}{2}}}\;,$
(4.48)

we find that any possible term, denoted $\Delta f$, that could be added to the
solution 4.44 ought to have no constant term in its expansion, that is,
$\Delta f_{0,0}=0$. The equations 4.47 imply recursion relations relating
$f_{m+1,0}$ and $f_{m,0}$, as well as $f_{0,n+1}$ and $f_{0,n}$, whereby we
find that $\Delta f$ must be zero. Thus, 4.44 is indeed the unique solution to
the differential equation that remains after ensemble averaging, and is the
result for the ensemble average of the Jacobi theta function 4.30 over
$O_{Q}(p,q,\mathbb{Z})\backslash O(p,q,\mathbb{R})$.

To show that the expression 4.44 indeed satisfies the correct modular
transformation property, we note that for

$\displaystyle
f(M,\tau,z)=e^{i\pi\left(\frac{c}{c\tau+d}Q_{L}(z)-\frac{c}{c\bar{\tau}+d}Q_{R}(z)\right)}$
(4.49)

we have

$\displaystyle
f(M_{1},\tau,z)f(M_{2},\tau_{M_{1}},z_{M_{1}\tau})=f(M_{2}M_{1},\tau,z)\;,$
(4.50)

which is the analogue of the composition law 2.42.

Although we shall not investigate this in detail, one can in principle go on
to study the ensemble average of orbifolded Narain CFTs with the inclusion of
chemical potentials for global symmetries in the partition function.

##  5\. Comments on the Holographic Bulk

We next comment on the bulk holographic dual. While we leave a detailed
analysis for future work, we will here comment on the salient features of the
bulk theory.222222See [8] for discussion of orbifolded holography for some
special examples of even self-dual $Q$.

Let us consider the holography for the orbifolded generalized Narain theory
after the ensemble average. First, the orbifold of the Narain theory by a
symmetry group $G$ will correspond to a gauging of the global $G$-symmetry of
the bulk. We expect that the resulting theory is still a TQFT, whose anyon
content will described by the enlarged discriminant group 3.34, which contains
the gauge-equivalence class of anyons of the original theory as well as newly
introduced anyons due to $G$-gauging. The modular transformations 4.14 of the
theory determine the action of the modular group on the Hilbert space of the
bulk theory on the boundary two-torus.

The expression of the Eisenstein series as a Poincarè sum 4.25 can be
interpreted as a sum over $\mathrm{PSL}(2,\mathbb{Z})$ black holes [39, 40],
which contain thermal AdS3 and the BTZ black hole [41] as special cases. As in
the discussion of the un-orbifolded case [19], the coefficients
$\gamma^{I}_{(\alpha,\beta;m,n)}(g)$ in 4.25 can be regarded as an overlap of
the two wave functions on the solid torus, which functions are determined by
the data $(\alpha,m)$ and $(\beta,n)$, and $g$ represents the gluing of the
two solid tori by an element $g$ of the mapping class group, where $g$ here
should be regarded as an element of
$\mathrm{PSL}(2,\mathbb{Z})/\Gamma_{\infty}=\mathrm{PSL}(2,\mathbb{Z})/\mathbb{Z}$.
The resulting overlap is the partition function of the bulk TQFT on the lens
space, where the choice of the lens space is determined by $g$ and we have in
general non-trivial Wilson line insertions each of the solid tori, resulting
in the Hopf link invariant. Thus the ensemble average of the boundary torus
partition function represents a sum over an infinite set of lens space
invariants of the bulk theory.

##  6\. Summary and Discussion

In this paper, we discussed a class of generalized Narain CFTs associated with
an even integral quadratic form, studied previously in [19]. We discussed
orbifolds of the generalized Narain CFTs, in a procedure similar to the
orbifolding of CFT associated with toroidal compactification of string theory.
We then discuss the ensemble averages of the orbifolded theories, and we
obtained a new generalization of Eisenstein series.

There are several obvious tasks to be pursued in the future. One problem is to
identify precise holographic duals for generalized Narain theories after
ensemble average, along the lines of Ref. [1, 2, 19]. One may also consider
finding roles of such theories in string theory and more generally quantum
gravity (cf. [14]). A direct extension of this paper is to odd quadratic
forms, where we expect a spin CFT, i.e., a CFT dependent on the choice of the
spin structure (See Ref. [19] discussions on the un-orbifolded case.).
Generalization of this work to asymmetric orbifolds [42] and CHL orbifolds
would be an obvious next step. A recent topic of interest seems to be the
study of relationships between code CFTs and quantum error corrections, see
e.g. [4, 43, 44, 1, 45, 46]. It would be an interesting undertaking to
identify the points of the moduli space where the generalized Narain CFT
becomes rational (cf. [27, 28, 47]).

## Acknowledgements

We would like to thank Matthew Dodelson for collaboration in early stages of
this project. We would also like to thank Florent Baume, Weiguang Cao, Craig
Lawrie, Jacob McNamara, Yuto Moriwaki, Caner Nazaroglu, and Yunqin Zheng for
useful discussions.

A.K.’s research is/was supported by the Schrödinger Junior Research Fellowship
(current), Riemann Fellowship (2022). A.K. thanks the hospitality of the
Riemann Center for Geometry and Physics, the Simons Center for Geometry and
Physics (during the Number Theory and Physics 2022 workshop), the Abdus Salam
ICTP, and the Pollica Center for Physics (during the workshop on new
connections between number theory and physics). M.Y. would like to thank KITP,
Santa Barbara for hospitality during the Integrable 22 workshop. A.K. and M.Y
thank the Isaac Newton Institute for Mathematical Sciences for support and
hospitality during the programme “Black Holes: Bridges betweeen number theory
and holographic quantum information”, and were supported by EPSRC Grant Nr.
EP/R014604/1. J.M.L. thanks Kavli Institute for the Physics and Mathematics of
the Universe for hospitality during the completion of a portion of this work.
A.K and M.A. thank Deutsches Elektronen-Synchrotron (DESY) for hospitality
during the completion of a portion of this work. M.A. and M.Y. are supported
in part by the JSPS Grant-in-Aid for Scientific Research (19H00689, 20H05860).
J.M.L. is supported by the Deutsche Forschungsgemeinschaft under Germany’s
Excellence Strategy - EXC 2121 “Quantum Universe”. M.Y. is also supported in
part by the JSPS Grant-in-Aid for Scientific Research (19K03820, 23H01168),
and by JST, Japan (PRESTO Grant No. JPMJPR225A, Moonshot R&D Grant No.
JPMJMS2061).

##  Appendix A Conventions and Notations

We list conventions, notations and standard definitions.

  * •

$\Lambda$: discrete data defining a generalized Narain theory, i.e. integer
lattice of signature $(p,q)$ endowed with a $(p+q)$-ary quadratic form $Q$.

  * •

$\Lambda^{*}$: dual lattice of $\Lambda$

  * •

$\mathscr{D}_{\Lambda}$: discriminant group of $Q$,
$\mathscr{D}_{\Lambda}=\Lambda^{*}/\Lambda$ and
$\lvert\mathscr{D}_{\Lambda}\rvert=\lvert\text{det}(Q)\rvert$

  * •

$\mathfrak{h}_{Q}$: representation space of $Q$, such that
$H\in\mathfrak{h}_{Q}\rightsquigarrow HQ^{-1}H=Q$

  * •

$H$: Majorant associated to quadratic form $Q$, such that $HQ^{-1}H=Q$

  * •

$\mathcal{E}$: generalized Narain vielbein defined by
$Q=\mathcal{E}^{T}\mathbbm{1}_{p,q}\mathcal{E}$ and defines
$H=\mathcal{E}^{T}\mathcal{E}$

  * •

$\mathcal{P}_{\Lambda}$: generalized Narain momentum lattice built from
$\Lambda$ and $\mathcal{E}$

  * •

$\mathcal{P}_{\Lambda}^{*}$: dual lattice of $\mathcal{P}_{\Lambda}$

  * •

$\mathrm{O}_{Q}(p,q;\mathbb{Z})$: T-duality group of entire generalized CFT,
i.e. elements preserving $Q$

  * •

$\mathrm{O}_{Q,\alpha}(p,q;\mathbb{Z})$: T-duality group associated to a
specific anyon, i.e. elements preserving $Q$ and the class
$[\alpha]\in\mathscr{D}_{\Lambda}$

  * •

$\mathfrak{p}$: element of $\mathcal{P}_{\Lambda}$

  * •

$\mathcal{M}_{\Lambda}$: generalized Narain moduli space

  * •

$\mathcal{P}_{\rm Narain}$: (standard) Narain lattice

  * •

$\mathcal{M}_{\rm Narain}$: (standard) Narain Moduli space

  * •

$\mathrm{O}(D,D;\mathbb{Z})$: T-duality group of the (standard) Narain CFT

##  Appendix B T-dualities

Here we prove a few statements in the main text.

######  Lemma B.1.

Every element of $\mathrm{O}_{Q}(p,q;\mathbb{Z})$ defines an automorphism of
$\Lambda^{*}$.

###### Proof.

Let $\Sigma\in\mathrm{O}_{Q}(p,q;\mathbb{Z})$. First, we note that for every
$\alpha_{1}\in\Lambda^{*}$, there exists another $\alpha_{2}\in\Lambda^{*}$
such that $\alpha_{1}=\Sigma\alpha_{2}$. This follows from the fact that
$\Sigma^{-1}$ (which is integral and unimodular) exists and

$Q(\Sigma^{-1}\alpha,\ell)=\alpha^{T}\Sigma^{T}{}^{-1}Q\ell=\alpha^{T}Q\Sigma\ell\in\mathbb{Z}$
(B.1)

for all $\ell\in\Lambda$, so that
$\alpha_{2}:=\Sigma^{-1}\alpha_{1}\in\Lambda^{*}$ for
$\alpha_{1}\in\Lambda^{*}$.

Next, we note that if $\alpha_{1}=\Sigma\alpha_{2}$,
$\alpha_{1}^{\prime}=\Sigma\alpha_{2}^{\prime}$, and
$\alpha_{1}\neq\alpha_{2}$, then $\alpha_{1}^{\prime}\neq\alpha_{2}^{\prime}$.
This follows by contradiction: assume
$\alpha_{1}^{\prime}=\alpha_{2}^{\prime}$. Then

$\alpha_{1}-\alpha_{2}=\Sigma^{-1}(\alpha_{1}^{\prime}-\alpha_{2}^{\prime})=0\;.$
(B.2)

This implies $\alpha_{1}=\alpha_{2}$, which is a contradiction. Thus
$\Sigma\in\mathrm{O}_{Q}(p,q;\mathbb{Z})$ defines a bijective map from
$\Lambda^{*}$ to itself.

∎

######  Lemma B.2.

Every element of $\mathrm{O}_{Q}(p,q;\mathbb{Z})$ defines an automorphism of
$\Lambda$.

###### Proof.

The proof is essentially identical to that of B.1. ∎

These give us the following corollaries immediately:

######  Corollary B.3.

If $\alpha\in\Lambda^{*}$ such that $[\alpha]=[0]$, then $[\Sigma\alpha]=[0]$
$\forall\;\Sigma\in\mathrm{O}_{Q}(p,q;\mathbb{Z})$.

######  Corollary B.4.

Let $\alpha_{1},\alpha_{2}\in\Lambda^{*}$ such that
$[\alpha_{1}]\neq[\alpha_{2}]$. Then
$[\Sigma\alpha_{1}]\neq[\Sigma\alpha_{2}]$. Furthermore,
$\Sigma\alpha_{1}\notin\Lambda$ unless $[\alpha_{1}]=[0]$.

######  Theorem B.5.

Then the elements of $\mathrm{O}_{Q}(p,q;\mathbb{Z})$ define automorphisms of
$\mathscr{D}_{\Lambda}$ that leave $[0]$ invariant.

###### Proof.

The proof follows from the above Lemmas and Corollaries. ∎

##  Appendix C Orbifold Jacobi Theta Functions

In this section, we derive the transformation properties of the orbifold theta
function 3.21 and the Jacobi theta function 4.30. While we include these for
completeness of the presentation, we note that essentially the same
transformations were derived previously in a recent mathematics paper [48]
(see also [49]).

Since $S$ and $T$ are generators of $\mathrm{SL}(2,\mathbb{Z})$, one can
verify the general transformation 3.37 by checking that (1) the formula
coincides with 3.26, 3.27 for $T,S$-transformations and that (2) composes
consistently under the composition of two $\mathrm{SL}(2,\mathbb{Z})$
matrices. The latter statement follows from 3.41, which is known for the un-
orbifolded case since the times of C. Siegel. In the following we will
nevertheless directly prove the modular transformation under a general element
of $\mathrm{SL}(2,\mathbb{Z})$.

###  C.1. Modular Transformations

We shall now consider general modular transformations for Jacobi theta
functions with characteristics, associated with an arbitrary even integral
lattice, denoted $I$, with indefinite signature $(p_{I},q_{I})$:

$\displaystyle\vartheta_{\delta,\eta}(\tau,z)=\sum_{y=\ell+\delta}\exp\left(i\pi\left\\{\tau
Q_{L}(y)-\bar{\tau}Q_{R}(y)\right\\}\right)e^{2\pi iQ(y,\eta)}e^{2\pi
iQ(z,y)}.$ (C.1)

Under a modular transformation by $[M]=\left[\left(\begin{array}[]{cc}a&b\\\
c&d\end{array}\right)\right]\in\mathrm{PSL}(2,\mathbb{Z})$:

$\displaystyle\tau$
$\displaystyle\rightarrow\tau_{M}:=\frac{a\tau+b}{c\tau+d}=\frac{a}{c}+c^{-2}\frac{-c}{c\tau+d}\;,$
(C.2) $\displaystyle z$ $\displaystyle\rightarrow
z_{M,\tau}:=Q^{-1}\left(\frac{1}{c\tau+d}Q_{L}-\frac{1}{c\bar{\tau}+d}Q_{R}\right)z,$

where we assumed $c\neq 0$. To simplify the notation, in this appendix we
denote $\vartheta_{I,(\delta,\eta)}$ by $\vartheta_{\delta,\eta}$.

We begin with the expression

$\displaystyle\vartheta^{I}_{\delta,\eta}(\tau_{M},z_{M,\tau})$
$\displaystyle=\sum_{\ell\in
I}e^{i\pi\left[\frac{a}{c}Q(\ell+\delta)+c^{-2}U(\ell+\delta)\right]}e^{2\pi
iQ(\ell+\delta,\eta)}e^{-\frac{2\pi i}{c}U(z,\ell+\delta)}\;,$ (C.3)

where we defined

$\displaystyle\tau_{M}Q_{L}-\bar{\tau}_{M}Q_{R}$
$\displaystyle=\frac{a}{c}Q+c^{-2}U\;,\quad
U:=\frac{-cQ_{L}}{c\tau+d}+\frac{cQ_{R}}{c\overline{\tau}+d}\;.$ (C.4)

Note that $U$ depends on the moduli $\tau,\overline{\tau}$. We now write
$\ell=c\ell^{\prime}+\ell_{c}$ in the sum C.3, where $\ell^{\prime}\in I$ and
$\ell_{c}\in I/(cI)$. After massaging the resulting expression, we
obtain232323Here $\cdot$ denotes a multiplication for matrices and vectors.
For example, in the expression $U^{-1}[Q\cdot(c\eta+a\delta)]$, the
combination $Q\cdot(c\eta+a\delta)$ denotes the product of a matrix $Q$ and a
vector $c\eta+a\delta$, and the resulting vector is then fed into the argument
of the quadratic form $U^{-1}$.

$\displaystyle\vartheta^{I}_{\delta,\eta}(\tau_{M},z_{M,\tau})$
$\displaystyle=\sum_{\ell_{c}\in
I/(cI)}e^{\left(i\pi\frac{a}{c}Q(\ell_{c}+\delta)\right)}e^{\left(-\frac{2\pi
ia}{c}Q(\ell_{c},\delta)-\frac{2\pi ia}{c}Q(\delta)\right)}$
$\displaystyle\hskip 56.9055pt\times\sum_{\ell^{\prime}\in I}e^{\left(i\pi
U\left[\ell^{\prime}+c^{-1}(\ell_{c}+\delta)+(U^{-1}Q)\cdot(c\eta+a\delta)-z\right]\right)}e^{-i\pi
U^{-1}[Q\cdot(c\eta+a\delta)]}$ $\displaystyle\hskip 56.9055pt\times e^{2\pi
iQ(c\eta+a\delta,z)}e^{-i\pi U(z,z)}.$ (C.5)

Performing a Poisson resummation in $\ell^{\prime}\in I$, we obtain242424After
a Poisson resummation, we obtain a sum over an element $\ell^{*}$ of the dual
lattice $I^{*}$, which we can write as $\ell^{*}=\ell+\gamma$ with $\ell\in
I,\gamma\in I^{*}/I$.

$\displaystyle\vartheta^{I}_{\delta,\eta}(\tau_{M},z_{M,\tau})$
$\displaystyle=[\textrm{det}(-iU)]^{-\frac{1}{2}}\sum_{\ell_{c}\in
I/(cI)}e^{\left(i\pi\frac{a}{c}Q(\ell_{c}+\delta)\right)}e^{\left(-\frac{2\pi
ia}{c}Q(\ell_{c},\delta)-\frac{2\pi ia}{c}Q(\delta)\right)}e^{-i\pi
U^{-1}[Q(c\eta+a\delta)]}$ $\displaystyle\times\sum_{\ell\in
I}\sum_{\gamma\in\mathscr{D}_{I}}e^{\left(-i\pi
U^{-1}[Q\cdot(\ell+\gamma)]-2\pi
i\left(Q\cdot(\ell+\gamma)\right)^{T}\cdot\left\\{c^{-1}(\ell_{c}+\delta)+(U^{-1}Q)\cdot(c\eta+a\delta)-z\right\\}\right)}\;.$
$\displaystyle\times e^{2\pi iQ(c\eta+a\delta,z)}e^{-i\pi U(z,z)}$ (C.6)

Since we have (as follows from the definition of $U$ in C.4)

$\displaystyle-U^{-1}[Q\cdot y]$
$\displaystyle=\bigg{(}\frac{d}{c}Q+(Q_{L}\tau-Q_{R}\bar{\tau})\bigg{)}[y]\;,$
(C.7)

and

$\displaystyle e^{-i\pi
U(z,z)}=e^{i\pi\left(\frac{c}{c\tau+d}Q_{L}(z)-\frac{c}{c\bar{\tau}+d}Q_{R}(z)\right)},$
(C.8)

we can rewrite

$\displaystyle\vartheta^{I}_{\delta,\eta}(\tau_{M},z_{M,\tau})=[\textrm{det}(-iU)]^{-\frac{1}{2}}\left(\sum_{\ell_{c}\in
I/(cI)}e^{\left(i\pi\frac{a}{c}Q(\ell_{c}+\delta)\right)}e^{\left(-\frac{2\pi
ia}{c}Q(\ell_{c},\delta)-\frac{2\pi ia}{c}Q(\delta)\right)}\right)$
$\displaystyle\times e^{i\pi\frac{d}{c}Q(c\eta+a\delta)}e^{i\pi(\tau
Q_{L}-\overline{\tau}Q_{R})(c\eta+a\delta)}e^{i\pi\left(\frac{c}{c\tau+d}Q_{L}(z)-\frac{c}{c\bar{\tau}+d}Q_{R}(z)\right)}$
$\displaystyle\times\sum_{\ell\in
I}\sum_{\gamma\in\mathscr{D}_{I}}e^{\left[i\pi\frac{d}{c}(\ell+\gamma)+i\pi(\tau
Q_{L}-\overline{\tau}Q_{R})(\ell+\gamma)-\frac{2\pi
i}{c}Q(\ell+\gamma,\ell_{c}+\delta)+\frac{2\pi
id}{c}Q(\ell+\gamma,c\eta+a\delta)+2\pi i(\tau
Q_{L}-\overline{\tau}Q_{R})(\ell+\gamma,c\eta+a\delta)\right]}\;$
$\displaystyle\times e^{2i\pi Q(\ell+\gamma+c\eta+a\delta,z)}.$ (C.9)

While the expression C.1 looks formidable, the moduli-dependent terms combine
nicely into

$\displaystyle e^{i\pi(Q_{L}\tau-
Q_{R}\bar{\tau})[\ell+\gamma+c\eta+a\delta]}e^{i\pi\left(\frac{c}{c\tau+d}Q_{L}(z)-\frac{c}{c\bar{\tau}+d}Q_{R}(z)\right)}\;,$
(C.10)

while the moduli-independent exponential terms combine into

$\displaystyle e^{2i\pi Q(\ell+\gamma+c\eta+a\delta,z)}e^{2i\pi
Q[\ell+\gamma+c\eta+a\delta,d\eta+b\delta]}\,e^{-i\pi
Q(a\delta+c\eta,b\delta+d\eta)}\left(\sum_{\ell_{c}\in
I/(cI)}e^{\frac{i\pi}{c}\left(aQ[\ell_{c}]-2Q[\ell_{c},\gamma]+dQ[\gamma]\right)}\right)\;.$
(C.11)

This combination is preserved when we shift $\gamma$ by an element of $I$, so
that we can set $\ell=0$ in C.11. This means we can bring C.11 outside the sum
over $\ell$, and we find that the sum over $\ell$ over the remaining
$\ell$-dependent expression gives rise to the Jacobi theta function
$\vartheta^{I}_{(\gamma+a\delta+c\eta,b\delta+d\eta)}(\tau,z)$. Finally, we
need to include the remaining factor

$\displaystyle[\textrm{det}(-iU)]^{-\frac{1}{2}}=\left[\textrm{det}\left(\frac{-icQ_{L}}{c\tau+d}\right)\right]^{-\frac{1}{2}}\left[\textrm{det}\left(\frac{icQ_{R}}{c\overline{\tau}+d}\right)\right]^{-\frac{1}{2}}=\frac{e^{-\frac{i\pi}{4}(p_{I}-q_{I})}c^{-\frac{p_{I}+q_{I}}{2}}\left(\textrm{det}\,Q\right)^{-\frac{1}{2}}}{(c\tau+d)^{-\frac{p_{I}}{2}}(c\overline{\tau}+d)^{-\frac{q_{I}}{2}}}\;.$
(C.12)

Shifting $\delta$ by an element $\gamma^{\prime}\in\mathscr{D}_{I}$, and
defining $\tilde{\gamma}=\gamma+a\gamma^{\prime}\in\Lambda^{*}/\Lambda$ allows
us to retrieve the most general modular transformation. By combining all the
factors above, we obtain

$\displaystyle\vartheta_{(\delta+\gamma^{\prime},\eta)}\left(\tau_{M},z_{M,\tau}\right)$
$\displaystyle=\frac{e^{-\frac{i\pi}{4}(p_{I}-q_{I})}c^{-\frac{p_{I}+q_{I}}{2}}\left(\textrm{det}\,Q\right)^{-\frac{1}{2}}}{(c\tau+d)^{-\frac{p_{I}}{2}}(c\overline{\tau}+d)^{-\frac{q_{I}}{2}}}\left(e^{-i\pi
abQ[\delta]-2i\pi bcQ[\delta,\eta]-i\pi cdQ[\eta]}\right)$ (C.13)
$\displaystyle\quad\times
e^{i\pi\left(\frac{c}{c\tau+d}Q_{L}(z)-\frac{c}{c\bar{\tau}+d}Q_{R}(z)\right)}\sum_{\tilde{\gamma}}\lambda_{\gamma^{\prime},\tilde{\gamma}}\vartheta_{\left(\tilde{\gamma}+c\eta+a\delta,d\eta+b\delta\right)}(\tau,z),$

where

$\displaystyle\lambda_{\gamma^{\prime},\tilde{\gamma}}:=\sum_{\ell_{c}\in
I/cI}e^{\frac{i\pi}{c}\left(aQ[\ell_{c}+\gamma^{\prime}]-2Q\left[\tilde{\gamma},\ell_{c}+\gamma^{\prime}\right]+dQ\left[\tilde{\gamma}\right]\right)}\;.$
(C.14)

In particular, setting $\delta=0=\eta$, we find the transformation formula for
a Jacobi theta function associated with a general even integral lattice

$\displaystyle\vartheta_{\gamma^{\prime}}\left(\tau_{M},z_{M,\tau}\right)=\frac{e^{-\frac{i\pi}{4}(p_{I}-q_{I})}c^{-\frac{p_{I}+q_{I}}{2}}\left(\textrm{det}\,Q\right)^{-\frac{1}{2}}}{(c\tau+d)^{-\frac{p_{I}}{2}}(c\overline{\tau}+d)^{-\frac{q_{I}}{2}}}\sum_{\gamma}\lambda_{\gamma^{\prime},\gamma}\vartheta_{\gamma}(\tau,z)e^{i\pi\left(\frac{c}{c\tau+d}Q_{L}(z)-\frac{c}{c\bar{\tau}+d}Q_{R}(z)\right)}.$
(C.15)

If we instead set $z=0$, we obtain 3.37.

We can see from 3.37 that the only effects of $\delta$ and $\eta$ are to
change the overall phases of the modular transformation. Note that in our
derivation we have not imposed any conditions on $\delta$ and $\eta$. If we
consider the S-transformations with $a=d=0$ and $c=-b=1$, we obtain

$\displaystyle\vartheta^{I}_{\delta,\eta}\left(-\frac{1}{\tau}\right)$
$\displaystyle=\frac{e^{-i\pi\frac{(p_{I}-q_{I})}{4}}}{\sqrt{\lvert\mathscr{D}_{I}\rvert}}\tau^{p_{I}/2}\bar{\tau}^{q_{I}/2}e^{2\pi
iQ[\eta,\delta]}\sum_{\gamma\in\mathscr{D}_{I}}\vartheta^{I}_{(\gamma+\eta,-\delta)}(\tau)\;.$
(C.16)

While we assumed $c\neq 0$ in the derivation, the case $c=0$ amounts to the
$T$-transformation of the form $\tau\rightarrow\tau+ab$ with $a^{2}=1$, and
$z\rightarrow az$. In this case, it is straightforward to show that

$\displaystyle\vartheta_{(\delta+\gamma^{\prime},\eta)}\left(\tau_{M},z_{M,\tau}\right)$
$\displaystyle=e^{\pi iabQ(\gamma^{\prime})}e^{-\pi
iabQ(\delta)}\vartheta_{(\delta+\gamma^{\prime},\eta+ab\delta)}\left(\tau,az\right)$
(C.17) $\displaystyle=e^{\pi iabQ(\gamma^{\prime})}e^{-\pi
iabQ(\delta)}\vartheta_{(a\delta+a\gamma^{\prime},a\eta+b\delta)}\left(\tau,z\right).$

In fact, the formula 3.37 works for the T-transformations with $a=b=d=1$ and
$c=0$:

$\displaystyle\vartheta^{I}_{\delta,\eta}(\tau+1)$ $\displaystyle=e^{-i\pi
Q[\delta]}\,\vartheta^{I}_{(\delta,\eta+\delta)}(\tau)\;,$ (C.18)

which formula can be verified directly from the definition of the theta
function.

###  C.2. Asymptotics at Cusps

At the cusp $\tau\to i\infty$, we find from the definition 3.21 that

$\displaystyle\vartheta_{I,(\delta,\eta)}(\tau\to i\infty)=\delta_{\delta\in
I}\;.$ (C.19)

We use the transformation law of the theta functions to determine the behavior
at another cusp $\tau=-d/c$. For this purpose we use 3.37 for
$M^{-1}=\begin{pmatrix}d&-b\\\ -c&a\end{pmatrix}$

$\displaystyle\vartheta^{I}_{(\delta,\eta)}\left(\frac{d\tau-b}{-c\tau+a}\right)$
$\displaystyle=\mu_{(\delta,\eta)\cdot
M^{-1}}\sum_{\gamma\in\mathscr{D}_{I}}\mathcal{U}_{\gamma}(M^{-1})\,\vartheta_{(\gamma+d\delta-c\eta,-b\delta+a\eta)}(\tau)\;,$
(C.20)

Let us define $\tau^{\prime}:=(d\tau-b)/(-c\tau+a)$, or equivalently
$\tau=(a\tau^{\prime}+b)/(c\tau^{\prime}+d)$, and consider taking the limit
$\tau\to i\infty$, which is equivalent to $\tau^{\prime}\to-d/c$. In this
limit, using C.19, we obtain252525 The expression on the right-hand side
depends only on the element $\mathrm{PSL}(2,\mathbb{Z})/\Gamma_{\infty}$,
namely the expression depends only on the entries $c,d$ of the
$\mathrm{PSL}(2,\mathbb{Z})$ matrix. To see this, let us pick up a modular
inverse $d^{*}$ of the integer $d$ via $dd^{*}=1\text{ mod }c$, so that
$\displaystyle\begin{split}a=ck+d^{*}\;,\quad
b=dk+\frac{dd^{*}-1}{c}\;,\quad(k\in\mathbb{Z})\;,\end{split}$ (C.21) where
$k$ labels an element of $\Gamma_{\infty}\simeq\mathbb{Z}$. The $k$-dependence
generates an extra factor of  $\displaystyle e^{i\pi
k(Q(d\delta-c\eta)-Q(\beta))}\;,$ (C.22) which is equal to the identity under
the delta-function constraint $\beta+d\delta-c\eta\in I$ (recall
$\beta\in\mathscr{D}_{I}$). This ensures that the behavior at the cusp is
independent of the upper two entries of the $\mathrm{PSL}(2,\mathbb{Z})$
matrix.

$\displaystyle\vartheta_{(\delta,\eta)}\left(\tau^{\prime}\to-\frac{d}{c}\right)$
$\displaystyle\to\frac{1}{\sqrt{\textrm{det}\,Q_{I}}}\,e^{-\frac{\pi
i}{4}(p_{I}-q_{I})}(c\tau^{\prime}+d)^{-\frac{p}{2}}(c\overline{\tau}^{\prime}+d)^{-\frac{q}{2}}\mu_{(\delta,\eta)\cdot
M^{-1}}\lambda_{0,-d\delta+c\eta}(M^{-1})\;,$ (C.23)

where we used $-c\tau+a=1/(c\tau^{\prime}+d)$. This matches the behavior of
the Eisenstein series 4.2 at the cusp.

As is clear from this discussion above, the only properties needed for the
derivation of the asymptotic behavior at the cups are (i) behavior at $\tau\to
i\infty$ as in C.19 as well as (ii) the modular transformation rule 3.21.
Since we can directly verify these two properties for the Eisenstein series
4.2, we can repeat the same argument to conclude that the Einstein series has
the same asymptotic as in C.23.

##  Appendix D Congruence Subgroup

In this appendix, we prove that the theta functions we have been working with
are modular forms for the congruence subgroup $\Gamma(N^{2}L)$, where $N$ is
the order of the orbifold action and $L$ is an integer multiple of the level
$L_{Q}$ of the quadratic form. Our discussion here seems to be new for $N>1$.
Our proof relies heavily on the result of [50, 51], which discuss the un-
orbifolded case $N=1$.262626For the special case $N=1$, [50] already noted
that the modular transformation formulas simplify for a special congruence
subgroup, and for example already considered $\Gamma(\textrm{lcm}(L_{Q},4))$
for $p+q$ odd. The focus there, however, was not necessarily to identify the
precise congruence subgroup where the multiplier system evaluates to $1$, and
we have not found an explicit statement in the literature that our theta
function is modular (without any multiplier system) for a concrete choice of
$\Gamma(L)$.

We shall focus on the case modular transformations where $c\neq 0$ first. Let
us reproduce the corresponding transformation law of the orbifold theta
functions:

$\displaystyle\vartheta_{(\alpha,0)+(\delta,\eta)}(\tau_{M})$
$\displaystyle=\sum_{\beta\in\mathcal{D}}\mathcal{U}_{\alpha,\beta}(M,\tau;(\delta,\eta))\,\vartheta_{\underbrace{(\beta,0)+(\delta,\eta)\cdot
M}_{\textrm{characteristics}}}\;,$ (D.1)
$\displaystyle\mathcal{U}_{\alpha,\beta}(M,\tau;(\delta,\eta))$
$\displaystyle=(c\tau+d)^{\frac{p}{2}}(c\overline{\tau}+d)^{\frac{q}{2}}$
$\displaystyle\times\underbrace{\exp\bigg{[}-i\pi(abQ[\delta]+2bcQ[\delta,\eta]+cdQ[\eta])\bigg{]}}_{\textrm{shift-
dependent
phase}}\underbrace{\lambda_{\alpha,\beta}(M)}_{\textrm{transformation
matrix}}\;.$ (D.2)

We wish to show that this reduces to the standard transformation for modular
functions when $M\in\Gamma(N^{2}L)$. We break the proof into three parts: the
shift-dependent phase, the theta function characteristics, and the
transformation matrix.

##### Shift-dependent Phase

We investigate the factor

$\displaystyle\exp\bigg{[}-i\pi(abQ[\delta]+2bcQ[\delta,\eta]+cdQ[\eta])\bigg{]}\;.$
(D.3)

For the factor of $\textrm{exp}(-i\pi(abQ[\delta]))$, we observe that it is
trivial since $b\equiv 0\textrm{ mod }N^{2}L$, $bQ[\delta]\in 2\mathbb{Z}$.
This is because $\delta\in I/N$ and from the definition of the level,
$LQ[\ell^{\prime}]\in 2\mathbb{Z}$ for $\ell^{\prime}\in I^{*}$ (recall that
$I\subset I^{*}$). Similarly, since $\eta\in I/N$ and $c\equiv 0\textrm{ mod
}N^{2}L$ the factor of $\textrm{exp}(-i\pi(cdQ[\eta]))$ is trivial as well.
Finally, the factor of $\textrm{exp}(-i\pi(2bcQ[\delta,\eta]))$ is trivial
since $c\equiv 0\textrm{ mod }N^{2}L$, so that $cQ[\delta,\eta]\in\mathbb{Z}$.

##### Characteristics

We look at the behaviour of the characteristics $(\delta,\eta)$ under a
modular transformation:

$\displaystyle(\delta,\eta)\rightarrow(\delta,\eta)\begin{pmatrix}a&b\\\
c&d\end{pmatrix}=(a\delta+c\eta,b\delta+d\eta)\;.$ (D.4)

In the definition of our theta functions, we have a lattice sum over
$I+\alpha+\delta$, $\alpha\in\mathcal{D}$ and $\delta\in I/N$. After the
transformation, we have the sum

$\sum_{\ell\in I+\alpha+a\delta+c\eta}\exp(i\pi\tau
Q_{L}(\ell)-i\pi\bar{\tau}Q_{R}(\ell))\exp(2i\pi Q_{I}(\ell,\delta b+\eta
d))\;.$ (D.5)

Now, since $a\delta\equiv\delta\textrm{ mod }I,d\eta\equiv\eta\textrm{ mod
}I,b\delta\in I,c\eta\in I$, the sum reduces to

$\sum_{\ell\in I+\alpha+\delta}\exp(i\pi\tau
Q_{L}(\ell)-i\pi\bar{\tau}Q_{R}(\ell))\exp(2i\pi Q_{I}(\ell,\eta))\;,$ (D.6)

and therefore the characteristics of the transformed theta function reduce to
those of the theta function prior to modular transformation.

##### Transformation Matrix

We now show that the matrix $\lambda^{\Lambda}_{\alpha,\beta}(M)$ becomes
trivial (i.e. unity) for elements of the congruence subgroup $\Gamma(N^{2}L)$.

Since we are interested in the evaluation of
$\lambda_{\alpha,\beta}^{\Lambda}(M)$, and since this factor was defined by
the modular transformation property of the un-orbifolded theta function, we
can take advantage of the literature for $N=1$.

In the following we use the quadratic residue symbol
$\displaystyle\left(\frac{a}{b}\right)$ as defined in [52], which is known as
the Kronecker symbol. This symbol is defined for an integer $a$ and an odd
integer $b\neq 0$, and coincides with the ordinary quadratic residue symbol
(Legendre symbol) when $b$ is an odd prime. This symbol has many interesting
properties, e.g. $\left(\dfrac{a}{b}\right)$ for a fixed $b$ is a character
modulo $b$ as a function of $a$. See [52] for further properties of this
symbol.

For $p+q$ even, it is known [51] that
$\lambda_{\alpha,\beta}(M)=\delta_{\alpha,\beta}$ for $M\in\Gamma(L_{Q})$. The
case of $p+q$ odd is more subtle: it was shown in [51] that

$\displaystyle\lambda_{\alpha,\beta}(M)=\left(\frac{-2}{d}\right)\left(\frac{c}{d}\right)\left(\frac{\det\,Q}{d}\right)\delta_{\alpha,\beta}$
(D.7)

for $M\in\Gamma(\textrm{lcm}(L_{Q},4))$. If we further impose $c\equiv
0\bmod\det\,Q$, we have
$\left(\dfrac{c}{d}\right)=\left(\dfrac{\det\,Q}{d}\right)$, which cancels the
two factors, leaving

$\displaystyle\left(\frac{-2}{d}\right)=\left(\dfrac{-1}{d}\right)\cdot\left(\dfrac{2}{d}\right)~{},$
(D.8)

which can be evaluated using the following identities

$\displaystyle\begin{split}\left(\dfrac{2}{d}\right)&=(-1)^{\frac{d^{2}-1}{8}}=\begin{cases}1,&d\equiv
1,7\mod 8\\\ -1,&d\equiv 3,5\mod 8\end{cases}\\\
\left(\dfrac{-1}{d}\right)&=(-1)^{\frac{d-1}{2}}=\begin{cases}1,&d\equiv 1\mod
4\\\ -1,&d\equiv 3\mod 4\end{cases}.\end{split}$ (D.9)

These symbols trivialize for $d\equiv 1\bmod 8$, and hence
$\lambda_{\alpha,\beta}(M)=\delta_{\alpha,\beta}$. The restrictions imposed on
$c,d$ can be realized by choosing $M$ from a congruence subgroup
$\Gamma(\textrm{lcm}(L_{Q},8,|\det\,Q|)=\Gamma(\textrm{lcm}(8,|\det\,Q|))$.

For future reference, let us state this as a Proposition:272727The implication
of this result is that if we consider a Narain moduli space of conformal field
theories of with difference in cleft and right moving central charges
$c_{L}-c_{R}=p-q\in 2\mathbb{Z}$ admitting an action of an order $N$ orbifold,
the theta function transforms as a modular form on $\Gamma(N^{2}L)$ (which we
momentarily refer to as the ‘even’ congruence subgroup), where $L$ is the
level of the Narain lattice. However, if the chiral central charge is odd,
then the theta function transforms on a smaller congruence subgroup contained
in the even congruence subgroup.

######  Proposition D.1.

$\lambda_{\alpha,\beta}(M)=\delta_{\alpha,\beta}$ if $M\in\Gamma(L_{Q})$ for
$p+q$ even and if $M\in\Gamma(\textrm{lcm}(8,|\det\,Q|))$ for $p+q$ odd.

######  Remark D.2.

D.1 demonstrates that the Siegel-Weil formula holds for a sufficiently small
principal congruence group $\Gamma\subset\Gamma(N^{2}L)$. It is however
possible that the congruence subgroup can be enlarged slightly such that the
Siegel-Weil theorem still holds, thereby permitting ensemble averages, but we
are unaware of justifiable constraints that allow us to do this.

#### The $c=0$ case

We have so far restricted ourselves to the case of $c\neq 0$. We also need to
show that the theta function is a modular function for $\Gamma(N^{2}L)$ even
when $c=0$. In this case, the modular transformation law takes the form

$\displaystyle\vartheta_{(\delta+\alpha,\eta)}\left(\tau_{M}\right)=e^{\pi
iabQ(\alpha)}e^{-\pi
iabQ(\delta)}\vartheta_{(\delta+\alpha,\eta+ab\delta)}\left(\tau\right),$
(D.10)

where $\alpha\in I^{*}/I$, and $\delta,\eta\in I/N$. Since $c=0$, and
$a,d=1\textrm{ mod }N^{2}L$ for an element of $\Gamma(N^{2}L)$, it must be the
case that $a=1$ and $d=1$ since $N>1$. In particular, $c\tau+d=1$, and the
theta function should be invariant under the modular transformation.

Now, since $b=0\textrm{ mod }N^{2}L$, and $LQ[\ell^{\prime}]\in 2\mathbb{Z}$
for $\ell^{\prime}\in I^{*}$, we find that the phases in D.10 are trivial.
Moreover,

$\displaystyle\vartheta_{(\delta+\alpha,\eta+ab\delta)}\left(\tau\right)=\sum_{y=\ell+\alpha+\delta}e^{\pi
i(\tau Q_{L}[y]-\bar{\tau}Q_{R}[y])}e^{2\pi iQ(y,\eta)}e^{2\pi
iQ(y,ab\delta)},$ (D.11)

and we find that

$\displaystyle e^{2\pi iQ(y,ab\delta)}=e^{2\pi
iQ(\ell+\alpha+\delta,ab\delta)}=1$ (D.12)

since $a=1$ and $b=0\textrm{ mod }N^{2}L$. Thus, we find that

$\displaystyle\vartheta_{(\delta+\alpha,\eta)}\left(\tau_{M}\right)=\vartheta_{(\delta+\alpha,\eta)}\left(\tau\right),$
(D.13)

where $M\in\Gamma(N^{2}L)$ with $c=0$. This completes a proof that the theta
function is a modular function for $\Gamma(N^{2}L)$.

##  Appendix E Transformation of Eta and Theta Functions

We reproduce here modular transformation properties of some modular forms used
in the text.

For $M=\begin{pmatrix}a&b\\\ c&d\end{pmatrix}\in\mathrm{SL}(2,\mathbb{Z})$,
the Dedekind eta function transforms as

$\eta(M\cdot\tau)=\epsilon(M)(c\tau+d)^{\frac{1}{2}}\eta(\tau)\;,$ (E.1)

where the multiplier system $\epsilon(M)$ is given by

$\displaystyle\epsilon(M):=\begin{cases}\exp\left({\dfrac{i\pi
b}{12}}\right),&c=0,\ d=1\\\
\exp\left(\dfrac{a+d}{24c}-\dfrac{s(d,c)}{2}-\dfrac{1}{8}\right),&c>0\end{cases}~{},$
(E.2)

where the $s(d,c)$ is the Dedekind sum given by

$\displaystyle s(d,c)=\sum_{n\textrm{ mod }c}((n/c))((dn/c))\;,$ (E.3)

and $((x))$ is the sawtooth function

$\displaystyle((x))$
$\displaystyle:=\begin{cases}x-[x]-\frac{1}{2}&x\in\mathbb{R}\backslash\mathbb{Z}\;,\\\
0&x\in\mathbb{Z}\;,\end{cases}$ (E.4)

with $[x]$ being the integer part of $x$.282828A study of Dedekind sums and
their relation to the study of the eta function can be found in [53].

Consider the Jacobi theta function with characteristics

$\displaystyle\vartheta\begin{bmatrix}\alpha\\\
\beta\end{bmatrix}(\tau):=\sum_{n\in\mathbb{Z}}e^{i\pi(n+\alpha)^{2}\tau}e^{2\pi
i(n+\alpha)\beta}.$ (E.5)

Under an element of $M=\begin{pmatrix}a&b\\\
c&d\end{pmatrix}\in\mathrm{SL}(2,\mathbb{Z})$ with $c>0$, we have the
transformation law

$\displaystyle\vartheta\begin{bmatrix}\alpha+\frac{1}{2}\\\
\beta+\frac{1}{2}\end{bmatrix}(\tau)=\frac{\epsilon(M)^{-3}\zeta(M,\alpha,\beta)^{-1}}{\sqrt{c\tau+d}}\,\vartheta\begin{bmatrix}a\alpha+c\beta+\frac{1}{2}\\\
b\alpha+d\beta+\frac{1}{2}\end{bmatrix}(M\cdot\tau)\;,$ (E.6)

where the characteristic-dependent phase is defined as in 4.29. This implies

$\frac{\vartheta\begin{bmatrix}\alpha+\frac{1}{2}\\\
\beta+\frac{1}{2}\end{bmatrix}(\tau)}{\eta(\tau)}=\epsilon(M)^{-2}\zeta(M,\alpha,\beta)^{-1}\frac{\vartheta\begin{bmatrix}a\alpha+c\beta+\frac{1}{2}\\\
b\alpha+d\beta+\frac{1}{2}\end{bmatrix}(M\cdot\tau)}{\eta(M\cdot\tau)}\;,$
(E.7)

## References

  * [1] N. Afkhami-Jeddi, H. Cohn, T. Hartman, and A. Tajdini, “Free partition functions and an averaged holographic duality,” arXiv:2006.04839 [hep-th].
  * [2] A. Maloney and E. Witten, “Averaging over Narain moduli space,” JHEP 10 (2020) 187, arXiv:2006.04855 [hep-th].
  * [3] A. Pérez and R. Troncoso, “Gravitational dual of averaged free CFT’s over the Narain lattice,” JHEP 11 (2020) 015, arXiv:2006.08216 [hep-th].
  * [4] A. Dymarsky and A. Shapere, “Solutions of modular bootstrap constraints from quantum codes,” arXiv:2009.01236 [hep-th].
  * [5] A. Dymarsky and A. Shapere, “Comments on the holographic description of Narain theories,” arXiv:2012.15830 [hep-th].
  * [6] V. Meruliya, S. Mukhi, and P. Singh, “Poincaré Series, 3d Gravity and Averages of Rational CFT,” arXiv:2102.03136 [hep-th].
  * [7] S. Datta, S. Duary, P. Kraus, P. Maity, and A. Maloney, “Adding Flavor to the Narain Ensemble,” arXiv:2102.12509 [hep-th].
  * [8] N. Benjamin, C. A. Keller, H. Ooguri, and I. G. Zadeh, “Narain to Narnia,” arXiv:2103.15826 [hep-th].
  * [9] V. Meruliya and S. Mukhi, “AdS3 Gravity and RCFT Ensembles with Multiple Invariants,” arXiv:2104.10178 [hep-th].
  * [10] L. Eberhardt, “Summing over Geometries in String Theory,” arXiv:2102.12355 [hep-th].
  * [11] J. Kames-King, A. Kanargias, B. Knighton, and M. Usatyuk, “The Lion, the Witch, and the Wormhole: Ensemble averaging the symmetric product orbifold,” arXiv:2306.07321 [hep-th].
  * [12] P. Saad, S. H. Shenker, and D. Stanford, “JT gravity as a matrix integral,” arXiv:1903.11115 [hep-th].
  * [13] D. Stanford and E. Witten, “JT Gravity and the Ensembles of Random Matrix Theory,” arXiv:1907.03363 [hep-th].
  * [14] M. Ashwinkumar, J. M. Leedom, and M. Yamazaki, “Duality Origami: Emergent Ensemble Symmetries in Holography and Swampland,” arXiv:2305.10224 [hep-th].
  * [15] J. J. Heckman, A. P. Turner, and X. Yu, “Disorder averaging and its UV discontents,” Phys. Rev. D 105 no. 8, (2022) 086021, arXiv:2111.06404 [hep-th].
  * [16] S. Collier and E. Perlmutter, “Harnessing S-duality in $\mathcal{N}$ = 4 SYM & supergravity as SL(2, $\mathbb{Z}$)-averaged strings,” JHEP 08 (2022) 195, arXiv:2201.05093 [hep-th].
  * [17] F. Baume, J. J. Heckman, M. Hübner, E. Torres, A. P. Turner, and X. Yu, “SymTrees and Multi-Sector QFTs,” arXiv:2310.12980 [hep-th].
  * [18] S. Gukov, E. Martinec, G. W. Moore, and A. Strominger, “Chern-Simons gauge theory and the AdS(3) / CFT(2) correspondence,” in From Fields to Strings: Circumnavigating Theoretical Physics: A Conference in Tribute to Ian Kogan, pp. 1606–1647.  3, 2004.  arXiv:hep-th/0403225.
  * [19] M. Ashwinkumar, M. Dodelson, A. Kidambi, J. M. Leedom, and M. Yamazaki, “Chern-Simons Invariants from Ensemble Averages,” arXiv:2104.14710 [hep-th].
  * [20] K. S. Narain, “New Heterotic String Theories in Uncompactified Dimensions $<$ 10,” Phys. Lett. B 169 (1986) 41–46.
  * [21] K. S. Narain, M. H. Sarmadi, and E. Witten, “A Note on Toroidal Compactification of Heterotic String Theory,” Nucl. Phys. B 279 (1987) 369–379.
  * [22] J. Dong, T. Hartman, and Y. Jiang, “Averaging over moduli in deformed WZW models,” JHEP 09 (2021) 185, arXiv:2105.12594 [hep-th].
  * [23] A. Antinucci, G. Galati, G. Rizi, and M. Serone, “Symmetries and topological operators, on average,” arXiv:2305.08911 [hep-th].
  * [24] H. P. Nilles, S. Ramos-Sanchez, A. Trautner, and P. K. S. Vaudrevange, “Orbifolds from Sp(4,Z) and their modular symmetries,” Nucl. Phys. B 971 (2021) 115534, arXiv:2105.08078 [hep-th].
  * [25] C. L. Siegel, “Lectures on Quadratic Forms,”. http://www.math.tifr.res.in/~publ/ln/tifr07.pdf.
  * [26] J. J. Sylvester, “A demonstration of the theorem that every homogeneous quadratic polynomial is reducible by real orthogonal substitutions to the form of a sum of positive and negative squares,” The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science 4 no. 23, (1852) 138–142.
  * [27] K. Wendland, “Moduli spaces of unitary conformal field theories,” other thesis, University of Bonn, 9, 2000.
  * [28] A. Kidambi, M. Okada, and T. Watari, “Towards Hodge Theoretic Characterizations of 2d Rational SCFTs,” arXiv:2205.10299 [hep-th].
  * [29] A. Kapustin and D. Orlov, “Vertex algebras, mirror symmetry, and D-branes: The Case of complex tori,” Commun. Math. Phys. 233 (2003) 79–136, arXiv:hep-th/0010293.
  * [30] M. Kontsevich and Y. Soibelman, “Homological mirror symmetry and torus fibrations,” in KIAS Annual International Conference on Symplectic Geometry and Mirror Symmetry, pp. 203–263.  11, 2000.  arXiv:math/0011041.
  * [31] Y. Moriwaki, “Full vertex algebra and non-perturbative current-current deformation of 2d CFT,” arXiv:2007.07327 [math.QA].
  * [32] A. Weil, “Sur certains groupes d’opérateurs unitaires,” Acta Math. 111 (1964) 143–211.
  * [33] G. W. Moore and N. Seiberg, “Classical and Quantum Conformal Field Theory,” Commun. Math. Phys. 123 (1989) 177.
  * [34] K. S. Narain, M. H. Sarmadi, and C. Vafa, “Asymmetric Orbifolds,” Nucl. Phys. B 288 (1987) 551.
  * [35] I. Senda and A. Sugamoto, “Orbifold Models and Modular Transformation,” Nucl. Phys. B 302 (1988) 291.
  * [36] J. W. Milnor and J. D. Stasheff, Characteristic classes.  Princeton University Press, Princeton, N. J.; University of Tokyo Press, Tokyo, 1974.  Annals of Mathematics Studies, No. 76.
  * [37] N. E. Steenrod, Cohomology operations.  Lectures by N. E. Steenrod written and revised by D. B. A. Epstein. Annals of Mathematics Studies, No. 50. Princeton University Press, Princeton, N.J., 1962.
  * [38] C. L. Siegel, “Indefinite quadratische Formen und Funktionentheorie. I,” Math. Ann. 124 (1951) 17–54.
  * [39] J. M. Maldacena and A. Strominger, “AdS(3) black holes and a stringy exclusion principle,” JHEP 12 (1998) 005, arXiv:hep-th/9804085.
  * [40] R. Dijkgraaf, J. M. Maldacena, G. W. Moore, and E. P. Verlinde, “A Black hole Farey tail,” arXiv:hep-th/0005003.
  * [41] M. Banados, C. Teitelboim, and J. Zanelli, “The Black hole in three-dimensional space-time,” Phys. Rev. Lett. 69 (1992) 1849–1851, arXiv:hep-th/9204099.
  * [42] K. S. Narain, M. H. Sarmadi, and C. Vafa, “Asymmetric orbifolds: Path integral and operator formulations,” Nucl. Phys. B 356 (1991) 163–207.
  * [43] A. Dymarsky and A. Shapere, “Quantum stabilizer codes, lattices, and CFTs,” JHEP 21 (2020) 160, arXiv:2009.01244 [hep-th].
  * [44] K. Kawabata, T. Nishioka, and T. Okuda, “Narain CFTs from quantum codes and their $\mathbb{Z}_{2}$ gauging,” arXiv:2308.01579 [hep-th].
  * [45] O. Aharony, A. Dymarsky, and A. D. Shapere, “Holographic description of Narain CFTs and their code-based ensembles,” arXiv:2310.06012 [hep-th].
  * [46] A. Barbar, A. Dymarsky, and A. D. Shapere, “Global Symmetries, Code Ensembles, and Sums Over Geometries,” arXiv:2310.13044 [hep-th].
  * [47] Y. Furuta, “On the Rationality and the Code Structure of a Narain CFT, and the Simple Current Orbifold,” arXiv:2307.04190 [hep-th].
  * [48] S. Zemel, “Jacobi forms of indefinite lattice index,” Res. Number Theory 7 no. 4, (2021) Paper No. 58, 22.
  * [49] R. E. Borcherds, “Automorphic forms with singularities on Grassmannians,” Invent. Math. 132 no. 3, (1998) 491–562.
  * [50] T. Shintani, “On construction of holomorphic cusp forms of half integral weight,” Nagoya Mathematical Journal 58 (1975) 83 – 126.
  * [51] T. Oda, “On modular forms associated with indefinite quadratic forms of signature $(2,n-2)$.,” Mathematische Annalen 231 (1977) 97–144.
  * [52] G. Shimura, “On modular forms of half integral weight,” Annals of Mathematics 97 no. 3, (1973) 440–481.
  * [53] H. Rademacher and E. Grosswald, Dedekind sums, vol. 16.  American Mathematical Soc., 1972.