# Flat equivariant gerbes: holonomies and dualities

Peng Cheng  Ilarion V. Melnikov  Ruben Minasian

###### Abstract

We examine the role of global topological data associated to choices of
holonomy for flat gauge fields in string compactification. Our study begins
with perturbative string compactification on compact flat manifolds preserving
$8$ supercharges in $5$ dimensions. By including non-trivial holonomy for
Wilson lines in the heterotic string and for the B-field gerbe in the type II
string we find worldsheet dualities that relate these backgrounds to other
string compactifications. While our simple examples allow for explicit
analysis, the concepts and some of the methods extend to a broader class of
compactifications and have implications for string dualities, perturbative and
otherwise.

##  1 Introduction

Anyone familiar with the cover of [1] knows that interactions between
mathematicians and physicists involve an aspect of translation: while the
former starts with global data that may have some local presentation in a set
of coordinates or representatives, the latter happily constructs quantities
from a combination of greek and latin letters, large and small. Yet the two
have much to say to each other, and over the years the language has remarkably
converged. Now physicists are happy to write papers on differential
cohomology, and mathematicians write texts on renormalization in quantum field
theory.

In this work we will be mostly concerned with structures where the usual
physics methodology is of no avail, and the gauge fields that we might
typically write will all be zero. Nevertheless, the global topological data
will have a profound effect on the physics. The examples that we study are
particularly simple: toroidal orbifold conformal theories and the associated
string compactifications.

Much of the fascination with orbifold compactification on $Y/G$ comes from the
presence of singular points, which correspond to points in $Y$ fixed by the
action of $G$ [2, 3]. Despite these singularities, the string theory on $Y/G$
can be well-behaved, and computations of various quantities from the
worldsheet perspective have lead to important insights into geometry, such as
the notion of stringy Hodge numbers and finer cohomological structures on
orbifolds and algebraic varieties (e.g. see [4] for an introduction).

The study of orbifolds has also led to insights into the moduli space of
string compactifications, for example by showing that certain limits of
compactification on smooth manifolds such as K3 or a Calabi-Yau $3$-fold where
the geometry becomes singular can nevertheless be understood entirely in
perturbative string theory. In addition, orbifolds have provided insights into
non-trivial dualities such as mirror symmetry. They have also been used
extensively in string phenomenology, and there are impressive classification
results, including the description of all symmetric orbifolds of the six-torus
${{\mathbb{T}}}^{6}$ that preserve spacetime supersymmetry [5].

In recent years orbifolds of toroidal compactifications have received renewed
interest in relation to the swampland program; see, for example, [6, 7, 8, 9,
10, 11]. This recent work involves asymmetric orbifolds that act on the
toroidal geometry by shifts, while having a non-trivial action on the gauge
sector. The results provide insights into the structure of disconnected
components in the moduli space of toroidal compactification that generalize
the CHL construction [12, 13] and are intimately related to the study of
heterotic compactifications with non-trivial flat connections [14, 15] and
related type II/M-theory/F-theory constructions with background RR fluxes and
frozen singularities [16, 17]. It will be useful to extend these efforts to
cases with less spacetime supersymmetry, where classification will be even
more challenging, but also the physics will be richer. Our work is just a
small indication of some of the structures that emerge in this larger setting.

In this paper we begin our study with $5$-dimensional compactifications on
$X_{5}={{\mathbb{T}}}^{5}/G$ that preserve half of spacetime supersymmetry,
and where $G={\mathbb{Z}}_{N}$ acts freely on ${{\mathbb{T}}}^{5}$.111As the
recent work [18] indicates, if one is willing to abandon spacetime
supersymmetry, then compact flat manifolds already become interesting in
compactifications to $7$ dimensions. To obtain a free and supersymmetric
action, we write ${{\mathbb{T}}}^{5}={{\mathbb{T}}}^{4}\times S^{1}$, and
choose $G$ to have a supersymmetric action on ${{\mathbb{T}}}^{4}$, so that
${{\mathbb{T}}}^{4}/G$ has holonomy
${\mathbb{Z}}_{N}\subset\operatorname{SU}(2)$. The orbifold $X_{5}$ is
nevertheless smooth because the action on ${{\mathbb{T}}}^{4}$ is combined
with an order $N$ shift on the $5$-th circle.

Since these quotients lack fixed points, we can usefully discuss their physics
from both supergravity and worldsheet perspectives. For example, we can
interpret these compactifications from a six-dimensional spacetime point of
view: it is possible to think of the theories as $S^{1}$ compactifications of
a $6$-dimensional theory, where a holonomy is turned on for a discrete
${\mathbb{Z}}_{N}$ gauge symmetry; the holonomy breaks half of the
supersymmetry and modifies the spectrum in other ways. This point of view
should be useful in fitting these compactifications in the framework of
F-theory/type II — heterotic duality, for example along the lines explored in
[19, 20].

Second, we show that in heterotic string theory these compactifications are
dual to more familiar compactifications on ${{\mathbb{T}}}^{4}/G\times S^{1}$
and are therefore connected to conventional compactification on K3. That this
should be the case was already suggested in [21], and we establish the claim
at the level of worldsheet CFT. In particular, we show that a small radius
limit of the $X_{5}$ compactification develops additional massless states and
a Higgs branch invisible in the supergravity description. We also note that
the duality exchanges spaces with different topologies, thus providing another
example of stringy geometry relating topologically distinct geometries.

The equivalence is based on the observation that there are orbifold actions on
the ${{\mathbb{T}}}^{5}$ worldsheet CFT, $G_{1}\simeq G$ and $G_{2}\simeq G$
that are related by a conjugation: there is an element
$t\in\operatorname{O{}}(21,5,{\mathbb{Z}})$ that gives an isomorphism
$t:G_{1}\to G_{2}$, with $t:g\mapsto t^{-1}gt$. While the quotient by $G_{1}$
leads to the smooth geometry $X_{5}$, the quotient by $G_{2}$ leads to
${{\mathbb{T}}}^{4}/G\times S^{1}$. The induced isomorphism on the moduli of
the two CFTs acts as a T-duality on the radius of the non-trivial $S^{1}$ in
combination with a shift of the Wilson line parameters by a lattice vector.

It is then natural to interpret the isomorphism as a T-duality of the $X_{5}$
CFT, but there is a subtlety in applying the standard paradigm of Buscher
rules for a non-linear sigma model, despite the fact that the $X_{5}$ CFT can
be perfectly well described by a large radius non-linear sigma model. The
trouble is that the natural circle fibration structure

${S^{1}}$${X_{5}}$${M}$

has $M={{\mathbb{T}}}^{4}/G$, i.e. the base is singular! Our results indicate
that T-duality extends to such singular fibrations, with the benefit of
enlarging the possibilities for topology change via T-duality beyond those
discussed in the context of principal torus bundles over smooth manifolds,
e.g. in [22, 23].

Next, we consider these backgrounds in the context of type II string theory
compactification and find that compactification on $X_{5}$ is T-dual to
compactification on ${{\mathbb{T}}}^{4}/G\times S^{1}$ equipped with a
topologically non-trivial flat $B$-field gerbe. This duality belongs to a
class of T-dual theories, where the geometry is a (possibly singular)
principal circle bundle $P_{g}\to M$, while the choice of gerbe is encoded in
a second circle bundle $P_{b}\to M$. The product bundle
$P_{g}\times_{M}P_{b}\to M$ is then the correspondence space [22] for
T-duality along the circle: the action of T-duality exchanges the two circle
bundles, so that the geometric fibration data is exchanged with the flat gerbe
data.

At the level of the orbifold CFT this illustrates, along the lines described
in [24], that the gerbe data gives a natural geometric meaning to certain
shift orbifold phases. One example of such phases is a choice of discrete
torsion [25, 26], but these are absent in all of our examples. As emphasized
in [24], the gerbe encodes more than the choice of discrete torsion, and our
work explores the physical consequences of this additional data.

The plan of the rest of the paper is as follows. We begin with a discussion of
the geometry of $X_{5}$ and construct the massless spectrum for heterotic
compactification on $X_{5}$ from a spacetime point of view. Next we study of
the details of the heterotic worldsheet and the duality between
compactification on $X_{5}$ and that on ${{\mathbb{T}}}^{4}/G\times S^{1}$. We
then turn to the type II string, where we study principal circle bundles over
${{\mathbb{T}}}^{4}/G$ with flat gerbe structure and explore T-dualities
between them. We end with a summary and discussion of future directions.
Several technical discussions are given in the appendix.

## Acknowledgements

IVM and MR would like to thank the Institut Henri Poincaré and its Research in
Paris program for hospitality and support while this work was being completed;
IVM’s work is also supported in part by NSF grant PHY-1914505, and he also
thanks the IPhT for additional support during his visit. IVM also thanks the
CTP at Queen Mary University of London for hospitality during the last stages
of this project. RM is partially supported by ERC grants
772408-Stringlandscape and 787320-QBH Structure. We thank R. Field, A. Font,
B. Fraiman, H. Parras de Freitas, M.R. Plesser, S. Sethi, and E. Sharpe for
useful communications and conversations, and we are especially grateful to P.
Vaudrevange, M. Ratz, and S. Ramos for sharing the classification results for
symmetric orbifolds of ${{\mathbb{T}}}^{6}$ with $N=2$ spacetime supersymmetry
that they obtained as part of the work in [5, 27].

##  2 Compactification on flat manifolds: a spacetime perspective

###  2.1 Essentials of compact flat geometries

A compact flat Riemannian manifold $X$ of dimension $n$ has ${\mathbb{R}}^{n}$
as its universal cover, so that $X={\mathbb{R}}^{n}/\pi_{1}$, where $\pi_{1}$
is the fundamental group of $X$. It is a classic result (reviewed in, for
example, [28]) that $\pi_{1}$ is a discrete and co-compact222A subgroup
$H\subset G$ of a Hausdorff topological group $G$ is said to be co-compact if
the coset $G/H$ is compact in the quotient topology. subgroup of
${\mathbb{R}}^{n}\rtimes\operatorname{O{}}(n)$, the isometry group of
${\mathbb{R}}^{n}$, with the following properties: $\pi_{1}$ is discrete,
contains no finite subgroups, and it contains a free abelian normal subgroup
$N\subset\pi_{1}$ generated by $n$ linearly independent translations, and
$G=\pi_{1}/N$ is a finite group. $G$ is the Riemannian holonomy group of $X$,
and it can be shown that every finite group $G$ can be the holonomy of some
compact flat Riemannian manifold [29].333It is also possible to show that a
finitely generated group $\pi_{1}$ is the fundamental group of some compact
flat manifold if and only if $\pi_{1}$ has no finite subgroups, and there is a
normal subgroup $N\subset\pi_{1}$ that is maximal abelian such that
$\pi_{1}/N$ is a finite group.

Since $\pi_{1}$ is a crystallographic group, it is possible to classify all
compact flat manifolds of dimension $n$. Reviews of the relevant computational
tools and algorithms are given in [30, 5]. As one might expect, the
classification becomes fairly elaborate for large $n$, and, as far as we are
aware, has only been carried out for $n\leq 6$. Fortunately, for our purposes
of finding supersymmetric compactifications, we need only consider a much
smaller class of manifolds.

The reason for the restriction is that while the spaces are flat, they in
general have non-trivial characteristic classes despite vanishing curvature.
There are classification results, complete through dimension $n\leq 6$, of
compact flat spin manifolds [31], as well as of compact flat Kähler manifolds
[32], which are defined to be manifolds with
$\pi_{1}\subset{\mathbb{C}}^{n/2}\rtimes\operatorname{U{}}(n/2)$. The latter
work also characterizes the compact flat Kähler manifolds with holonomy
$G\subset\operatorname{SU}(n/2)$, i.e. the flat Calabi-Yau manifolds.444In
this definition of “Calabi-Yau” we allow for both
$G\subset\operatorname{SU}(n/2-1)$ and $b_{1}(X)\neq 0$—properties that are
often explicitly excluded by definition in discussion of Calabi-Yau $3$-folds.
The results of the classification include the following:

  1. 1.

for $n=4$ there are $8$ topological types of Kähler manifolds, with
$G\in\\{1,{\mathbb{Z}}_{2},{\mathbb{Z}}_{3},{\mathbb{Z}}_{4},{\mathbb{Z}}_{6}\\}$,
and the only Calabi-Yau manifold is ${{\mathbb{T}}}^{4}$ with $G=1$;

  2. 2.

for $n=6$ there are $174$ topological types of flat compact Kähler manifolds,
and $13$ types of Calabi-Yau manifolds with
$G\in\\{1,{\mathbb{Z}}_{2},{\mathbb{Z}}_{3},{\mathbb{Z}}_{4},{\mathbb{Z}}_{6},{\mathbb{Z}}_{2}\times{\mathbb{Z}}_{2},D_{8}\\}$,
where $D_{8}$ is the dihedral group of order $8$.

The Hodge numbers of these manifolds are also determined in [32]. For example,
the authors obtain the following result for Calabi-Yau $X$ with $b_{1}(X)=2$:

$\displaystyle\begin{matrix}G&&1&&{\mathbb{Z}}_{2}&&{\mathbb{Z}}_{3}&&{\mathbb{Z}}_{4}&{\mathbb{Z}}_{6}\\\
\\#\text{of types}&&1&&2&&2&&2&1\end{matrix}$ (2.1)

Each of these manifolds can be used as a background for supersymmetric
compactifications of string theory or supergravity, and although the class is
not large, the relative simplicity makes such compactifications interesting,
especially, as we will see, in the realm of stringy geometry where the volume
of some cycle becomes comparable to the string scale.

From the classification results we see that $n=6$ is the smallest dimension
with non-trivial compact flat Calabi-Yau manifolds, and these have a natural
sub-class, where $X$ is isometric to $X=X_{5}\times S^{1}$, and $X_{5}$ is
compact flat manifold with $G={\mathbb{Z}}_{N}$ for $N\in\\{1,2,3,4,6\\}$. It
is easy to construct examples of such $X_{5}$, as we now discuss.

Let $(z_{1},z_{2};\theta)$ denote coordinates on
${{\mathbb{T}}}^{2}\times{{\mathbb{T}}}^{2}\times S^{1}$ with the familiar
identifications

$\displaystyle(z_{1},z_{2};\theta)\sim(z_{1}+m_{1}+n_{1}\tau_{1},z_{2}+m_{2}+n_{2}\tau_{2},\theta+2\pi
m_{3})$ (2.2)

for $m_{1,2,3},n_{1,2}\in{\mathbb{Z}}$, and with $\tau_{1},\tau_{2}$ being the
complex structure parameters for the ${{\mathbb{T}}}^{2}$ factors. Every
${{\mathbb{T}}}^{2}$ has a reflection symmetry ${\mathbb{Z}}_{2}:z\mapsto-z$;
when $\tau=i$ the ${{\mathbb{T}}}^{2}$ admits a ${\mathbb{Z}}_{4}$ action
${\mathbb{Z}}_{4}:z\mapsto iz$, and when $\tau=e^{2\pi i/6}$ there is a
${\mathbb{Z}}_{3}$ action $z\mapsto e^{2\pi i/3}z$. In this latter case there
is also a ${\mathbb{Z}}_{6}={\mathbb{Z}}_{2}\times{\mathbb{Z}}_{3}$ action
$z\mapsto e^{2\pi i/6}z$. Using these symmetries (after tuning the $\tau_{i}$
appropriately), we obtain some familiar ${\mathbb{Z}}_{N}$ quotients of
${{\mathbb{T}}}^{4}$ that are consistent with its hyper-Kähler structure:

$\displaystyle(z_{1},z_{2})\mapsto(\zeta_{N}z_{1},\zeta_{N}^{-1}z_{2})~{},$
(2.3)

where $\zeta_{N}=e^{2\pi i/N}$. Of course such a quotient leads to a singular
orbifold ${{\mathbb{T}}}^{4}/{\mathbb{Z}}_{N}$, and the corresponding CFTs
have been studied extensively in the literature, e.g. [33, 34, 35], with a
focus on $6$ dimensional heterotic vacua and aspects of string duality. All of
these can be understood as degenerations of a K3 surface [36, 37], giving a
connection between these orbifold CFTs and compactification on a smooth large
radius geometry.

Now it is a simple matter to construct our smooth $X_{5}$. For each
$N\in\\{2,3,4,6\\}$ (and tuning $\tau_{i}$ appropriately for $N>2$), we take
the $G={\mathbb{Z}}_{N}$ action on $X_{5}$ with generator

$\displaystyle
g(z_{1},z_{2},\theta)=\left(\zeta_{N}z_{1},\zeta_{N}^{-1}z_{2},\theta+{\textstyle\frac{2\pi}{N}}\right)~{}.$
(2.4)

The Calabi-Yau flat geometries naturally fit into a larger class of spaces,
the supersymmetric orbifolds of ${{\mathbb{T}}}^{6}$ classified in [5]. The
authors of [5] provided us with the list of orbifolds that preserve $N=2$
spacetime supersymmetry, and when we further restrict attention to freely-
acting orbifolds that leave one $S^{1}$ invariant, we recover exactly the
class of geometries just described, and these geometries will play a key role
in our investigation.

#### Two fibration structures

There are two useful perspectives on the geometry of $X_{5}$ as a fibration.
First, we can think of the projection $\mu_{\text{cir}}:X_{5}\to S^{1}$, with
$\mu_{\text{cir}}(z_{1},z_{2},\theta)=N\theta$, where the fiber
$\mu_{\text{cir}}^{-1}(\varphi)\simeq{{\mathbb{T}}}^{4}$, and the fibers
$\mu_{\text{cir}}^{-1}(\varphi)$ and $\mu_{\text{cir}}^{-1}(\varphi+2\pi)$ are
related by the ${\mathbb{Z}}_{N}$ action on ${{\mathbb{T}}}^{4}$. This
structure will be useful when we think of compactification on $X_{5}$ in two
steps: first, we compactify string theory on ${{\mathbb{T}}}^{4}$, and then we
compactify the resulting theory further on an additional circle with a non-
trivial holonomy for the $G={\mathbb{Z}}_{N}$ action.

Second, we have the projection
$\mu_{\text{orb}}:X_{5}\to{{\mathbb{T}}}^{4}/{\mathbb{Z}}_{N}$, where the map
simply forgets the last coordinate and maps the torus coordinates to the
corresponding equivalence class:
$\mu_{\text{orb}}(z_{1},z_{2},\theta)=[z_{1},z_{2}]$. In this case, the fiber
is a circle, but with an identification that jumps at the fixed points of the
${{\mathbb{T}}}^{4}/{\mathbb{Z}}_{N}$ orbifold: for a generic point in the
base we identify $\theta\sim\theta+2\pi$, while at each fixed point
$p\in{{\mathbb{T}}}^{4}/{\mathbb{Z}}_{N}$ we have
$\theta\sim\theta+2\pi/n(p)$, where $n(p)$ is the order of the stabilizer
subgroup of the point $p$.

###  2.2 Heterotic compactification to six dimensions: set up

We begin our study of heterotic compactification on $X_{5}$ with a review of
the well-known construction of heterotic theory compactified on
${{\mathbb{T}}}^{4}$. This compactification yields a $d=6$ (1,1) supergravity
theory with massless content consisting of the (1,1) supergravity multiplet
and a number of vector multiplets. Each vector multiplet contains $4$ scalars
that transform in the adjoint of the gauge algebra. At a generic point in the
scalar moduli space
$\operatorname{Gr}(20,4)/\operatorname{O{}}(\Gamma_{20,4})$ the gauge algebra
is $\operatorname{\mathfrak{u}}(1)^{\oplus 20}$.555Here
$\operatorname{Gr}(20,4)$ is the coset
$\operatorname{SO}(20,4,{\mathbb{R}})/\operatorname{SO}(20,{\mathbb{R}})\times\operatorname{SO}(4,{\mathbb{R}})$,
and $\operatorname{O{}}(\Gamma_{20,4})$ is the group of lattice isomorphisms
of the even self-dual lattice $\Gamma_{20,4}$, often written as
$\operatorname{O{}}(20,4,{\mathbb{Z}})$. At special points the gauge algebra
can be enhanced, and the enhanced symmetries and the corresponding loci in the
moduli space have been recently studied in [7]. For our purposes it will be
sufficient to consider the locus where the gauge algebra is
$\operatorname{\mathfrak{e}}_{8}\oplus\operatorname{\mathfrak{u}}(1)^{\oplus
4}\oplus\operatorname{\mathfrak{e}}_{8}$: this locus has a standard RNS
worldsheet realization, which will make it easy to describe the constructions
we wish to consider. In this section we will summarize some of the relevant
details, and in the next section we will apply them to some classic examples.

We work in light-cone gauge on a Euclidean worldsheet with coordinates
$z,\overline{z}$, with worldsheet supersymmetry on the right (anti-
holomorphic) side of the string. In addition to the degrees of freedom for the
Minkowski directions, the internal CFT consists of the scalar fields
$\Phi^{i}(z,\overline{z})$ and their right-moving Majorana-Weyl superpartners,
$8$ left-moving Weyl fermions, and a level $1$
$\operatorname{\mathfrak{e}}_{8}$ current algebra for the “hidden
$\operatorname{E}_{8}$” that will play a spectator role in our analysis.

We find it convenient to break up the left-moving fermions into $2$ Weyl
fermions $\gamma^{1,2}(z)$ and their conjugates $\overline{\gamma}^{1,2}(z)$,
and $6$ more Weyl fermions $\xi^{I}(z)$, $\overline{\xi}^{I}(z)$; similarly,
we organize the right-moving fermions into a Weyl pair
$\psi^{1,2}(\overline{z})$, with conjugates
$\overline{\psi}^{1,2}(\overline{z})$. This gives a decomposition of the
corresponding level $1$ current algebras

$\displaystyle\operatorname{\mathfrak{so}}(16)_{L}$
$\displaystyle\supset\operatorname{\mathfrak{so}}(12)_{L}\oplus\operatorname{\mathfrak{su}}(2)_{L}\oplus\operatorname{\mathfrak{su}}(2)_{L}^{\prime}~{},$
$\displaystyle\operatorname{\mathfrak{so}}(4)_{R}$
$\displaystyle\simeq\operatorname{\mathfrak{su}}(2)_{R}\oplus\operatorname{\mathfrak{su}}(2)_{R}^{\prime}~{}.$
(2.5)

The fermions transform in the following representations. On the holomorphic
side

$\displaystyle\xi$
$\displaystyle\in({{\boldsymbol{12}}},{{\boldsymbol{1}}},{{\boldsymbol{1}}})~{},$
$\displaystyle\gamma,\overline{\gamma}$
$\displaystyle\in({{\boldsymbol{1}}},{{\boldsymbol{2}}},{{\boldsymbol{2}}})~{},$
(2.6)

while on the anti-holomorphic side we have
$\psi,\overline{\psi}\in({{\boldsymbol{2}}},{{\boldsymbol{2}}})$ of
$\operatorname{\mathfrak{su}}(2)_{R}\oplus\operatorname{\mathfrak{su}}(2)^{\prime}_{R}$.

This structure is a special case of a (0,4) SCFT necessary [38] for the
preservation of (1,0) supersymmetry in ${\mathbb{R}}^{1,5}$. As our ultimate
aim will be study compactifications with $8$ supercharges, we will now
describe some features in this more general setting, focusing on the
identification of states in the worldsheet theory with massless fermions in
spacetime.

The connection is established through the current algebra, which is now of the
form
$\operatorname{\mathfrak{so}}(12)_{L}\oplus\operatorname{\mathfrak{su}}(2)_{L}$
on the left and $\operatorname{\mathfrak{su}}(2)_{R}$ on the right.666In
general (0,4) theories the $\operatorname{\mathfrak{su}}(2)_{L}$ symmetry
factor will be reduced to $\operatorname{\mathfrak{u}}(1)_{L}$ or be entirely
absent. It will be present in all theories we consider and will simplify the
analysis. The latter is the R-symmetry of the N=4 superconformal algebra
(SCA), and it contains $\operatorname{\mathfrak{u}}(1)_{R}$ an R-symmetry for
an N=2 subalgebra of the N=4 SCA with current $\overline{J}$ and operator
charges labeled by $\overline{q}$. We will also choose
$\operatorname{\mathfrak{u}}(1)_{L}\subset\operatorname{\mathfrak{su}}(2)_{L}$
with current $J$ and charges $q$ to label our states, and we similarly define
the currents and charges $J^{\prime},q^{\prime}$ for
$\operatorname{\mathfrak{u}}(1)^{\prime}_{L}\subset\operatorname{\mathfrak{su}}(2)^{\prime}_{L}$
and $\overline{J}^{\prime},\overline{q}^{\prime}$ for
$\operatorname{\mathfrak{u}}(1)^{\prime}_{R}\subset\operatorname{\mathfrak{su}}(2)^{\prime}_{R}$.
The currents give simple expressions for the left- and right-fermion numbers
that are necessary for the GSO projections: we have $(-1)^{F_{\gamma}}=e^{i\pi
J_{0}}$, while $(-1)^{F_{\psi}}=e^{i\pi\overline{J}_{0}}$.

We can now use the familiar rules—see e.g. [39, 40]—to identify worldsheet
states with massless multiplets in spacetime.

  1. 1.

The identity operator of the internal CFT gives rise to a (1,0) supergravity
multiplet and a (1,0) tensor multiplet.777Details of the multiplet structure
can be found in the recent pedagogical review [41].

  2. 2.

The spacetime gauge bosons arise in two ways: every holomorphic current gives
rise to a spacetime gauge boson, and every anti-holomorphic operator with
weight $\overline{h}=1/2$ (i.e. a free fermion) gives rise to an abelian gauge
boson. The latter, when present, complete the (1,0) gravity multiplet and the
(1,0) tensor multiplet to a (1,1) supergravity multiplet.

In all of our examples the holomorphic current algebra will be of the form
$\operatorname{\mathfrak{e}}_{8}\oplus\operatorname{\mathfrak{u}}(1)^{\oplus
k}\oplus\operatorname{\mathfrak{g}}$, and in most of our examples
$\operatorname{\mathfrak{g}}\supset\operatorname{\mathfrak{e}}_{7}\oplus\operatorname{\mathfrak{u}}(1)_{L}^{\prime}$.
Note that the “linearly realized” currents
$\operatorname{\mathfrak{so}}(12)\oplus\operatorname{\mathfrak{su}}(2)_{L}\oplus\operatorname{\mathfrak{u}}(1)^{\prime}_{L}$
are completed to $\operatorname{\mathfrak{e}}_{7}$ by additional currents from
the left-moving Ramond sector in the
$({{\boldsymbol{32}}},{{\boldsymbol{1}}})\oplus({{\boldsymbol{12}}},{{\boldsymbol{2}}})$
representation.

In some of our examples the $\operatorname{\mathfrak{u}}(1)^{\prime}_{L}$
current will be enhanced to $\operatorname{\mathfrak{su}}(2)_{L}^{\prime}$,
and when there are $h=1/2$ holomorphic operators transforming in
${{\boldsymbol{2}}}$ of $\operatorname{\mathfrak{su}}(2)_{L}^{\prime}$, the
left-moving Ramond sector contains currents in the representation

$\displaystyle({{\boldsymbol{32}}},{{\boldsymbol{2}}},{{\boldsymbol{1}}})\oplus({{\boldsymbol{12}}},{{\boldsymbol{2}}},{{\boldsymbol{1}}})\oplus({{\boldsymbol{32^{\prime}}}},{{\boldsymbol{1}}},{{\boldsymbol{2}}})\oplus({{\boldsymbol{12}}},{{\boldsymbol{1}}},{{\boldsymbol{2}}})~{},$
(2.7)

that complete $\operatorname{\mathfrak{g}}$ to
$\operatorname{\mathfrak{g}}=\operatorname{\mathfrak{e}}_{8}$.

Finally, the $\operatorname{\mathfrak{u}}(1)^{\oplus k}$ factor arises from
additional currents neutral with respect to $\operatorname{\mathfrak{g}}$. For
example in ${{\mathbb{T}}}^{4}$ compactification, $k=4$ with the currents
$i\partial\Phi^{i}$.

  3. 3.

The remaining spacetime massless fields reside in (1,0)
${\textstyle\frac{1}{2}}$–hypermultiplets. These are in one to one
correspondence with the $\overline{\text{NS}}$ chiral primary states with
$\overline{q}=1$ and holomorphic weight $h=1$. Their gauge transformations are
determined by the left-moving sector: each worldsheet state with $q=0$ leads
to an $\operatorname{\mathfrak{e}}_{7}$–neutral
${\textstyle\frac{1}{2}}$–hypermultiplet, while each worldsheet state with
$q=1$ leads to a state transforming in ${{\boldsymbol{56}}}$ of
$\operatorname{\mathfrak{e}}_{7}$.

Let us apply these rules to the ${{\mathbb{T}}}^{4}$ example, focusing on the
(1,0) vector and hypermultiplets. The R-charge assignments to the right-moving
fermions are

$\displaystyle\begin{matrix}~{}&&\psi^{1}&&\psi^{2}&&\overline{\psi}^{1}&&\overline{\psi}^{2}\\\
\overline{q}&&+1&&+1&&-1&&-1\\\
\overline{q}^{\prime}&&+1&&-1&&-1&&+1\end{matrix}$ (2.8)

We can easily see that the spectrum is consistent with (1,1) spacetime
supersymmetry: for every holomorphic current $J$ we have the operators
$J\psi^{1}$ and $J\psi^{2}$ that together give rise to a (1,0) hypermultiplet
transforming in the same way as the current $J$ under the gauge symmetry.

###  2.3 ${{\mathbb{T}}}^{4}/{\mathbb{Z}}_{N}$ compactifications

We now turn our attention to (1,0) theories that are built by considering
symmetric orbifolds ${{\mathbb{T}}}^{4}/{\mathbb{Z}}_{N}$. In each case, we
take the ${\mathbb{Z}}_{N}$ action to be just the action described in (2.3),
extending it to act on the worldsheet fermions in the standard left-
right–symmetric fashion: the generator $g$ of the ${\mathbb{Z}}_{N}$ action is
set to be

$\displaystyle g=\underbrace{\exp\left[{\textstyle\frac{2\pi
i}{N}}J^{\prime}_{0}\right]}_{=g_{\gamma}}~{}~{}\underbrace{\exp\left[{\textstyle\frac{2\pi
i}{N}}\overline{J}^{\prime}_{0})\right]}_{=g_{\psi}}~{}.$ (2.9)

The ${\mathbb{Z}}_{N}$ action is diagonally embedded in
$\operatorname{U{}}(1)_{L}^{\prime}\times\operatorname{U{}}(1)_{R}^{\prime}$.
For $N=2$ the action is contained in the center of each corresponding
$\operatorname{SU}(2)$ factor, so that the
$\operatorname{\mathfrak{su}}(2)_{L}^{\prime}\oplus\operatorname{\mathfrak{su}}(2)_{R}^{\prime}$
algebra is left invariant, while for $N>2$ the invariant subalgebra is
$\operatorname{\mathfrak{u}}(1)_{L}^{\prime}\oplus\operatorname{\mathfrak{u}}(1)_{R}^{\prime}$.
We are of course by no means the first to consider these orbifolds—see, for
example [42, 33, 34, 43, 35] and references therein for earlier work and some
asymmetric generalizations. However, we will recall some of the details and
give a view that will be useful for what follows.

#### Untwisted massless states

To describe the untwisted massless states, we just need to apply the
projection onto ${\mathbb{Z}}_{N}$–invariant states. We first observe that for
all $N>0$ the special anti-holomorphic states with $\overline{h}=1/2$ are
projected out, and therefore the spacetime supersymmetry is reduced. However,
since the projection keeps the $\operatorname{\mathfrak{su}}(2)_{R}$ current
algebra and leaves the vacuum of the internal theory invariant, we are
guaranteed (1,0) supersymmetry. This makes the preceding discussion of the
multiplet structure well-adapted to understand the projection.

The invariant content of the (1,1) supergravity multiplet is exactly the (1,0)
supergravity and tensor multiplets. To analyze the invariant content of the
(1,1) vector multiplets, we use the decomposition

$\displaystyle\text{ (1,1) vector}=\text{(1,0) vector}\oplus\text{ (1,0)
hyper}~{}.$ (2.10)

The ${\mathbb{Z}}_{N}$ action on each multiplet is a combination of an
$\operatorname{SU}(2)^{\prime}_{L}$ gauge symmetry, an
$\operatorname{SU}(2)^{\prime}_{R}$ R-symmetry, as well as
$G_{\text{ab}}\simeq{\mathbb{Z}}_{N}$, which acts on the abelian (1,1) vector
multiplets corresponding to the $\operatorname{\mathfrak{u}}(1)^{\oplus 4}$
factor according to (2.3). To describe that action, we package the $4$ vector
multiplets into two complex combinations that transform according to

$\displaystyle{{\boldsymbol{1}}}_{\zeta_{N}}\oplus{{\boldsymbol{1}}}_{\zeta_{N}^{-1}}~{}$
(2.11)

under the action of $G_{\text{ab}}$.

Consider first the (1,1) vector multiplets corresponding to the hidden
$\operatorname{\mathfrak{e}}_{8}$ symmetry. While the (1,0) vector
submultiplets of these are left invariant, every
${\textstyle\frac{1}{2}}$-hyper is projected out, so we are left with just the
$(1,0)$ vector multiples of $\operatorname{\mathfrak{e}}_{8}$.

Next, we examine the $(1,1)$ abelian vectors: now each $(1,0)$ vector is
projected out, but because of the additional action in
$\operatorname{SU}(2)_{R}^{\prime}$, the $(1,0)$ hypermultiplet content is
more interesting: under $\operatorname{SU}(2)_{R}^{\prime}\times
G_{\text{ab}}$ the ${\textstyle\frac{1}{2}}$–hypermultiplets transform as

$\displaystyle{{\boldsymbol{2}}}_{\zeta_{N}}\oplus{{\boldsymbol{2}}}_{\zeta_{N}^{-1}}~{},$
(2.12)

or decomposing further with respect to $\operatorname{U{}}(1)_{R}^{\prime}$,
as

$\displaystyle{{\boldsymbol{1}}}_{1,\zeta_{N}}\oplus{{\boldsymbol{1}}}_{-1,\zeta_{N}}\oplus{{\boldsymbol{1}}}_{1,\zeta_{N}^{-1}}\oplus{{\boldsymbol{1}}}_{-1,\zeta_{N}^{-1}}~{}.$
(2.13)

Thus, if $N=2$, then all of these are invariant, while if $N>2$ half of these
are invariant. So, we find $4$ neutral hypers for $N=2$ and $2$ neutral hypers
for $N>2$.

To discuss the $\operatorname{\mathfrak{e}}_{8}$ vectors, we decompose

$\displaystyle\operatorname{\mathfrak{e}}_{8}$
$\displaystyle\supset\operatorname{\mathfrak{e}}_{7}\oplus\operatorname{\mathfrak{u}}(1)^{\prime}_{L}$
$\displaystyle{{\boldsymbol{248}}}$
$\displaystyle={{\boldsymbol{133}}}_{0}\oplus{{\boldsymbol{1}}}_{0}\oplus{{\boldsymbol{56}}}_{+1}\oplus{{\boldsymbol{56}}}_{-1}\oplus{{\boldsymbol{1}}}_{+2}\oplus{{\boldsymbol{1}}}_{-2}~{}.$
(2.14)

From this we see that for $N>2$ the invariant $(1,0)$ vectors transform in the
adjoint of
$\operatorname{\mathfrak{e}}_{7}\oplus\operatorname{\mathfrak{u}}(1)^{\prime}$,
while for $N=2$ we find the adjoint of
$\operatorname{\mathfrak{e}}_{7}\oplus\operatorname{\mathfrak{su}}(2)^{\prime}_{L}$.
To obtain the invariant ${\textstyle\frac{1}{2}}$-hypers, we tensor this with
${{\boldsymbol{2}}}$ of $\operatorname{\mathfrak{su}}(2)_{R}^{\prime}$, or
equivalently with ${{\boldsymbol{1}}}_{+1}\oplus{{\boldsymbol{1}}}_{-1}$ of
$\operatorname{\mathfrak{u}}(1)_{R}^{\prime}$:

$\displaystyle{{\boldsymbol{248}}}$
$\displaystyle={{\boldsymbol{133}}}_{0,+1}\oplus{{\boldsymbol{133}}}_{0,-1}\oplus{{\boldsymbol{1}}}_{0,+1}\oplus{{\boldsymbol{1}}}_{0,-1}$
$\displaystyle\quad\oplus{{\boldsymbol{56}}}_{+1,+1}\oplus{{\boldsymbol{56}}}_{+1,-1}\oplus{{\boldsymbol{56}}}_{-1,+1}\oplus{{\boldsymbol{56_{-1,+1}}}}$
$\displaystyle\quad\oplus{{\boldsymbol{1}}}_{+2,+1}\oplus{{\boldsymbol{1}}}_{+2,-1}\oplus{{\boldsymbol{1}}}_{-2,+1}\oplus{{\boldsymbol{1}}}_{-2,-1}~{}.$
(2.15)

The invariant ${\textstyle\frac{1}{2}}$-hypers are those with
$q^{\prime}+\overline{q}^{\prime}=0\mod N$. The result is the following
invariant spectrum from the untwisted sector (as promised, we ignore the
hidden $\operatorname{\mathfrak{e}}_{8}$):

$\displaystyle{\mathbb{Z}}_{N}$ gauge symmetry hypers
$\displaystyle{\mathbb{Z}}_{2}$
$\displaystyle\operatorname{\mathfrak{e}}_{7}\oplus\operatorname{\mathfrak{su}}(2)_{L}^{\prime}$
$\displaystyle{{\boldsymbol{1}}}_{0}^{\oplus
4}\oplus({{\boldsymbol{56}}},{{\boldsymbol{2}}})$
$\displaystyle{\mathbb{Z}}_{3}$
$\displaystyle\operatorname{\mathfrak{e}}_{7}\oplus\operatorname{\mathfrak{u}}(1)_{L}^{\prime}$
$\displaystyle{{\boldsymbol{1}}}^{\oplus
2}_{0}\oplus{{\boldsymbol{1}}}_{2}\oplus{{\boldsymbol{56}}}_{1}$
$\displaystyle{\mathbb{Z}}_{4,6}$
$\displaystyle\operatorname{\mathfrak{e}}_{7}\oplus\operatorname{\mathfrak{u}}(1)_{L}^{\prime}$
$\displaystyle{{\boldsymbol{1}}}^{\oplus 2}_{0}\oplus{{\boldsymbol{56}}}_{1}$
(2.16)

#### Twisted sector contributions

A simple approach to work out the twisted sector contributions is to model
each fixed point of the orbifold by ${\mathbb{C}}^{2}/{\mathbb{Z}}_{N}$ and
use free field techniques to calculate the quantum numbers of states in the
twisted sector. It is then not too difficult to read off the twisted sector
states that yield spacetime massless states. We will not have need for details
of the construction, so we will simply quote the results from the literature
[33, 35]. The twisted sectors make no contribution to the gauge symmetry, but
they do produce additional hypermultiplets arranged as follows:

$\displaystyle\begin{array}[]{ccl}{\mathbb{Z}}_{N}&&\text{twisted
hypers}\\\\[5.69054pt]
{\mathbb{Z}}_{2}&&({{\boldsymbol{2}}},{{\boldsymbol{1}}})^{\oplus
32}\oplus({{\boldsymbol{1}}},{{\boldsymbol{56}}})^{\oplus 8}\\\\[5.69054pt]
{\mathbb{Z}}_{3}&&{{\boldsymbol{56}}}^{\oplus
9}_{-1/3}\oplus{{\boldsymbol{1}}}^{\oplus
45}_{2/3}\oplus{{\boldsymbol{1}}}^{\oplus 18}_{-4/3}\\\\[5.69054pt]
{\mathbb{Z}}_{4}&&{{\boldsymbol{56}}}_{-1/2}^{\oplus
4}\oplus{{\boldsymbol{1}}}_{3/2}^{\oplus
8}\oplus{{\boldsymbol{1}}}_{1/2}^{\oplus
24}\oplus{{\boldsymbol{56}}}_{0}^{\oplus
5}\oplus{{\boldsymbol{1}}}_{1}^{\oplus 32}\\\\[5.69054pt]
{\mathbb{Z}}_{6}&&{{\boldsymbol{1}}}^{\oplus
8}_{1/3}\oplus{{\boldsymbol{1}}}^{\oplus
2}_{-5/3}\oplus{{\boldsymbol{56}}}_{-2/3}\oplus{{\boldsymbol{1}}}^{\oplus
22}_{2/3}\oplus{{\boldsymbol{1}}}^{\oplus
10}_{-4/3}\oplus{{\boldsymbol{56}}}^{\oplus
5}_{-1/3}\oplus{{\boldsymbol{1}}}_{1}^{\oplus
22}\oplus{{\boldsymbol{56}}}_{0}^{\oplus 3}\end{array}$ (2.21)

The reader can check that in all cases the six-dimensional anomaly cancelation
condition $N_{\text{hyper}}-N_{\text{vector}}=244$ is satisfied.

Each of these theories has a Higgs branch: we can Higgs the
$\operatorname{\mathfrak{su}}(2)_{L}^{\prime}$ for $N=2$ and the
$\operatorname{\mathfrak{u}}(1)_{L}^{\prime}$ for $N>2$ to obtain a spectrum
with gauge algebra $\operatorname{\mathfrak{e}}_{7}$ and hypermultiplets in
$({{\boldsymbol{56}}})^{\oplus 10}$—which is exactly the massless spectrum of
heterotic compactification on K3 with standard embedding. We expect this
result since each of these singularities is a degeneration limit of the K3
surface.888The limit in the non-linear sigma model’s moduli space is subtle
due to a choice of $B$-field on the collapsing two-cycles of the degenerating
K3 surface: see [36, 37] for further discussion.

###  2.4 Compactification on ${{\mathbb{T}}}^{4}/{\mathbb{Z}}_{N}\times S^{1}$

Having obtained the $d=6$ theory, it is a simple matter to compactify further
on a circle. If we treat the circle as completely decoupled, then the massless
spectrum is obtained by standard Kaluza-Klein reduction [41]:

  1. 1.

the reduction of the (1,0) supergravity and tensor multiplets leads to a
(minimal) $d=5$ supergravity multiplet and $2$ abelian vector multiplets999One
of these is obtained by dualizing a $d=5$ abelian tensor multiplet.;

  2. 2.

each $d=6$ vector multiplet in representation ${{\boldsymbol{r}}}$ of the
gauge algebra reduces to a $d=5$ vector multiplet, which now has a real scalar
transforming in the same representation;

  3. 3.

similarly, each $d=6$ hypermultiplet reduces to a $d=5$ hypermultiplet in the
same representation of the gauge algebra.

All in all, the gauge algebra is now $\operatorname{\mathfrak{u}}(1)^{\oplus
2}\oplus\operatorname{\mathfrak{e}}_{8}\oplus\operatorname{\mathfrak{g}}$,
where the first factor is due to the vector multiplets obtained from the
reduction of supergravity and tensor (1,0) multiplets, and the hypermultiplet
spectrum is unmodified.

The new feature in $d=5$ is the existence of a Coulomb branch. Giving a
generic set of expectation values to the scalars in the vector multiplets
breaks the gauge group to $\operatorname{\mathfrak{u}}(1)^{\oplus 18}$ and
leaves $4$ neutral hypermultiplets in the ${\mathbb{Z}}_{2}$ orbifold and just
$2$ neutral hypermultiplets for ${\mathbb{Z}}_{3,4,6}$. Observing that under
$\operatorname{\mathfrak{e}}_{7}\supset\operatorname{\mathfrak{e}}_{6}\oplus\operatorname{\mathfrak{u}}(1)$

$\displaystyle{{\boldsymbol{56}}}={{\boldsymbol{27}}}_{+1}\oplus{{\overline{\boldsymbol{27}}}}_{-1}\oplus{{\boldsymbol{1}}}_{+3}\oplus{{\boldsymbol{1}}}_{-3}~{},$
(2.22)

we see that for the ${\mathbb{Z}}_{3,4,6}$ compactifications there is a
Coulomb branch with unbroken
$\operatorname{\mathfrak{g}}=\operatorname{\mathfrak{u}}(1)^{2}\oplus\operatorname{\mathfrak{e}}_{8}\oplus\operatorname{\mathfrak{e}}_{6}\oplus\operatorname{\mathfrak{u}}(1)^{\prime}$
where all of the twisted sector matter is lifted, and there are $2$ neural
massless hypermultiplets. A special feature of the ${\mathbb{Z}}_{3}$ example
is that it is possible to lift all of the twisted sector states by just going
on the $\operatorname{\mathfrak{u}}(1)^{\prime}$ Coulomb branch, where the
full gauge algebra $\operatorname{\mathfrak{g}}$ is preserved.

###  2.5 Compactification on $X_{5}$

Having reviewed the well-known compactifications on
${{\mathbb{T}}}^{4}/{\mathbb{Z}}_{N}\times S^{1}$, we now consider
compactification on $X_{5}$. From the worldsheet point of view, the new
ingredient relative to the preceding discussion is a ${\mathbb{Z}}_{N}$ shift
orbifold of the circle. By itself this theory is easy to understand: starting
with a CFT for a circle of radius $r$, the shift orbifold is a CFT of radius
$r/N$ [44]. In the full quotient CFT for $X_{5}=({{\mathbb{T}}}^{4}\times
S^{1})/{\mathbb{Z}}_{N}$, we just need to combine the twisted sectors of the
${{\mathbb{T}}}^{4}/{\mathbb{Z}}_{N}$ and $S^{1}/{\mathbb{Z}}_{N}$ CFTs and
impose the orbifold projection. With a view to later developments, we will
first describe the construction of the $S^{1}$ shift orbifold in some detail.

#### The ${\mathbb{Z}}_{N}$ shift orbifold of the circle

Consider a compact $c=1$ boson at a generic radius $r$. Splitting the
worldsheet field $\Phi(z,\overline{z})$ into left- and right-moving
components, we have the defining OPEs

$\displaystyle\Phi_{L}(z)\Phi_{L}(w)$
$\displaystyle\sim-{\textstyle\frac{1}{r^{2}}}\log(z-w)~{},$
$\displaystyle\Phi_{R}(\overline{z})\Phi_{R}(\overline{w})$
$\displaystyle\sim-{\textstyle\frac{1}{r^{2}}}\log(\overline{z}-\overline{w})~{}.$
(2.23)

The theory enjoys a Kac-Moody
$\operatorname{U{}}(1)^{\text{shift}}_{L}\times\operatorname{U{}}(1)^{\text{shift}}_{R}$
symmetry with currents

$\displaystyle J$ $\displaystyle=ir\partial\Phi_{L}~{},$
$\displaystyle\overline{J}$ $\displaystyle=ir\bar{\partial}\Phi_{R}~{},$
(2.24)

and the (Kac-Moody) primary states $|p\rangle$ are labeled by the momentum and
winding modes $n,w\in{\mathbb{Z}}$. More precisely,

$\displaystyle p=w\boldsymbol{e}+n\boldsymbol{e}^{\ast}\in\Gamma_{1,1}~{},$
(2.25)

with $\Gamma_{1,1}\subset{\mathbb{R}}^{1,1}$ the even self-dual lattice
spanned by two lattice vectors $\boldsymbol{e}$ and $\boldsymbol{e}^{\ast}$
satisfying
$\boldsymbol{e}.\boldsymbol{e}=\boldsymbol{e}^{\ast}.\boldsymbol{e}^{\ast}=0$
and $\boldsymbol{e}.\boldsymbol{e}^{\ast}=1$. Here $a.b$ denotes the inner
product on ${\mathbb{R}}^{1,1}$ induced by the bilinear pairing on the
lattice. Note that for our discussions of lattices here and in what follows,
we will take ${\mathbb{R}}^{n_{L},n_{R}}$ to have Lorentzian metric

$\displaystyle\eta$ $\displaystyle=\begin{pmatrix}-{\mathbbm{1}}_{n_{L}}&0\\\
0&+{\mathbbm{1}}_{n_{R}}\end{pmatrix}~{}.$ (2.26)

Our theory has a one-dimensional moduli space, which we can think of as a
specification of a spacelike $1$-plane $\Pi\subset{\mathbb{R}}^{1,1}$. We can
always choose a basis vector for $\Pi$

$\displaystyle\boldsymbol{\pi}$
$\displaystyle=\boldsymbol{e}+r^{2}\boldsymbol{e}^{\ast}~{},$ (2.27)

so that
$\widetilde{\boldsymbol{\pi}}=\boldsymbol{e}-r^{2}\boldsymbol{e}^{\ast}$ spans
the orthogonal complement $\Pi_{\perp}$.

With that preparation, we write the operator corresponding to the state
$|p\rangle$ as

$\displaystyle{\cal V}_{p}$
$\displaystyle=~{}:~{}\boldsymbol{C}(p)\exp\left[{\textstyle\frac{i}{\sqrt{2}}}(\boldsymbol{\pi}.p)\Phi_{R}(\overline{z})+{\textstyle\frac{i}{\sqrt{2}}}(\widetilde{\boldsymbol{\pi}}.p)\Phi_{L}(z)\right]~{}:~{}.$
(2.28)

$\boldsymbol{C}(p)$ is an operator constructed from the momentum zero modes
that we will discuss further below; this “cocycle operator” is necessary to
ensure proper commutation relations of the vertex operators. Using the OPEs,
it is now straightforward to see that ${\cal V}_{p}$ carries
$\operatorname{U{}}(1)^{\text{shift}}_{L}\times\operatorname{U{}}(1)^{\text{shift}}_{R}$
charges

$\displaystyle q_{L}^{\text{sh}}$
$\displaystyle=\frac{\widetilde{\boldsymbol{\pi}}.p}{r\sqrt{2}}~{},$
$\displaystyle q_{R}^{\text{sh}}$
$\displaystyle=\frac{\boldsymbol{\pi}.p}{r\sqrt{2}}~{}$ (2.29)

and weights

$\displaystyle h_{L}$
$\displaystyle={\textstyle\frac{1}{2}}(q_{L}^{\text{sh}})^{2}={\textstyle\frac{1}{4r^{2}}}\left(n-wr^{2}\right)^{2}~{},$
$\displaystyle h_{R}$
$\displaystyle={\textstyle\frac{1}{2}}(q_{R}^{\text{sh}})^{2}={\textstyle\frac{1}{4r^{2}}}\left(n+wr^{2}\right)^{2}~{}.$
(2.30)

The spin of the operator is

$\displaystyle s(p)=h_{L}-h_{R}=-{\textstyle\frac{1}{2}}p.p=-nw~{}.$ (2.31)

Note that in these conventions the self-dual radius is $r=1$, with the
T-duality map being $r\mapsto 1/r$ and $(n,w)\mapsto(w,n)$.

With this preparation, we define the action of the shift symmetry
${\mathbb{Z}}_{N}^{\text{shift}}$, taking its generator $g_{\text{sh}}$ to act
as

$\displaystyle g_{\text{sh}}|p\rangle$ $\displaystyle=e^{2\pi
in/N}|p\rangle~{}.$ (2.32)

Since the momentum quantum number $n$ is conserved, this is clearly a symmetry
of the spectrum, of the OPE, and of the correlation functions. Moreover, we
can represent this action in terms of the conserved shift currents (or,
rather, the corresponding conserved charges):

$\displaystyle g_{\text{sh}}$ $\displaystyle=e^{\frac{2\pi i}{N}Q}~{},$ (2.33)

with the charge $Q$ given by

$\displaystyle
Q={\textstyle\frac{r}{\sqrt{2}}}\left(J_{L,0}^{\text{sh}}+J_{R,0}^{\text{sh}}\right)~{}.$
(2.34)

The orbifold projection is then onto states with $Q\in N{\mathbb{Z}}$.

This is a key simplification in the orbifold analysis, since it allows us to
directly construct the twisted Hilbert space—see [40] for a recent pedagogical
discussion.101010A recent discussion of the more general situation is given in
[45]. To carry this out, we find a twist field of the form

$\displaystyle\Sigma_{k}(z,\overline{z})=\exp\left[-i{\textstyle\frac{r^{2}}{\sqrt{2}}}\tau_{k}\Phi_{R}-i{\textstyle\frac{r^{2}}{\sqrt{2}}}\widetilde{\tau}_{k}\Phi_{L}\right]~{},$
(2.35)

where the parameters $\tau_{k}$ and $\widetilde{\tau}_{k}$ are chosen so that
the OPE of ${\cal V}_{p}(z,\overline{z})$ and $\Sigma_{k}(z,\overline{z})$ has
the correct monodromy, i.e. so that under a continuous rotation $z\to
e^{i\theta}z$, we obtain

$\displaystyle{\cal V}_{p}(e^{2\pi i}z,e^{-2\pi
i}\overline{z})\Sigma_{k}(0)=e^{{\textstyle\frac{2\pi ikn}{N}}}{\cal
V}_{p}(z,\overline{z})~{}.\Sigma_{k}(0)$ (2.36)

The field $\Sigma$ then gives the $k$-th twisted ground state
$|0;k\rangle=\lim_{z,\overline{z}\to 0}\Sigma(z,\overline{z})|0\rangle$, and
the full Hilbert space in the $k$-th twisted sector is constructed from
$|p;k\rangle$—obtained by acting further with the ${\cal V}_{p}$ on the
$|0;k\rangle$—by acting with all possible oscillators.

In our example a quick computation shows that choosing

$\displaystyle\tau_{k}$ $\displaystyle=-{\textstyle\frac{k}{N}}~{},$
$\displaystyle\widetilde{\tau}_{k}$ $\displaystyle={\textstyle\frac{k}{N}}~{}$
(2.37)

produces the correct monodromy. Moreover, it is straightforward to read off
the shifts in the weights and charges due to the twist, i.e. the weights and
charges of the state $|p;k\rangle$:

$\displaystyle q^{\text{sh}}_{L}(p;k)$
$\displaystyle=\frac{\widetilde{\pi}.p-\widetilde{\tau}_{k}r^{2}}{r\sqrt{2}}~{},$
$\displaystyle q^{\text{sh}}_{R}(p;k)$
$\displaystyle=\frac{\pi.p-\tau_{k}r^{2}}{r\sqrt{2}}~{},$ (2.38)

and the weights are

$\displaystyle h_{L}(p;k)$
$\displaystyle={\textstyle\frac{1}{2}}(q_{L}^{\text{sh}}(p;k))^{2}~{},$
$\displaystyle h_{R}(p;k)$
$\displaystyle={\textstyle\frac{1}{2}}(q_{R}^{\text{sh}}(p;k))^{2}~{}.$ (2.39)

Not only do we know the twisted Hilbert spaces, but we also know how to
implement the projection onto invariant states: we just need to project onto
states with

$\displaystyle
Q(p;k)=\frac{r}{\sqrt{2}}\left(q_{L}^{\text{sh}}(p;k)+q_{R}^{\text{sh}}(p;k)\right)\in
N{\mathbb{Z}}~{}.$ (2.40)

For our circle shift orbifold it is easy to check that $Q(p;k)=Q(p)$, and also
that the spin satisfies

$\displaystyle s(p;k)=h_{L}(p;k)-h_{R}(p;k)=s(p)-\frac{k}{N}Q(p;k)~{},$ (2.41)

so that ${\mathbb{Z}}_{N}$–invariant states are guaranteed to have integer
spin.

Carrying out the construction for the $N-1$ twisted sectors, we find the
expected structure: the projection sets the momentum modes to be valued in
$N{\mathbb{Z}}$, as is consistent with a circle of radius $r/N$, while the
$k$-th twisted sector adds in the winding modes with
$w\in{\textstyle\frac{k}{N}}+{\mathbb{Z}}$, which are the “extra” winding
modes for a circle of radius $r/N$ relative to that of radius $r$.

#### The $X_{5}$ orbifold

Having understood the shift orbifold in detail, we could easily construct the
full partition function for the $X_{5}$ CFT. The only modification to our
previous discussion of the ${{\mathbb{T}}}^{4}/{\mathbb{Z}}_{N}$
compactification is to treat the right-moving superpartner of
$\Phi(z,\overline{z})$ as part of the internal CFT rather than belonging to
the CFT describing the ${\mathbb{R}}^{1,5}$ degrees of freedom.111111There are
some subtleties in applying the usual RNS rules in odd-dimensional
compactification, but they can be easily avoided by introducing an additional
spectator circle with its superpartner; see e.g. [46].

If our interest is in the massless spectrum, then the effect of the extra
shift at generic radius is simple to understand: the untwisted massless states
are exactly those of the ${{\mathbb{T}}}^{4}/{\mathbb{Z}}_{N}\times S^{1}$
CFT, while all of the twisted sector states are massive. This observation was
the starting point for our study, since it suggests a close relationship
between the two theories.

#### A spacetime picture

There is another way to think about the $X_{5}$ compactification, inspired by
the $\mu_{\text{cir}}$ fibration described in section 2.1

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We start in the $d=6$ (1,1) obtained by heterotic compactification on
${{\mathbb{T}}}^{4}$. The worldsheet symmetries we identified in our
discussion of the ${{\mathbb{T}}}^{4}/{\mathbb{Z}}_{N}$ orbifold are
interpreted as spacetime gauge symmetries, and in particular there is a
discrete gauge symmetry $G={\mathbb{Z}}_{N}$ which is a subgroup

$\displaystyle G\subset
G_{\text{ab}}\times\operatorname{SU}(2)^{\prime}_{L}\times\operatorname{SU}(2)^{\prime}_{R}~{},$
(2.42)

the latter action being part of the spacetime R-symmetry group of the $d=6$
(1,1) theory. Given the presence of this discrete spacetime gauge symmetry,
when we compactify the theory on a circle with coordinate
$\theta\sim\theta+2\pi$, we can choose to turn on a holonomy for $G$. That is,
we modify the periodicity conditions on the fields as

$\displaystyle\phi(x,\theta+2\pi)=g\cdot\phi(x,\theta)~{}.$ (2.43)

The holonomy will have the effect that only $G$–invariant fields will have
zero modes in the $\theta$ expansion, so that the low energy theory obtained
in this way will have the same massless spectrum as compactification on
$X_{5}$. Because $G$ involves an $\operatorname{SU}(2)^{\prime}_{R}$ action,
half of the gravitinos will be lifted in the process, reducing supersymmetry
from $16$ to $8$ supercharges.

Our point of view is that the orbifold construction of the heterotic string on
$X_{5}$ is a UV completion of circle compactification of the spacetime theory
with a $G$-holonomy. Note we are not suggesting a spacetime interpretation
relating compactification based on a CFT to that based on the orbifold CFT.
Instead, the relationship is between a compactification based on a freely
acting orbifold and a circle compactification of a theory from one dimension
higher. In the appendix we present a toy bosonic string model illustrating
this structure for the ${\mathbb{Z}}_{2}$ quotient.

It is clear that the existence of a UV completion is a non-trivial condition
because not every global discrete symmetry of the heterotic CFT leads to a
modular-invariant orbifold, yet we expect such symmetries to still correspond
to gauge symmetries in the spacetime theory. Thus, there should in general be
obstructions to turning on non-trivial holonomy for a discrete spacetime gauge
symmetry $G$. We hope to return to a study of these in the future.

##  3 The heterotic worldsheet and duality

In the previous section we presented two classes of $d=5$ heterotic
compactifications that preserve $8$ supercharges. While we saw that their
spectra are closely related—in fact identical when restricted to the massless
untwisted states—in general the theories appear to be distinct. As a stark
difference we noted that ${{\mathbb{T}}}^{4}/{\mathbb{Z}}_{N}\times S^{1}$
compactifications can be deformed to $\text{K3}\times S^{1}$ by going on the
Higgs branch, while no such Higgs branch appears to be present in the $X_{5}$
compactifications.

Nevertheless, we will now show that by enlarging the moduli space to include
Wilson lines we can establish an isomorphism between these classes of
theories. Thus, not only are these compactifications connected in moduli
space, they are in fact identical.

###  3.1 Bosonic construction

Turning on Wilson line parameters is best described in the bosonized
construction of the heterotic worldsheet CFT on ${{\mathbb{T}}}^{d}$ [47, 48].
Grouping the left-moving fermions into $16$ Weyl fermions $\xi^{a}$,
$\overline{\xi}^{a}$, so that the currents
$J^{a}=~{}:\xi^{a}\overline{\xi}^{a}:$ generate a Cartan algebra of
$\operatorname{\mathfrak{e}}_{8}\oplus\operatorname{\mathfrak{e}}_{8}$, we
bosonize these as $J^{a}=i\partial{\cal X}^{a}_{L}$. The bosonic OPEs now
depend on the metric $g_{ij}$ on ${{\mathbb{T}}}^{d}$:

$\displaystyle\Phi^{i}_{L}(z)\Phi^{j}_{L}(w)$
$\displaystyle\sim-g^{ij}\log(z-w)~{},$
$\displaystyle\Phi^{i}_{R}(\overline{z})\Phi^{j}_{R}(\overline{w})$
$\displaystyle\sim-g^{ij}\log(\overline{z}-\overline{w})~{},$
$\displaystyle{\cal X}_{L}^{a}(z){\cal X}_{L}^{b}(w)$
$\displaystyle\sim-\delta^{ab}\log(z-w)~{}.$ (3.1)

The vertex operators are labeled by
$p\in\Gamma_{d+16,d}\subset{\mathbb{R}}^{d+16,d}$. The lattice isomorphism
$\Gamma_{d+16,d}\simeq(\Gamma_{1,1})^{5}+\Gamma_{8}+\Gamma_{8}$—see e.g. [49,
7] for details of the isomorphism relevant to the relation between the
$\operatorname{E}_{8}\times\operatorname{E}_{8}$ and
$\operatorname{Spin}(32)/{\mathbb{Z}}_{2}$ heterotic strings—induces an
isomorphism ${\mathbb{R}}^{d+16,d}\simeq
R^{d,d}\oplus{\mathbb{R}}^{8}\oplus{\mathbb{R}}^{8}$, and we can use this to
pick a basis that isometrically respects this splitting. For the
${\mathbb{R}}^{d,d}$ factor we choose lattice vectors
$\boldsymbol{e}_{i},\boldsymbol{e}^{\ast i}$ satisfying

$\displaystyle\boldsymbol{e}_{i}.\boldsymbol{e}^{\ast j}$
$\displaystyle=\delta^{j}_{i}~{},$
$\displaystyle\boldsymbol{e}_{i}.\boldsymbol{e}_{j}$ $\displaystyle=0~{},$
$\displaystyle\boldsymbol{e}^{\ast i}.\boldsymbol{e}^{\ast j}$
$\displaystyle=0~{},$ (3.2)

while for each of the $\Gamma_{8}$ factors we choose the set of simple roots
$\boldsymbol{\alpha}_{I}$, $I=1,\ldots,8$, of
$\operatorname{\mathfrak{e}}_{8}$, with
$\boldsymbol{\alpha}_{I}\cdot\boldsymbol{\alpha}_{J}$ the Killing metric,
taken to be negative in our conventions. The $\boldsymbol{\alpha}_{I}$ are
encoded in the Dynkin diagram written in terms of the standard orthonormal
basis for ${\mathbb{R}}^{8}$, denoted by $\boldsymbol{v}_{a}$, with
$a=1,\ldots,8$ and $\boldsymbol{v}_{a}\cdot\boldsymbol{v}_{b}=-\delta_{ab}$:

To accommodate the second (our hidden) $\operatorname{\mathfrak{e}}_{8}$
factor, we will let the indices $I,a$ run through $9,\ldots,16$ as well, but
this will play no role in our analysis.

With the lattice set up complete, a point in the Narain moduli space
corresponds to a choice of spacelike $d$-plane $\Pi$ spanned by the vectors
$\boldsymbol{\pi}_{i}$, which can be taken to be of the form121212We follow,
with some small adjustments, the presentation given in [37]

$\displaystyle\boldsymbol{\pi}_{i}$
$\displaystyle=\boldsymbol{e}_{i}+(g_{ij}+b_{ij}-{\textstyle\frac{1}{2}}A_{i}\cdot
A_{j})\boldsymbol{e}^{\ast j}+A_{i}~{},$ $\displaystyle
A_{i}=A_{i}^{I}\boldsymbol{\alpha}_{I}=A_{i}^{a}\boldsymbol{v}_{a}~{}.$ (3.3)

In addition to the torus metric $g_{ij}$, this encodes the choice of constant
$B$-field $b_{ij}$ and Wilson lines $A_{i}^{a}$ valued the Cartan subalgebra
of $\operatorname{\mathfrak{e}}_{8}$.

The spacelike vectors satisfy
$\boldsymbol{\pi}_{i}.\boldsymbol{\pi}_{j}=2g_{ij}$, and it is easy to find a
basis for the orthogonal complement $\Pi_{\perp}$, a timelike $(d+16)$–plane.
We take it to be spanned by

$\displaystyle\widetilde{\boldsymbol{\pi}}_{i}$
$\displaystyle=\boldsymbol{\pi}_{i}-2g_{ij}\boldsymbol{e}^{\ast j}~{},$
$\displaystyle\widetilde{\boldsymbol{\pi}}_{a}$
$\displaystyle=\boldsymbol{v}_{a}+A^{a}_{i}\boldsymbol{e}^{\ast i}~{}.$ (3.4)

These two sets of vectors are orthogonal to the $\boldsymbol{\pi}_{i}$ and
each other, and satisfy
$\widetilde{\boldsymbol{\pi}}_{i}.\widetilde{\boldsymbol{\pi}}_{j}=-2g_{ij}$
and
$\widetilde{\boldsymbol{\pi}}_{a}.\widetilde{\boldsymbol{\pi}}_{b}=-\delta_{ab}$.
Extending our discussion of the compact boson, we arrive at the vertex
operators

$\displaystyle{\cal V}_{p}$
$\displaystyle=~{}:~{}\boldsymbol{C}(p)\exp\left[{\textstyle\frac{i}{\sqrt{2}}}(\boldsymbol{\pi}_{i}.p)\Phi^{i}_{R}(\overline{z})+{\textstyle\frac{i}{\sqrt{2}}}(\widetilde{\boldsymbol{\pi}}_{i}.p)\Phi^{i}_{L}(z)+i(\widetilde{\boldsymbol{\pi}}_{a}.p){\cal
X}_{L}^{a}(z)\right]~{}:~{}$ (3.5)

for all

$\displaystyle p$
$\displaystyle=w^{i}\boldsymbol{e}_{i}+n_{i}\boldsymbol{e}^{\ast
i}+\lambda\in\Gamma_{16+d,d}~{},$ $\displaystyle\lambda$
$\displaystyle\in\Gamma_{8}+\Gamma_{8}~{}.$ (3.6)

The weights of these operators are given by a generalization of (2.30):

$\displaystyle h_{R}(p)$
$\displaystyle={\textstyle\frac{1}{4}}(\boldsymbol{\pi}_{i}.p)g^{ij}(\boldsymbol{\pi}_{j}.p)~{},$
$\displaystyle h_{L}(p)$
$\displaystyle={\textstyle\frac{1}{4}}(\widetilde{\boldsymbol{\pi}}_{i}.p)g^{ij}(\widetilde{\boldsymbol{\pi}}_{j}.p)+\textstyle\sum_{a}{\textstyle\frac{1}{2}}(\widetilde{\boldsymbol{\pi}}_{a}.p)^{2}~{},$
(3.7)

and the spin satisfies

$\displaystyle s(p)=h_{L}(p)-h_{R}(p)$
$\displaystyle=-{\textstyle\frac{1}{2}}p.p=-n^{i}w_{i}-{\textstyle\frac{1}{2}}\lambda\cdot\lambda~{}.$
(3.8)

Since $\Gamma_{8}$ is an even lattice the last term is an integer for all
$\lambda$, so that $s(p)\in{\mathbb{Z}}$.

Finally, to complete the heterotic construction we need to add the right-
moving superpartners of the $\Phi^{i}$, the fermions $\psi^{i}(\overline{z})$.
For all of our applications we will be able to group these into the two Weyl
fermions $\psi^{1,2}$, $\overline{\psi}^{1,2}$ for the ${{\mathbb{T}}}^{4}$
directions, and the extra Majorana-Weyl $\psi^{5}$ for the additional circle
direction.

###  3.2 Two orbifolds

Starting with this presentation of the heterotic string on
${{\mathbb{T}}}^{5}$, we now consider the special locus where the
${{\mathbb{T}}}^{4}$ factor admits a $G={\mathbb{Z}}_{N}$ symmetry with
generator $g$. The ${\mathbb{Z}}_{N}$ symmetry leads to significant
simplification in the CFT moduli: both $g_{i5}$ and $b_{i5}$ must be zero for
$i\neq 5$, and the $A_{i\neq 5}$ may be set to zero as well.131313Note the
$A_{i\neq 5}$ Wilson lines are not required to be set to zero by our
${\mathbb{Z}}_{N}$ symmetry because there is an accompanying action on the
left-moving fermions, or equivalently on the ${\cal X}_{a}$. In the orbifold
theory these symmetry-preserving Wilson line parameters describe
hypermultiplet expectation values.

To understand how $G$ acts on the states, we reconsider the actions described
in our RNS discussion. These were the geometric orbifold action on
${{\mathbb{T}}}^{4}$, accompanied by its supersymmetric extension to the
right-moving fermions, the translation on $S^{1}$, and the action on the left-
moving fermions $\gamma^{1,2}$. We will now translate each of these into the
bosonic description.

First, the geometric action on the ${{\mathbb{T}}}^{4}$ bosons induces an
action on the associated vectors $\vec{p}\in\Gamma_{d,d}$. If the $\Phi^{i}$
coordinates transform as

$\displaystyle g_{T}\cdot\Phi^{i}=R^{i}_{j}\Phi^{j}~{},$ (3.9)

then

$\displaystyle g_{T}\cdot w^{i}$ $\displaystyle=R^{i}_{j}w^{j}~{},$
$\displaystyle g_{T}\cdot n_{i}$ $\displaystyle=(R^{-1})^{j}_{i}n_{i}~{}.$
(3.10)

This preserves the spin of every ${\cal V}_{p}$, and, because it is a torus
isometry, it preserves the weights as well. The right-moving fermions also
transform accordingly, and as we discussed around equation (2.9), it is
convenient to represent the action by $g_{\psi}=\exp[{\textstyle\frac{2\pi
i}{N}}\overline{J}^{\prime}_{0}]$.

Second, the translation in the circle direction is precisely the shift
symmetry $g_{\text{sh}}$ described in section 2.5. It leaves the lattice
vectors invariant and acts on states by a ${\mathbb{Z}}_{N}$-valued phase.

Finally, we have the action on the $\gamma$ fermions, represented by
$g_{\gamma}=\exp[{\textstyle\frac{2\pi i}{N}}J^{\prime}_{0}]$. This is the
only one that is not quite obvious in the bosonized description, but because
we can write it in terms of the current $J^{\prime}$, finding the
corresponding action on the ${\cal X}^{a}$ is not too difficult. Taking a look
at our Dynkin diagram, we see that

$\displaystyle J^{\prime}$ $\displaystyle=i\partial{\cal X}^{7}-i\partial{\cal
X}^{8}~{},$ (3.11)

i.e. it is the current that corresponds to the extended root of the diagram.
Thus, $g_{\gamma}$ turns out to be another shift action in the bosonized
description:

$\displaystyle g_{\gamma}|p\rangle$ $\displaystyle=e^{{\textstyle\frac{2\pi
i}{N}}(\boldsymbol{v}_{7}-\boldsymbol{v}_{8}).p}|p\rangle~{}.$ (3.12)

We can see this gives the expected structure for the unbroken symmetry with
$A=0$. In this case the $\operatorname{\mathfrak{e}}_{8}$ currents that
survive the projection are the Cartan currents $i\partial{\cal X}^{a}$, as
well as the roots $\boldsymbol{\alpha}_{I}$ orthogonal to
$\boldsymbol{v}_{7}-\boldsymbol{v}_{8}$. For $N>2$ this gives rise to the
$\operatorname{\mathfrak{e}}_{7}\oplus\operatorname{\mathfrak{u}}(1)^{\prime}$
current algebra, and in the special case of $N=2$ the state corresponding to
the $\boldsymbol{v}_{7}-\boldsymbol{v}_{8}$ root is left invariant as well,
leading to the
$\operatorname{\mathfrak{u}}(1)^{\prime}\to\operatorname{\mathfrak{su}}(2)^{\prime}$
enhancement.

As above, we have two ${\mathbb{Z}}_{N}$ symmetries by which we can orbifold:
$G$, generated by $g=g_{T}g_{\psi}g_{\gamma}g_{\text{sh}}$, leads to the
$X_{5}$ orbifold, while $G^{\prime}$, generated by
$g^{\prime}=g_{T}g_{\psi}g_{\gamma}$, produces the
${{\mathbb{T}}}^{4}/{\mathbb{Z}}_{N}\times S^{1}$ orbifold.

Both of the symmetries are consistent with turning on a Wilson line $A_{5}$
along the shift circle, and by including this degree of freedom, we will
demonstrate that, despite appearances, the two orbifolds are equivalent. We
will do this by finding an element $t$ of the T-duality group
$\operatorname{O{}}(\Gamma_{17,1})\subset\operatorname{O{}}(\Gamma_{21,5})$
such that

$\displaystyle g^{\prime}=t^{-1}gt~{}.$ (3.13)

To give this construction, we will have to delve a little bit into the
structure of the heterotic T-duality group.

###  3.3 Elements of $\operatorname{O{}}(\Gamma_{17,1})$

The T-duality group $\operatorname{O{}}(\Gamma_{16+d,d})$ arises from lattice
isomorphisms, and each such isomorphism induces a (possibly trivial) action on
the moduli. A discussion of its generators is given in [50, 7]. In our
discussion two elements will play a role: the T-duality transformation on the
$5$-th circle, as well as a shift of the Wilson line associated to the circle
by a lattice vector in $\Gamma_{8}$.

Typically, the induced action on the moduli is complicated. For example, a
T-duality on a single circle usually involves a non-trivial action on the
$g_{ij}$, $b_{ij}$ and $A_{i}$ components [50, 7]. However, since we assume
the ${\mathbb{Z}}_{N}$ symmetry, the moduli are restricted, so that we may as
well think of our transformations as living in
$\operatorname{O{}}(\Gamma_{17,1})$.

A vector in $\Gamma_{17,1}$ has the form

$\displaystyle p$
$\displaystyle=w\boldsymbol{e}+n\boldsymbol{e}^{\ast}+\lambda~{},$ (3.14)

and using (3.7) with $g_{55}=r^{2}$, and $A_{5}=A$, we find the weight and
spin

$\displaystyle h_{R}(p)$
$\displaystyle={\textstyle\frac{1}{4r^{2}}}\left(\pi.p\right)^{2}~{},$
$\displaystyle s(p)$
$\displaystyle=h_{L}(p)-h_{R}(p)=-{\textstyle\frac{1}{2}}p.p=-nw-{\textstyle\frac{1}{2}}\lambda\cdot\lambda~{},$
(3.15)

with

$\displaystyle\pi.p$
$\displaystyle=n+\left(r^{2}-{\textstyle\frac{1}{2}}A\cdot
A\right)w+A\cdot\lambda~{}.$ (3.16)

The T-duality action on the lattice is simply the exchange $(n,w)\to(w,n)$.
Let us call this action $g_{1}(p)$. Clearly $g_{1}(p).g_{1}(p)=p.p$ for all
$p$, which is necessary and sufficient for $g_{1}$ to be a lattice
isomorphism. Moreover, we can find an induced action the moduli $(r,A)$, so
that

$\displaystyle(g_{1}^{\ast}h_{R})(p)=h_{R}(g_{1}(p))~{}:$ (3.17)
$\displaystyle(r^{\prime},A^{\prime})$ $\displaystyle=g_{1}^{\ast}(r,A)~{},$
$\displaystyle r^{\prime}$
$\displaystyle=\frac{r}{r^{2}-{\textstyle\frac{1}{2}}A\cdot A}~{},$
$\displaystyle A^{\prime}$
$\displaystyle=\frac{A}{r^{2}-{\textstyle\frac{1}{2}}A\cdot A}~{}.$ (3.18)

The lattice vector shift of the Wilson line arises through a more elaborate
lattice isomorphism, depending on a choice of lattice vector
$\rho\in\Gamma_{8}+\Gamma_{8}$:

$\displaystyle
g_{2}(p)=p+(\rho\cdot\lambda-{\textstyle\frac{1}{2}}\rho\cdot\rho
w)\boldsymbol{e}^{\ast}-\rho w~{},$ (3.19)

or more explicitly,

$\displaystyle
g_{2}(n,w,\lambda)=(n+\rho\cdot\lambda-{\textstyle\frac{1}{2}}\rho\cdot\rho
w,w,\lambda-\rho w)~{}.$ (3.20)

This action also preserves the lattice since $g_{2}(p).g_{2}(p)=p.p$ , and the
induced map on the moduli is simply

$\displaystyle(r^{\prime},A^{\prime})$
$\displaystyle=g_{2}^{\ast}(r,A)=(r,A+\rho)~{}.$ (3.21)

Having reviewed the form of these basic maps, we will now use them to
construct an isomorphism between the two orbifold theories, i.e. the
compactification on $X_{5}$ and the ${{\mathbb{T}}}^{4}/Z_{N}\times S^{1}$
orbifold.

###  3.4 Heterotic isomorphism

Let $t=g_{1}g_{2}$. We claim that this combination of
$\operatorname{O{}}(\Gamma_{17,1})$ elements leads to the desired equivalence
(3.13) for an appropriate choice of lattice vector $\rho$. Since $t$ does not
act on the ${{\mathbb{T}}}^{4}$ bosons or right-moving fermions, and our two
actions $g$ and $g^{\prime}$ act in the same way on those degrees of freedom,
it is sufficient to check the claim for $p\in\Gamma_{17,1}$.

We set the $\rho$ to be proportional to the root
$\boldsymbol{\alpha}_{6}=\boldsymbol{v}_{6}-\boldsymbol{v}_{7}$:

$\displaystyle\rho$ $\displaystyle=(1-N)\boldsymbol{\alpha}_{6}~{},$ (3.22)

and we define for convenience
$\boldsymbol{\gamma}=\boldsymbol{v}_{7}-\boldsymbol{v}_{8}$. Note that
$\boldsymbol{\gamma}\cdot\rho=1-N$. We now compute

$\displaystyle t|n,w,\lambda\rangle$
$\displaystyle=|w,n+\rho\cdot\lambda-{\textstyle\frac{1}{2}}\rho\cdot\rho
w,\lambda-\rho w\rangle~{},$ (3.23)

and therefore

$\displaystyle t^{-1}gt|n,w,\lambda\rangle$
$\displaystyle=\exp\left[{\textstyle\frac{2\pi
i}{N}}\left(w+\boldsymbol{\gamma}\cdot(\lambda-\rho
w)\right)\right]|n,w,\lambda\rangle=e^{{\textstyle\frac{2\pi
i}{N}}\boldsymbol{\gamma}\cdot\lambda}|n,w,\lambda\rangle=g^{\prime}|n,w,\lambda\rangle~{}.$
(3.24)

In the second equality the phase dependence on the winding mode $w$ drops out
precisely because $\boldsymbol{\gamma}\cdot\rho=1-N$. Thus, despite
appearances, the two orbifold CFTs are isomorphic!

#### Special features of $N=3$

As we remarked above, the correspondence between the theories is particularly
nice when $N=3$, since there it is possible to match the massless spectra by
turning on a single Wilson line for the
$\operatorname{\mathfrak{u}}(1)^{\prime}$ vector multiplet. We can see these
special features from the point of view of our isomorphism as well. Precisely
when $N=3$ we can set the Willson line shift to be $\rho=\boldsymbol{\gamma}$,
and the winding mode will again drop out from the phase factor.

In this case, then, we can identify the $X_{5}$ theory with radius $r$ and
circle Wilson line $A_{5}=a\boldsymbol{\gamma}$ with the
${{\mathbb{T}}}^{4}/{\mathbb{Z}}_{3}\times S^{1}$ theory with radius
$r^{\prime}$ and circle Wilson line $A_{5}=a^{\prime}\boldsymbol{\gamma}$ via

$\displaystyle r$ $\displaystyle={\textstyle\frac{r^{\prime}}{r^{\prime
2}+(a^{\prime}-1)^{2}}}~{},$ $\displaystyle a$
$\displaystyle={\textstyle\frac{a^{\prime}-1}{r^{\prime
2}+(a^{\prime}-1)^{2}}}~{}.$ (3.25)

As expected, the duality is a stringy one. For example, setting
$a^{\prime}=1$, which corresponds to the $a=0$ $X_{5}$ theory analyzed above,
we find that a large radius circle on the
${{\mathbb{T}}}^{4}/{\mathbb{Z}}_{3}\times S^{1}$ side corresponds to $X_{5}$
with a small shift circle.

Note that the theory with $a^{\prime}=1$ is equivalent to $a^{\prime}=0$ in
the ${{\mathbb{T}}}^{5}$ theory, but that is not the case in the orbifold,
where instead the equivalence is $a^{\prime}\sim a^{\prime}+3$. This is
closely related to the appearance of the fractional
$\operatorname{U{}}(1)^{\prime}_{L}$ charges in the twisted sector states, and
it is thus not surprising that such massless states are lifted for
$a^{\prime}=1$.

The $a^{\prime}=0$ locus is also interesting, since it corresponds to the
${{\mathbb{T}}}^{4}/{\mathbb{Z}}_{3}\times S^{1}$ theory with the massless
hypermultiplets coming from the twisted sectors. It would be surprising if we
could identify it in the $X_{5}$ description at large radius, and indeed this
is not so. When written in terms of the $X_{5}$ moduli, the $a^{\prime}=0$
locus corresponds to the semicircle with $r\geq 0$ and

$\displaystyle(2r)^{2}+(1+2a)^{2}=1~{},$ (3.26)

so that $r\leq 1/2$.

###  3.5 Cocycle subtleties and their resolution

The reader who has made it this far may wonder if we have not introduced more
formalism than we need to describe our results: we have not used the structure
of the vertex operators ${\cal V}_{p}$ in any detail, and the operators
$\boldsymbol{C}(p)$, which are typically relegated to a last section or
appendix have played no role in our discussion. We will now explain the need
for these terms and point out that in general their presence can lead to
subtleties in the analysis of CFT symmetries. We will also see, however, that
in this case we are lucky: all of the potential subtle factors drop out, and
the duality conclusions reached in the previous section remain unmodified.
Nevertheless, since the general observations here may lead to subtleties in
closely related discussions, we will include them. In addition to the standard
textbook references [51, 52], which supply some of the background, our
thinking about these issues was guided by the work [53].

#### A class of CFT symmetries

We are interested in discussing a class of CFT symmetries of Narain
compactification that are realized as the following $G$ action on the vertex
operators. For every $g\in G$ there is a map $\varphi_{g}:\Gamma\to\Gamma$ and
a factor $U(g,p)\in{\mathbb{C}}$ such that

$\displaystyle g\circ{\cal V}_{p}=U(g,p){\cal V}_{\varphi_{g}(p)}~{}.$ (3.27)

Not all CFT symmetries can be realized in this fashion. For example, the
$\operatorname{SU}(2)\times\operatorname{SU}(2)$ symmetries of the compact
boson at self-dual radius take a more general form and also mix the ${\cal
V}_{p}$ with non-Kac-Moody primary operators such as $\partial\Phi$ and
$\bar{\partial}\Phi$.

We wish the $g$-action to be invertible and to be consistent with the OPE, and
it must preserve the weights of the operators. Therefore it must be that for
every $g$ the factor $U(g,p)\in{\mathbb{C}}^{\ast}$, and $\varphi_{g}$ is a
lattice isomorphism. We will insist that the action is unitary, which means
$U(g,p)$ is a pure phase. Since we also want composition to be consistent with
the group product structure, i.e.

$\displaystyle(g_{2}g_{1})\circ{\cal V}_{p}=g_{2}\circ\left(g_{1}\circ{\cal
V}_{p}\right)~{},$ (3.28)

We also learn that the phases must obey

$\displaystyle U(g_{2}g_{1},p)=U(g_{2},g_{1}(p))U(g_{1},p)~{},$ (3.29)

and similarly the $\varphi_{g}$ should satisfy
$\varphi_{g_{1}g_{2}}(p)=\varphi_{g_{1}}(\varphi_{g_{2}}(p))$. We now see that
$\varphi$ must be a map to the group of automorphisms of the lattice, i.e.
$\varphi:G\to\operatorname{O{}}(\Gamma)$. The resulting subgroup of
$\operatorname{O{}}(\Gamma)$

$\displaystyle G_{\Gamma}\simeq G/\ker(\varphi)$ (3.30)

is in general smaller than $G$: for example, the circle shift symmetry has
$G={\mathbb{Z}}_{N}$ and $G_{\Gamma}=1$.

These are sensible constraints determined by the group structure, but there
are further constraints on the factors $U(g,p)$, and this is where the
cocycles make an appearance.

#### A look at the cocycles

The factors $\boldsymbol{C}(p)$ are introduced to resolve an issue with
commutation properties of the naive vertex operators ${\cal
V}_{p}^{\text{naive}}$ which lack these factors: the OPE of two such operators
has a non-trivial monodromy as we transport one operator around the other,
which leads to

$\displaystyle:{\cal V}^{\text{naive}}_{p_{1}}(-z/2):~{}:{\cal
V}^{\text{naive}}_{p_{2}}(z/2):~{}=e^{i\pi p_{1}.p_{2}}:{\cal
V}^{\text{naive}}_{p_{2}}(z/2):~{}:{\cal V}^{\text{naive}}_{p_{1}}(-z/2):~{}.$
(3.31)

Thus, the operators appear to anticommute whenever $p_{1}.p_{2}$ is odd. To
resolve this, the operators are modified to include the $\boldsymbol{C}(p)$,
which are chosen so that

$\displaystyle{\cal V}_{q}\boldsymbol{C}_{p}$
$\displaystyle=\varepsilon(q,p)\boldsymbol{C}_{p}{{\cal V}_{q}}~{},$
$\displaystyle\boldsymbol{C}_{p}\boldsymbol{C}_{q}$
$\displaystyle=\boldsymbol{C}_{p+q}~{}$ (3.32)

for some phase $\varepsilon(q,p)$. Including these factors will remove the
unwanted factor $e^{i\pi p_{1}.p_{2}}$ if the phases obey

$\displaystyle\varepsilon(p_{2},p_{1})=(-1)^{p_{1}.p_{2}}\varepsilon(p_{1},p_{2})~{}.$
(3.33)

Associativity of the OPE places further non-trivial conditions on the phases:

$\displaystyle\varepsilon(p_{1},p_{3})\varepsilon(p_{2},p_{3})$
$\displaystyle=\varepsilon(p_{1}+p_{2},p_{3})~{},$ (3.34)

and

$\displaystyle\varepsilon(p_{1},p_{2})\varepsilon(p_{1}+p_{2},p_{3})=\varepsilon(p_{1},p_{2}+p_{3})\varepsilon(p_{2},p_{3})~{}.$
(3.35)

The second of these justifies the name of cocycle, i.e. $\varepsilon\in
H^{2}(\Gamma,\operatorname{U{}}(1))$. Consider the group homomorphisms
$\Gamma\times\Gamma\to\operatorname{U{}}(1)$, denoted by $c(p_{1},p_{2})$. The
coboundary map to the group of homomorphisms
$\Gamma\times\Gamma\times\Gamma\to\operatorname{U{}}(1)$ is taken to be

$\displaystyle\delta_{32}c(p_{1},p_{2},p_{3})$
$\displaystyle=\frac{c(p_{2},p_{3})}{c(p_{1}+p_{2},p_{3})}\frac{c(p_{1},p_{2}+p_{3})}{c(p_{1},p_{2})}~{},$
(3.36)

Similarly, starting with homomorphisms $\Gamma\to\operatorname{U{}}(1)$
denoted by $f(p)$, we have

$\displaystyle\delta_{21}f(p_{1},p_{2})$
$\displaystyle=\frac{f(p_{1})f(p_{2})}{f(p_{1}+p_{2})}~{},$ (3.37)

and the reader can check $\delta_{32}(\delta_{21}f)=1$. The condition (3.35)
is the statement that $\delta_{32}\varepsilon=1$, i.e. that $\varepsilon$ is a
cocycle that defines a class in $H^{2}(\Gamma,\operatorname{U{}}(1))$. On the
other hand, the condition (3.33) shows that $\varepsilon(p_{1},p_{2})$ cannot
be a coboundary.

As shown in the references [51, 52, 53], it is possible to choose the
$\boldsymbol{C}(p)$ so that $\varepsilon$ take the following form:

$\displaystyle\varepsilon(p_{1},p_{2})$
$\displaystyle=\exp\left[{\textstyle\frac{i\pi}{2}}(n_{1i}w_{2}^{i}-w_{1}^{i}n_{2i})+i\pi\Omega(\lambda_{1},\lambda_{2})\right]~{},$
(3.38)

where the bilinear antisymmetric form $\Omega(\lambda_{1},\lambda_{2})$ is
obtained by writing $\lambda_{1,2}=\lambda_{1,2}^{I}\boldsymbol{\alpha}_{I}$
in terms of the simple roots and then setting

$\displaystyle\Omega(\lambda_{1},\lambda_{2})$
$\displaystyle={\textstyle\frac{1}{2}}\sum_{I>J}\left(\lambda_{1}^{I}\lambda_{2}^{J}-\lambda_{2}^{I}\lambda_{1}^{J}\right)\boldsymbol{\alpha}_{I}\cdot\boldsymbol{\alpha}_{J}~{}.$
(3.39)

#### Cocycles and symmetry phases

Finally we come to our key point: the cocycle factors and the phases $U(g,p)$
are intertwined, and the consistency of the OPE with the symmetry action, i.e.

$\displaystyle~{}:g\circ{\cal V}_{p}(z_{1}):~{}:g\circ{\cal
V}_{q}(z_{2}):~{}=~{}g\circ\left(~{}:{\cal V}_{p}(z_{1}):~{}:{\cal
V}_{q}(z_{2}):~{}\right)~{}$ (3.40)

require the phase factors to obey [53]

$\displaystyle\frac{U(g,p+q)}{U(g,p)U(g,q)}$
$\displaystyle=\frac{\varepsilon(\varphi_{g}(p),\varphi_{g}(q))}{\varepsilon(p,q)}~{}.$
(3.41)

There are two ways to read this equation. On one hand, it shows that the ratio
of cocycles
$\frac{\varepsilon(\varphi_{g}(p),\varphi_{g}(q))}{\varepsilon(p,q)}$ is a
coboundary, and $U(g,p)$ is the trivializing cycle. On the other hand,
whenever the ratio is not $1$, it gives an obstruction to choosing $U(g,p)$ to
be a homomorphism from the lattice to $\operatorname{U{}}(1)$.

There are two obvious questions about this result: how does the it depend on
the choice of cocycle? to what extent does it determine the phases $U(g,p)$?
We leave it to the reader to check that a modification of $\varepsilon$ by a
coboundary $f(p)$ modifies $U(g,p)\to U(g,p)f(g(p))/f(p)$, and the new
$U(g,p)$ so obtained is consistent with the group structure. As far as the
second question goes, it is clear that if $U_{1}(g,p)$ and $U_{2}(g,p)$ both
solve (3.41) with the same $\varepsilon$, then their ratio $U_{1}/U_{2}$ is a
homomorphism from $\Gamma$ to $\operatorname{U{}}(1)$ for all $g$.

#### Some examples

Let us consider some simple examples that illustrate the appearance (and non-
appearance) of non-trivial phase factors $U(g,p)$. By non-trivial we mean here
factors that are not in $\operatorname{Hom}(\Gamma,\operatorname{U{}}(1))$ and
whose appearance is necessitated by (3.41).

By far the simplest example is any action where $\varphi_{g}(p)=p$ for all
$p$. In this case, $U(g,p)$ must be in
$\operatorname{Hom}(\Gamma,\operatorname{U{}}(1))$. The circle shift orbifold
yields a simple example. A bit more interesting is any geometric symmetry that
arises from a $\operatorname{GL}(d,{\mathbb{Z}})$ change of coordinates on the
torus, such as our symmetry $g_{T}$ in (3.10), acting on the winding and
momentum modes as in (3.10). However, in this case too we find
$\varepsilon(\varphi_{g}(p),\varphi_{g}(q))=\varepsilon(p,q)$, so that again
the phase can be taken to be trivial.

For a more interesting example in $\Gamma_{d,d}$ we can consider integral
shifts of the $b$-field background, for which

$\displaystyle\varphi_{g}(n_{i},w^{i})=(n_{i}+\Theta_{ij}w^{j},w^{i})~{},$
(3.42)

where $\Theta_{ij}$ is an arbitrary integral antisymmetric matrix. In this
case

$\displaystyle\frac{\varepsilon(g(p_{1}),g(p_{2}))}{\varepsilon(p_{1},p_{2})}=\exp\left[i\pi\textstyle\sum_{i>j}\Theta_{ij}(w_{1}^{i}w_{2}^{j}-w_{2}^{i}w_{1}^{j})\right]\neq
1~{},$ (3.43)

and a non-trivial phase is required to satisfy (3.41). A choice that works is
to take

$\displaystyle U(g,p)$
$\displaystyle=\exp\left[i\pi\textstyle\sum_{i>j}\Theta_{ij}w^{i}w^{j}\right]~{}.$
(3.44)

Since the $g$–action preserves winding numbers $U(g,g(p))=U(g,p)$, so that
$U(g^{k},p)=U(g,p)^{k}$, which is consistent with the group structure.

For a more elaborate example, we consider the Wilson line shift in
$\Gamma_{17,1}$ relevant to our construction:

$\displaystyle\varphi_{g}(n,w,\tau)=(n+\rho\cdot\lambda-{\textstyle\frac{1}{2}}\rho\cdot\rho
w,w,\lambda-\rho w)~{}.$ (3.45)

In this case a somewhat lengthy computation, making creative use of mod $2$
conditions in the exponential, leads to

$\displaystyle\frac{\varepsilon(\varphi_{g}(p_{1}),\varphi_{g}(p_{2}))}{\varepsilon(p_{1},p_{2})}=\exp\left[i\pi\left(w_{1}S(\rho,\lambda_{2})+w_{2}S(\rho,\lambda_{1})\right)\right]~{},$
(3.46)

where the bilinear map $S$ is

$\displaystyle
S(\rho,\lambda)=\sum_{I}\rho^{I}\lambda^{I}-\sum_{I>J}\boldsymbol{\alpha}_{I}\cdot\boldsymbol{\alpha}_{J}\rho^{I}\lambda^{J}\in{\mathbb{Z}}~{}.$
(3.47)

The naive guess $U(g,p)=e^{i\pi wS(\rho,\lambda)}$, while solving (3.41), does
not satisfy the group property

$\displaystyle U(g^{2},p)=U(g,g(p))U(g,p)~{}.$ (3.48)

We can fix this by multiplying the naive guess by an element of
$\operatorname{Hom}(\Gamma,\operatorname{U{}}(1))$. We find that

$\displaystyle U(g,p)$ $\displaystyle=\exp\left[i\pi
wS(\rho,\lambda+\rho/2)\right]~{}$ (3.49)

satisfies all of the desired properties.

#### Duality unmodified

We just discussed at length that symmetry actions of the sort we used to prove
the duality between $X_{5}$ and ${{\mathbb{T}}}^{4}/{\mathbb{Z}}_{N}\times
S^{1}$ heterotic compactifications in general come with extra phase factors.
Since the whole point of our duality claim was a match of phase factors for
two group actions, this is a non-trivial concern.

To allay this worry, we write the correct form for the symmetry action on the
states, taking into account the possible extra phases. In keeping with the
spirit of this section, we distinguish in our notation between the action on
the states and the action on the lattice vectors, so that we have, coming back
to (3.13),

$\displaystyle t^{-1}gt|p\rangle$
$\displaystyle=U(t,p)U(g,\varphi_{t}(p))U(t^{-1},\varphi_{gt}(p))|\varphi_{t^{-1}gt}(p)\rangle~{}.$
(3.50)

The group multiplication properties imply the relation

$\displaystyle
1=U(t^{-1}t,\varphi_{t^{-1}}(p))=U(t^{-1},p)U(t,\varphi_{t}^{-1}(p))~{},$
(3.51)

so that

$\displaystyle U(t^{-1},p)=\frac{1}{U(t,\varphi_{t}^{-1}(p))}~{}.$ (3.52)

But, since we have $\varphi_{t^{-1}gt}(p)=p$, the phase factor becomes

$\displaystyle t^{-1}gt|p\rangle$
$\displaystyle=U(t,p)U(g,\varphi_{t}(p))\frac{1}{U(t,p)}|\varphi_{t^{-1}gt}(p)\rangle=U(g,\varphi_{t}(p))|p\rangle~{},$
(3.53)

but that is exactly the factor we showed to be equal to $U(g^{\prime},p)$.

We neglected two points in this discussion: the phases associated to the
right-moving fermions, and the ${\mathbb{Z}}_{N}$ action on
${{\mathbb{T}}}^{4}$. As we saw above, the latter does not lead to any
additional phase factors, and the former is identical in both theories.

##  4 Gerbes and duality in type II compactification

We have seen so far that in the context of the heterotic string T-duality can
relate compactifications on $X_{5}$ to more familiar compactifications on
${{\mathbb{T}}}^{4}/{\mathbb{Z}}_{N}\times S^{1}$, but the relationship cannot
be seen at the level of the NS sector and
$\operatorname{O{}}(d,d,{\mathbb{Z}})$ transformations—the Wilson line
background is essential. It is therefore not surprising that a similar
construction is unavailable in type II compactifications: there is no
transformation $t\in\operatorname{O{}}(d,d,{\mathbb{Z}})$ such that (3.13)
holds.

Nevertheless, we can still ask: what is the T-dual of $X_{5}$? We will argue
that the dual geometry is indeed that of
${{\mathbb{T}}}^{4}/{\mathbb{Z}}_{N}\times S^{1}$, but this geometry is
equipped with a choice of a flat $B$-field, which leads to drastic
modifications in the spectrum of the theory.

Some of the backgrounds that we discuss involve $B$-field data that is pure
holonomy: the connection $B=0$, but the gerbe background is nevertheless
topologically non-trivial. To describe these global structures we will need to
introduce some machinery of abelian gerbes, and in our context these arise as
$G$-equivariant topologically trivial gerbes on ${{\mathbb{T}}}^{5}$.

It is not hard to see why such structures should arise naturally in our
context. We are interested in geometries that are presented as quotients
${{\mathbb{T}}}^{5}/G$ and that also have a presentation as a circle fibration

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In each case the fibration is flat, meaning that the transition functions can
be taken to be constant, and the connection set identically to zero.
Nevertheless, the fibration is topologically non-trivial, with holonomy
encoding the topological data. On the other hand, T-duality in general
exchanges the topology of the circle fibration for that of the $B$-field [22],
and we might expect this to be the case in our context as well. We will
develop this picture, but to motivate some of the developments, we will make a
small digression on T-duality with $H$-flux, following [22].

#### Correspondence spaces and vertical gerbes

Suppose $X$ is the total space of a principal circle bundle $\pi_{g}:P_{g}\to
M$. This space has a nowhere vanishing vector field associated to the circle
fiber, which we denote $V$ and a dual form $\Theta_{g}$ that satisfies

$\displaystyle d\Theta_{g}=\pi^{\ast}_{g}(F_{g})~{},$ (4.1)

where $F_{g}$ is the curvature of the circle bundle. An $H$-flux over $X$ is
said to be T-dualizable if $dH=0$ and $V\llcorner H=\pi^{\ast}(F_{b})$ for
some closed $2$-form $F_{b}$. This form is necessarily integral and defines
(up to torsion) another principal circle bundle $\pi_{b}:P_{b}\to M$ with
$1$-form $\Theta_{b}$ satisfying $d\Theta_{b}=\pi^{\ast}_{b}(F_{b})$.

When this is the case, $H$ decomposes as

$\displaystyle
H=\Theta_{g}\wedge\pi^{\ast}_{g}(F_{b})-\pi^{\ast}_{g}(\Omega)~{},$ (4.2)

where $\Omega$ is a $3$-form on $M$ that satisfies $d\Omega=F_{g}\wedge
F_{b}$, and we can also define the T-dual $H$-flux on the total space of
$P_{b}$ by

$\displaystyle\widehat{H}=\Theta_{b}\wedge\pi^{\ast}_{b}(F_{g})-\pi^{\ast}_{b}(\Omega)~{}.$
(4.3)

This $H$-flux is clearly T-dualizable, and it can be argued that the exchange
of fibrations and fluxes implements the action of T-duality on various
generalized (in general twisted) cohomology theories such as K-theory, as well
as reproducing the Buscher rules [22].141414The reference [22] treats the
general case of a principal torus fibration, but for us the circle fibration
will be sufficient.

The correspondence space is the geometric setting for these actions: it is the
space $Z=P_{g}\times_{M}P_{b}$, which fits into the commutative diagram

${Z}$ ${P_{g}}$ ${P_{b}}$ ${M}$
$\scriptstyle{p_{g}}$$\scriptstyle{p_{b}}$$\scriptstyle{\pi_{g}}$$\scriptstyle{\pi_{b}}$
(4.4)

The pullbacks of the two fluxes $p_{g}^{\ast}(H)$ and
$p_{b}^{\ast}(\widehat{H})$ differ by an exact form:
$d\left(p_{b}^{\ast}(\Theta_{b})\wedge p_{g}^{\ast}(\Theta_{g})\right)$.

Turning the construction around, given two principal circle bundles over $M$
with curvatures $F_{g}$ and $F_{b}$ satisfying $[F_{g}\wedge F_{b}]=0\in
H^{4}(M,{\mathbb{R}})$, we can choose $H$-fluxes on each of them that will be
exchanged by T-duality.

A special case of the construction that is of particular importance for us is
to take $H=\Theta_{g}\wedge\pi^{\ast}_{g}(F_{b})$—this is the analogue of what
we will call a “vertical” gerbe. While elegant, this comes at a heavy price:
the curvature forms have to satisfy $F_{g}\wedge F_{b}=0$ pointwise, and not
merely in cohomology.

Ordinarily the discussion of string backgrounds with $H$–flux is complicated
by basic questions about existence of a such a solution in string theory. Our
constructions will not have this difficulty since $H=0$, and the
correspondence space will give an elegant encoding of T-dual theories with
flat principal circle bundles. But we will also see torsional echoes of the
$F_{g}\wedge F_{b}=0$ constraint: not every pair of circle bundles over
$M={{\mathbb{T}}}^{4}/G$ can be a correspondence space. To describe those
results, we will now delve into equivariance.

###  4.1 Equivariant flat line bundles

Let $X$ be a compact smooth manifold with an action by a discrete group $G$,
i.e. there are diffeomorphisms $\varphi_{a}:X\to X$ obeying
$\varphi_{a}(\varphi_{b}(x))=\varphi_{ab}(x)$ for all $a,b\in G$. We are
interested in describing flat line bundles over the quotient $M=X/G$, and this
can accomplished by describing $G$–equivariant line bundles on $L\to X$.

Recall that a line bundle is flat if and only if its curvature is zero. In
this case it is possible to choose a trivialization such that the transition
functions are constant, and the connection is identically zero. Since line
bundles are topologically classified by $c_{1}(L)\in H^{2}(X,{\mathbb{Z}})$,
the first Chern class of a flat line bundle is a torsion class. When
$H^{2}(X,{\mathbb{Z}})$ (or, equivalently, $H_{1}(X,{\mathbb{Z}})$) is
torsion-free, then every flat line bundle is trivial.

A line bundle $\pi:L\to X$ is $G$-equivariant if there is a lift of the
$G$-action on $X$ to a $G$-action on $L$ compatible with the projection $\pi$,
where $G$ acts linearly on the fiber directions. The quotient $L/G$ is then
naturally a line bundle over $M=X/G$. While in general a bundle may not admit
a lift, every trivial bundle does admit a lift. In our setting, where
$X={{\mathbb{T}}}^{d}$, every flat line bundle is trivial: we can always find
a lift of the $G$ action, and the characterization of flat line bundles over
$M$ is then equivalent to the characterization of these lifts.151515This
situation is not special to the torus: for any $M$ we can set $X$ to be its
universal cover, with $G$ the group of deck transformations; in this case we
find the explicit characterization of flat line bundles on $M$ via
representations $\rho:H_{1}(M,{\mathbb{Z}})\to\operatorname{U{}}(1)$[54].

The reader may wish to consult the appendix B.1 for a review of some general
aspects of such constructions, but in this section we will stick to the
concrete setting relevant for our examples: $G={\mathbb{Z}}_{N}$,
$X={{\mathbb{T}}}^{d}$, and $L=X\times{\mathbb{C}}$ is the trivial line bundle
with trivial transition functions and a torus-invariant connection $A$.

We lift the $G$ action on $X$ to a $G$ action on the total space $L$ by
specifying circle-valued functions $R^{a}:X\to S^{1}$ that obey161616As
described in more detail in the appendix B.1, for a general line bundle $L\to
X$ the $R^{a}$ need not be globally defined circle-valued functions, but when
$L$ is trivial, the $R^{a}$ can be taken to be globally defined.

$\displaystyle R^{ba}=R^{a}\varphi_{a}^{\ast}(R^{b})~{}.$ (4.5)

Note that this composition is automatically associative:
$R^{(cb)a}=R^{c(ba)}$.

The $G$-action on a point $(x,\xi)\in X\times{\mathbb{C}}$ is

$\displaystyle\psi$ $\displaystyle:L\to L~{},$
$\displaystyle\psi(x,\xi)=(\varphi_{a}(x),R^{a}(x)\xi)~{}.$ (4.6)

We also demand that the $G$ action preserves the connection, which
requires171717We take our connections to be real.

$\displaystyle\varphi_{a}^{\ast}(A)=A-id\log R^{a}~{}.$ (4.7)

We assume that $A$ is a torus-invariant $1$-form on ${{\mathbb{T}}}^{d}$, and
so we will naturally demand that $d\log R^{a}$ is torus-invariant as well.

When $G={\mathbb{Z}}_{N}$ and $X={{\mathbb{T}}}^{d}$, we can describe the
$R^{a}$ explicitly. Think of
${{\mathbb{T}}}^{d}={\mathbb{R}}^{d}/{\mathbb{Z}}^{d}$ with coordinates
$\vec{x}\sim\vec{x}+\vec{m}$, where $\vec{m}\in{\mathbb{Z}}^{d}$. Let $a$ be
the generator of ${\mathbb{Z}}_{N}$ which acts by a linear transformation
$a:\vec{x}\to\varphi_{a}(\vec{x})$ on the torus coordinates, and denote by
$R^{n}$ the function corresponding to the element $a^{k}$, defined by

$\displaystyle R^{1}$ $\displaystyle=e^{2\pi
i\left({\textstyle\frac{k}{N}}+\vec{k}\cdot\vec{x}\right)}~{},$ $\displaystyle
R^{n}$ $\displaystyle=R^{1}\varphi_{a}^{\ast}\left(R^{n-1}\right)~{},\quad
k=2,\ldots,N-1~{}.$ (4.8)

These are circle-valued functions for $\vec{k}\in{\mathbb{Z}}^{d}$, and when
$k\in{\mathbb{Z}}$ we find $R^{N}=1$, so that this indeed gives a
representation of ${\mathbb{Z}}_{N}$.

There is a further equivalence on these lifts, because given a circle-valued
function $h:X\to S^{1}$ we can make a global gauge transformation which acts
by

$\displaystyle R^{1}\to R^{1}\frac{\varphi_{a}^{\ast}(h)}{h}~{},$ (4.9)

and choosing $h=e^{2\pi i\vec{r}\cdot\vec{x}}$, we find an equivalence
$\vec{k}\sim\vec{k}+M\vec{r}$, where

$\displaystyle M=(\varphi_{a}^{T}-{\mathbbm{1}})~{}.$ (4.10)

A constant equivariant connection that is consistent with this action is

$\displaystyle A_{\vec{k}}=2\pi(M^{-1}\vec{k})\cdot d\vec{x}~{}.$ (4.11)

Thus, the gauge-inequivalent lifts of the ${\mathbb{Z}}_{N}$ action on
${{\mathbb{T}}}^{d}$ are characterized by $k\in\\{0,1,\ldots,N-1\\}$ and
$\vec{k}\in{\mathbb{Z}}^{d}/\operatorname{im}M$. It is not hard to show that
these are in 1:1 correspondence with representations
$\rho:H_{1}({{\mathbb{T}}}^{d}/G,{\mathbb{Z}})\to\operatorname{U{}}(1)$, which
topologically characterize flat line bundles on the quotient
$M={{\mathbb{T}}}^{d}/G$ [54], and this shows that the line bundles on the
quotient are in general non-trivial: the first Chern class, while torsion, is
non-zero.

The total space of the associated circle bundle $P_{g}\to M$ is then also easy
to understand: it is the quotient of ${{\mathbb{T}}}^{d+1}$ by the
${\mathbb{Z}}_{N}$ action generated by

$\displaystyle
a(\vec{x},x_{d+1})=\left(\varphi_{a}(\vec{x}),x_{d+1}+{\textstyle\frac{k}{N}}+\vec{k}\cdot\vec{x}\right)~{},$
(4.12)

with

$\displaystyle k$ $\displaystyle\in\\{0,1,\ldots,N-1\\}~{},$
$\displaystyle\vec{k}$
$\displaystyle\in{\mathbb{Z}}^{d}/\operatorname{im}M~{}.$ (4.13)

Note that the compact flat manifolds described above arise naturally in this
setting: taking $d=4$ and setting $\vec{k}=0$ and $k=1$, we obtain exactly the
compact flat manifolds $X_{5}$. More generally, we will denote these spaces by
$X_{k,\vec{k}}$ . We also note that this same classification of equivariant
line bundles was obtained in [15] by calculating equivariant cohomology
groups.

#### (Orbifold) diffeomorphism types of $X_{k,\vec{k}}$

Although each choice of $k$ and $\vec{k}$ as above yields distinct circle
bundles over ${{\mathbb{T}}}^{4}/{\mathbb{Z}}_{N}$, Not all possible choices
of $k,\vec{k}$ give distinct geometries for the total space.181818We say that
two orbifolds $X/G_{1}$ and $X/G_{2}$ are diffeomorphic if their $G$–actions
are related by a diffeomorphism of $X$.

As an example, consider the ${\mathbb{Z}}_{2}$ quotient, where
$\varphi_{a}(\vec{x})=-\vec{x}$, and we can take the parameters
$k\in\\{0,1\\}$, and $k_{i}\in\\{0,1\\}$. To characterize the inequivalent
orbifolds we can make changes of coordinates on ${{\mathbb{T}}}^{5}$ as
follows.

Suppose $\vec{k}\neq 0$. Then, because $k_{i}\in\\{0,1\\}$, there is a
$\operatorname{SL}(4,{\mathbb{Z}})$ change of coordinates $\vec{y}=A\vec{x}$,
such that $y_{4}=\vec{k}\cdot\vec{x}$. So, without loss of generality we set
$\vec{k}\cdot\vec{x}=x_{4}$. Now define a new coordinate
$y_{4}=x_{4}+{\textstyle\frac{k}{2}}$. Since $a(y_{4})=-y_{4}+2k\sim-y_{4}$,
the quotient is equivalent to the action

$\displaystyle a(\vec{x},x_{5})=(-\vec{x},x_{5}+x_{4})~{}.$ (4.14)

Combining this with the remaining possibilities, we obtain $3$ diffeomorphism
classes for the ${\mathbb{Z}}_{2}$ quotients, represented by

orbifold quotient action $\displaystyle
X_{0,0}={{\mathbb{T}}}^{4}/{\mathbb{Z}}_{2}\times S^{1}$
$\displaystyle(-\vec{x},x_{5})$ $\displaystyle X_{1,0}~{}\text{(smooth)}$
$\displaystyle(-\vec{x},x_{5}+{\textstyle\frac{1}{2}})$ $\displaystyle
X_{0,\vec{k}}$ $\displaystyle(-\vec{x},x_{5}+x_{4})$ (4.15)

A similar treatment can be given for the remaining ${\mathbb{Z}}_{N}$ actions,
with similar results. We relegate the full list to the appendix and here just
summarize the key features.

  1. 1.

For $N=3,4$ there are the analogous $3$ classes to the ones we found for
${\mathbb{Z}}_{2}$, with the replacement
${\textstyle\frac{1}{2}}\to{\textstyle\frac{1}{N}}$.

  2. 2.

For ${\mathbb{Z}}_{4}$ there is an additional class $X_{2,0}$.

  3. 3.

For ${\mathbb{Z}}_{6}$ it is possible to set $\vec{k}=0$ by a change of
coordinates, and there are just four inequivalent quotients with $0\leq k\leq
3$:

orbifold quotient action $\displaystyle X_{k,0}$
$\displaystyle(\varphi_{a}(\vec{x}),x_{5}+{\textstyle\frac{k}{6}})~{}.$ (4.16)

These results are consistent with the orbifold classification [5].

#### Invariant form associated to circle isometry

As we saw above, the discussion of T-duality on a circle bundle $P_{g}$
involves the nowhere vanishing $1$-form $\Theta$. In our construction a
natural guess is to set $\Theta=dx_{d+1}$, but when $\vec{k}\neq 0$ the
differential $dx_{d+1}$ is not $G$–invariant. We can compensate for its
transformation by using the connection $A_{\vec{k}}$:

$\displaystyle\Theta=dx_{d+1}-{\textstyle\frac{1}{2\pi}}A_{\vec{k}}~{}.$
(4.17)

$\Theta$ is not necessarily an integral form on ${{\mathbb{T}}}^{d+1}$.
Denoting a homology cycle $\gamma\in H_{1}({{\mathbb{T}}}^{d+1},{\mathbb{Z}})$
by $(\vec{m},m)$, we find

$\displaystyle\int_{\gamma}\Theta_{d+1}=m-\vec{m}\cdot M^{-1}\cdot\vec{k}~{}.$
(4.18)

###  4.2 Vertical equivariant flat gerbes

Just as $G$–equivariant line bundles on $X$ can be used to describe line
bundles on the quotient $M=X/G$, $G$–equivariant gerbes on $X$ describe gerbes
on $M$, and the data that specifies the lift of the $G$–action to the gerbe
enters the computation of the partition function for the orbifold CFT
associated to the quotient [26, 24]. We provide a detailed view of the
construction in appendix B.2, while in this section we describe the gerbes on
$X$ and the $G$–action relevant to our examples.

In a nutshell, the $G$–action on a gerbe over $X$ is encoded in a set of gerbe
gauge transformations, one for each element $a\in G$. When
$H^{2}(X,{\mathbb{Z}})$ is torsion-free and the gerbe is topologically trivial
and flat, meaning $H=dB=0$, the relevant gerbe gauge transformations reduce to
a choice of line bundles ${\cal L}^{a}\to X$ equipped with connections ${\cal
A}^{a}$ such that ${\cal L}^{b}\otimes\varphi_{b}^{\ast}({\cal
L}^{a})\simeq{\cal L}^{ab}$, and

$\displaystyle\varphi_{a}^{\ast}(B)$ $\displaystyle=B+d{\cal A}^{a}~{},$
$\displaystyle{\cal A}^{ab}$ $\displaystyle={\cal
A}^{b}+\varphi_{b}^{\ast}({\cal A}^{a})+id\log h^{a,b}~{}.$ (4.19)

The $h^{a,b}$ are gauge transformations that realize the bundle isomorphism,
and associativity of the group product requires these to satisfy

$\displaystyle\varphi_{c}^{\ast}(h^{a,b})h^{ab,c}=h^{a,bc}h^{b,c}~{}.$ (4.20)

Given any solution to these constraints and a representative $\lambda^{a,b}$
of a class in $H^{2}(G,\operatorname{U{}}(1))$, we can obtain another solution
by setting $h^{a,b}_{\text{new}}=\lambda^{a,b}h^{a,b}$. On the other hand,
multiplying $h^{a,b}$ by a group coboundary can be absorbed into gauge
transformations on each of the bundles ${\cal L}^{a}$ and gives the same
$G$–action on the gerbe. This choice of $[\lambda]\in
H^{2}(G,\operatorname{U{}}(1))$ is the discrete torsion of [25]. Since
$H^{2}({\mathbb{Z}}_{N},\operatorname{U{}}(1))=1$ this degree of freedom is
absent in all of our constructions. However, as emphasized in [26],
$G$–equivariance allows for additional topological data associated to the
holonomy of the gerbe.

We are interested in a particular class of gerbes defined by the following
data: a manifold $Y=X\times S^{1}$ equipped with a closed $1$-form $\Theta$
and a flat line bundle that is the pull-back of a bundle $L\to X$ with flat
connection $\Lambda$, which is itself a global $1$-form on $X$.191919Recall
that every flat line bundle admits a trivialization with constant transition
functions, and the connections compatible with this choice are global
$1$-forms on $X$ In this case, we obtain a flat “vertical” gerbe by setting

$\displaystyle B=\Lambda\wedge\Theta~{}.$ (4.21)

Now suppose further that there is a $G$ action on $Y$ such that $\Theta$ is
$G$–invariant, while $L$ is $G$–equivariant with the lift described by a
choice of phases $\widetilde{R}^{a}$. In this case the vertical gerbe may
admit a $G$–action as well. To explore when this is the case, we simply
compute

$\displaystyle\varphi_{a}^{\ast}(B)-B=d\left(-i\log(\widetilde{R}^{a})\Theta\right)~{},$
(4.22)

so that a naive solution to (4.19) is to set

$\displaystyle{\cal A}^{a}=-i\log(\widetilde{R}^{a})\Theta~{}.$ (4.23)

There are three reasons why this construction is naive. First, the ${\cal
A}^{a}$ so defined may not be a connection on a line bundle ${\cal L}^{a}$
because ${\textstyle\frac{1}{2\pi}}d{\cal A}^{a}$ is not properly quantized.
Second, it may not be possible to find $h^{a,b}$ to satisfy (4.19) and (4.20).

Turning to our specific interest with $Y={{\mathbb{T}}}^{d+1}$,
$G={\mathbb{Z}}_{N}$ acting as in (4.12), and $\Theta$ as in (4.17), we
set—$\beta$ and $\vec{\beta}$ are parameters analogous to $k$ and $\vec{k}$—

$\displaystyle\widetilde{R}^{1}$ $\displaystyle=e^{2\pi
i\left({\textstyle\frac{\beta}{N}}+\vec{\beta}\cdot\vec{x}\right)}~{},$
$\displaystyle\Lambda$ $\displaystyle=2\pi(M^{-1}\vec{\beta})\cdot
d\vec{x}~{},$ (4.24)

and find

$\displaystyle{\cal A}^{1}$
$\displaystyle=2\pi\left({\textstyle\frac{\beta}{N}}+\vec{\beta}\cdot\vec{x}\right)\left(dx_{d+1}-(M^{-1}\vec{k})\cdot
d\vec{x}\right)~{}.$ (4.25)

Because $\Theta$ is not an integrally quantized form, the requirement that
${\textstyle\frac{1}{2\pi}}d{\cal A}^{1}$ is an integral form on
${{\mathbb{T}}}^{d}$ is non-trivial. Up to gauge transformations of the bundle
$L$, we find that the only solutions are: $\vec{\beta}=0$ or $\vec{k}=0$ (any
$N$), or $N=2$ and $\vec{\beta}=\vec{k}$. For any such solutions we have

$\displaystyle{\textstyle\frac{1}{2\pi}}d{\cal A}^{1}$
$\displaystyle=\vec{\beta}\cdot d\vec{x}\wedge dx_{d+1}~{}.$ (4.26)

To explore the remaining consistency conditions, we will denote by ${\cal
A}^{p}$ the connection corresponding to the group element $a^{p}$, with $0\leq
p<N$, and similarly for the diffeomorphism $\varphi_{p}$. Writing a sum of two
such integers as $p_{1}+p_{2}=m_{1}+m_{2}N$, with $0\leq m_{1}<N$, consistency
with the group law (4.19) requires

$\displaystyle{\cal A}^{m_{1}}-{\cal A}^{p_{1}}-\varphi^{\ast}_{p_{1}}({\cal
A}^{p_{2}})=id\log h^{p_{1},p_{2}}~{}.$ (4.27)

Using the explicit form of the ${\cal A}^{p}$ we find

$\displaystyle{\cal A}^{m_{1}}-{\cal A}^{p_{1}}-\varphi^{\ast}_{p_{1}}({\cal
A}^{p_{2}})=-2\pi m_{2}\beta\Theta=im_{2}d\log f~{},$ (4.28)

where

$\displaystyle f=\exp\left[-2\pi
i\beta\left(x_{d+1}-(M^{-1}\vec{k})\cdot\vec{x}\right)\right]~{}.$ (4.29)

To describe the $h^{p_{1},p_{2}}$, we define

$\displaystyle\sigma(p)$ $\displaystyle=\begin{cases}0~{},&p<N~{};\\\
1~{},&p\geq N~{}.\end{cases}~{},$ $\displaystyle\widetilde{h}(p)$
$\displaystyle=f^{\sigma(p)}~{},$ (4.30)

as well as a remainder $\overline{p}$ such that $\overline{p}=p\mod N$, and
$0\leq\overline{p}<N$. With that preparation, we can now write the general
solution for the $h^{p_{1},p_{2}}$ as

$\displaystyle h^{p_{1},p_{2}}=\widetilde{h}(p_{1}+p_{2})c^{p_{1},p_{2}}~{},$
(4.31)

where the $c^{p_{1},p_{2}}$ are constants. With this form the constraint
(4.20) is

$\displaystyle\frac{\varphi^{\ast}_{p_{3}}(\widetilde{h}(p_{1}+p_{2}))\widetilde{h}(\overline{p_{1}+p_{2}}+p_{3})}{\widetilde{h}(p_{1}+\overline{p_{2}+p_{3}})\widetilde{h}(p_{2}+p_{3})}=\frac{c^{p_{1},\overline{p_{2}+p_{3}}}c^{p_{2},p_{3}}}{c^{p_{1},p_{2}}c^{\overline{p_{1}+p_{2}},p_{3}}}~{}.$
(4.32)

Using the explicit form of $f$, we find

$\displaystyle\varphi_{p_{3}}^{\ast}(\widetilde{h}(p))=\exp\left[-2\pi
i{\textstyle\frac{\beta k}{N}}p_{3}\sigma(p)\right]\widetilde{h}(p)~{},$
(4.33)

so that (4.20) requires

$\displaystyle\frac{\widetilde{h}(p_{1}+p_{2})\widetilde{h}(\overline{p_{1}+p_{2}}+p_{3})}{\widetilde{h}(p_{1}+\overline{p_{2}+p_{3}})\widetilde{h}(p_{2}+p_{3})}=\exp\left[2\pi
i{\textstyle\frac{\beta
k}{N}}\sigma(p_{1}+p_{2})p_{3}\right]\frac{c^{p_{1},\overline{p_{2}+p_{3}}}c^{p_{2},p_{3}}}{c^{p_{1},p_{2}}c^{\overline{p_{1}+p_{2}},p_{3}}}~{}.$
(4.34)

The reader can check that the left-hand-side is not only $x$-independent, but
in fact equals $1$. It is also possible to show that the phase $\exp\left[2\pi
i{\textstyle\frac{\beta k}{N}}\sigma(p_{1}+p_{2})p_{3}\right]$ gives a non-
trivial cocycle in
$H^{3}({\mathbb{Z}}_{N},\operatorname{U{}}(1))\simeq{\mathbb{Z}}_{N}$ whenever
$\beta k\neq 0\mod N$, in which case it is not possible to choose the
$c^{p_{1},p_{2}}$ to satisfy the consistency conditions.202020This is easy to
see explicitly for $N=2$: setting $p_{1}=p_{3}=1$ and $p_{2}=0$, we find the
requirement $c^{0,1}=c^{1,0}$, while taking $p_{1}=p_{2}=p_{3}=1$ leads to
$e^{i\pi\beta k}c^{1,0}=c^{0,1}$.  On the other hand, when $\beta k=0\mod N$,
$c^{p_{1},p_{2}}$ must be a cocycle, and since
$H^{2}({\mathbb{Z}}_{N},\operatorname{U{}}(1))=1$, it must in fact be a group
coboundary, so that we can set $c^{p_{1},p_{2}}=1$ by constant gauge
transformations on the individual bundles ${\cal L}^{a}$.

Finally, we should take care of an ambiguity in our description: the
parameters $(k,\vec{k})$ and $(\beta,\vec{\beta})$ are not uniquely defined.
For any $\vec{r}\in{\mathbb{Z}}^{4}$ there are changes of coordinates on
$T^{4}$

$\displaystyle\vec{x}\to\vec{y}$
$\displaystyle=\vec{x}+\widetilde{M}^{-1}\vec{r}~{},$
$\displaystyle~{}\widetilde{M}$ $\displaystyle=\varphi_{a}-{\mathbbm{1}}~{},$
(4.35)

which preserve the ${\mathbb{Z}}_{N}$ action on ${{\mathbb{T}}}^{4}$, i.e.
$a(\vec{y})=\varphi_{a}(\vec{y})+\vec{r}\sim\varphi_{a}(\vec{y})$ and lead to
shifts in the parameters $\beta$ and $k$:

$\displaystyle k$ $\displaystyle\to
k+N\vec{k}\cdot\widetilde{M}^{-1}\vec{r}~{},$ $\displaystyle\beta$
$\displaystyle\to\beta+N\vec{\beta}\cdot\widetilde{M}^{-1}\vec{r}~{},$ (4.36)

and these shifts do not in general preserve $\beta k\mod N$. Since the
consistency of the equivariant gerbe should not depend on which representative
we choose, we demand that the condition $\beta k=0\mod N$ is preserved for all
$\vec{r}\in{\mathbb{Z}}^{4}$.

#### Summary: vertical flat ${\mathbb{Z}}_{N}$–equivariant gerbes on
${{\mathbb{T}}}^{5}$

With these results in hand, we can now use the language of correspondence
spaces to describe the possible ${\mathbb{Z}}_{N}$–equivariant flat vertical
gerbes on ${{\mathbb{T}}}^{5}$ with ${\mathbb{Z}}_{N}$ action generated by
(4.12). Each correspondence space $Z_{k,\vec{k};\beta,\vec{\beta}}$ is a
double circle fibration over ${{\mathbb{T}}}^{d}/{\mathbb{Z}}_{N}$ with action
generated by

$\displaystyle
a(\vec{x},x_{5},x_{6})=(\varphi_{a}(\vec{x}),x_{5}+{\textstyle\frac{k}{N}}+\vec{k}\cdot\vec{x},x_{6}+{\textstyle\frac{\beta}{N}}+\vec{\beta}\cdot\vec{x})~{}.$
(4.37)

subject to conditions

  1. 1.

$(\beta+N\vec{\beta}\cdot\widetilde{M}^{-1}\vec{r})(k+N\vec{k}\cdot\widetilde{M}^{-1}\vec{r})=0\mod
N$ for all $\vec{r}\in{\mathbb{Z}}^{4}$ ;

  2. 2.

$\vec{k}=0$ or $\vec{\beta}=0$ for $N\geq 2$, or $\vec{\beta}=\vec{k}\neq 0$
for $N=2$.

There is a geometric perspective on these consistency conditions. The
${\mathbb{Z}}_{N}$–equivariant gerbe on ${{\mathbb{T}}}^{5}$ defines a gerbe
on the quotient $X={{\mathbb{T}}}^{5}/{\mathbb{Z}}_{N}$, and for flat gerbes
this amounts to an assignment of holonomies to homology $2$-cycles in $X$,
subject to the requirement that any two homologous cycles must be assigned the
same holonomy. Using the computations of holonomies assigned to cycles in
${{\mathbb{T}}}^{4}/{\mathbb{Z}}_{N}$ by equivariant line bundles given in
[55], we checked that our two consistency conditions are the “upstairs” view
of this requirement.

We can also understand the conditions from the point of view of the orbifold
CFT. As we will see shortly, and as is discussed in greater length in [24] and
reviewed in detail in the appendix, a choice of gerbe amounts to dressing the
orbifold twisted sectors with additional phases, which are constrained by
unitarity, modular invariance, and factorization properties of the CFT [25].

###  4.3 T-duality

Having described a class of ${\mathbb{Z}}_{N}$–equivariant flat gerbes on
${{\mathbb{T}}}^{5}$, we return to our original question: what is the T-dual
of type II compactification on $X_{k,\vec{k}}$? The answer is captured by the
correspondence space. In the previous section we constructed what one might
call “T-dualizable” flat gerbes on ${\cal G}_{\beta,\vec{\beta}}\to
X_{k,\vec{k}}$, and the geometries fit into the correspondence space

${Z_{k,\vec{k};\beta,\vec{\beta}}}$ ${X_{k,\vec{k}}}$
${X_{\beta,\vec{\beta}}}$ ${{{\mathbb{T}}}^{4}/{\mathbb{Z}}_{N}}$
$\scriptstyle{p_{g}}$$\scriptstyle{p_{b}}$$\scriptstyle{\pi_{g}}$T-dual$\scriptstyle{\pi_{b}}$
(4.38)

As we reviewed above, a key result in the study of T-duality with $H$–flux is
that $H$ is T-dualizable if and only if the dual flux $\widehat{H}$ is
T-dualizable [22]. The consistency conditions on the gerbes are also symmetric
under the exchange $(k,\vec{k})\leftrightarrow(\beta,\vec{\beta})$.

So, we find that the smooth geometry $X_{5}=X_{1,0}$ with trivial gerbe is
T-dual to the singular geometry ${{\mathbb{T}}}^{4}/{\mathbb{Z}}_{N}\times
S^{1}$ equipped with a flat vertical gerbe with a
${\mathbb{Z}}_{N}$–equivariant description on ${{\mathbb{T}}}^{5}$ with
$\beta=1$ and $\vec{\beta}=0$.

Similarly, if we start with $X_{0,0}={{\mathbb{T}}}^{4}/{\mathbb{Z}}_{N}\times
S^{1}$ but now turn on a gerbe with $\beta=0$ and $\vec{\beta}\neq 0$, the
T-dual geometry is $X_{0,\vec{\beta}}$ with trivial gerbe. While both spaces
are singular, the singularity structure is quite different. For example, in
the $N=2$ case the singular locus of $X_{0,0}$ consists of $(\vec{x},x_{5})$,
where $x_{1,2,3,4}\in\\{0,{\textstyle\frac{1}{2}}\\}$, while $x_{5}$ is
arbitrary. On the other hand, the singular locus for $X_{0,\vec{\beta}}$
requires that in addition $x_{4}=0$: instead of $16$ singular circles, the are
just $8$.

###  4.4 Equivariant gerbes and orbifold CFT

So far we have defined a class of gerbes and used the correspondence space to
suggest a class of T-dual orbifold geometries. In this section we will connect
that perspective with an explicit worldsheet computation.

The notion that equivariant gerbes enter the construction of the orbifold
partition function was developed in [26, 24], and the second reference showed
that pure holonomy equivariant gerbes with $B=0$ produce the kinds of shift
orbifold factors that are necessary to relate the $X_{5}$ and
${{\mathbb{T}}}^{4}/{\mathbb{Z}}_{N}\times S^{1}$ orbifolds. We will now
review those ideas and illustrate in an example that they do lead to the
expected phases.

Let $Z_{0,0}$ denote the genus $1$ partition function of a CFT which admits
the action of a discrete abelian group $G$ with $0$ the additive identity
element. The orbifold partition function then takes the familiar form

$\displaystyle{\cal Z}$ $\displaystyle=\frac{1}{|G|}\sum_{a,b\in
G}Z_{a,b}~{},$ $\displaystyle Z_{a,b}$ $\displaystyle=\operatorname{Tr}_{{\cal
H}_{b}}\rho_{b}(a){\textbf{q}}^{L_{0}-c/24}{\overline{{\textbf{q}}}}^{\overline{L}_{0}-\overline{c}/24}~{},$
(4.39)

where $\rho_{b}(a)$ is a representation of the $G$-action on the Hilbert space
${\cal H}_{b}$, i.e. the space of states twisted by the element $b$. In
general the choice of representation $\rho_{b}$ is not unique. For example, if
$\rho_{b}(a)$ leads to a consistent choice of modular-invariant and unitary
partition function, then so does

$\displaystyle\widetilde{\rho}_{b}(a)=\rho_{b}(a)\frac{\omega^{a,b}}{\omega^{b,a}}$
(4.40)

for any representative of a class $[\omega]\in
H^{2}(G,\operatorname{U{}}(1))$—this is the discrete torsion of [25].

The observation from [26] is that the equivariant gerbe data for a
$G={\mathbb{Z}}_{N}$ quotient enters the choice of $\rho_{b}(a)$ through the
pull-back of the holonomy of the gerbe to the string worldsheet:

$\displaystyle
P_{a,b}=\exp\left[i\int\Phi^{\ast}(B)\right]\exp\left[i\int_{0}^{1}\Phi^{\ast}({\cal
A}^{b})-i\int_{0}^{\tau}\Phi^{\ast}({\cal
A}^{a})\right]\frac{h^{a,b}(x)}{h^{b,a}(x)}~{}$ (4.41)

in the Lagrangian computation of $Z_{a,b}$. Here $\Phi:T^{2}\to X$ is the map
from the worldsheet torus with complex structure $\tau=\tau_{1}+i\tau_{2}$ to
the targetspace with

$\displaystyle\Phi(0)$ $\displaystyle=x~{},$ $\displaystyle\Phi(1)$
$\displaystyle=\varphi_{b}(x)~{},$ $\displaystyle\Phi(\tau)$
$\displaystyle=\varphi_{a}(x)~{},$ $\displaystyle\Phi(1+\tau)$
$\displaystyle=\varphi_{ab}(x)~{}.$ (4.42)

The integration contours in the $z$ plane of the worldsheet torus are along
the left and bottom boundaries of the parallelogram:

$0$$1$$\tau+1$$\tau$$z$ (4.43)

The expression $P_{a,b}$ is not easy to interpret on a general targetspace.
However, our interest is in $X={{\mathbb{T}}}^{d}/{\mathbb{Z}}_{N}\times
S^{1}$, where the interpretation is straightforward in the situation when
$B=0$. Focusing on the bosonic part of the CFT, the original theory’s
configuration space decomposes into sectors labeled by their periodicities
under shifts of $z\to z+1$ and $z\to z+\tau$. The $x_{5}$ field corresponding
to $S^{1}$, which is neutral under the orbifold action, has configurations

$\displaystyle x_{5}(z,\overline{z})$
$\displaystyle={\textstyle\frac{i}{2\tau_{2}}}\left(w-\widetilde{n}\overline{\tau}\right)z+{\textstyle\frac{i}{2\tau_{2}}}\left(-w+\widetilde{n}\tau\right)\overline{z}~{}.$
(4.44)

While $w$ has the interpretation as the winding quantum number,
$\widetilde{n}$ is not the momentum mode $n$ of the Hamiltonian
formulation—the two are related through a Poisson resummation. In constructing
the path integral of $Z_{a,b}$ we must also consider fluctuations around this
classical solution, but those do not enter the phase factor $P_{a,b}$.

In our examples above, the gerbes with $B=0$ have ${\cal
A}^{a}={\textstyle\frac{2\pi a}{N}}dx^{5}$ and $h^{a,b}=h^{b,a}$, which leads
to phases

$\displaystyle P_{a,b}=\exp\left[{\textstyle\frac{2\pi
i}{N}}\left(aw-b\widetilde{n}\right)\right]~{}.$ (4.45)

The contribution to the circle partition function can be evaluated by
including the integration over the fluctuations around the semi-classical
solution [44]. Including the phase $P_{a,b}$ the result is (our conventions
are given in appendix A)

$\displaystyle
Z^{\text{cir}}_{a,b}={\textstyle\frac{1}{\eta\overline{\eta}}}\sum_{w}e^{2\pi
i{\textstyle\frac{aw}{N}}}{\textstyle\frac{\sqrt{\tau_{2}}}{r}}\sum_{\widetilde{n}}\exp\left[-S_{0}(\widetilde{n},w,\tau,r)-2\pi
i{\textstyle\frac{b}{N}}\widetilde{n}\right]~{},$ (4.46)

where $S_{0}$ is the classical action evaluated on the solution (4.44). Making
a Poisson resummation on $\widetilde{n}$, we therefore obtain

$\displaystyle
Z^{\text{cir}}_{a,b}={\textstyle\frac{1}{\eta\overline{\eta}}}\sum_{w}e^{2\pi
i{\textstyle\frac{aw}{N}}}{\textstyle\frac{\sqrt{\tau_{2}}}{r}}\sum_{n}\int_{-\infty}^{\infty}dx\exp\left[-S_{0}(x,w,\tau,r)-2\pi
i{\textstyle\frac{b}{N}}x-2\pi ixn\right]~{}.$ (4.47)

When $b=0$, this leads to the usual expression for the circle partition
function with an extra phase:

$\displaystyle
Z^{\text{cir}}_{a,0}={\textstyle\frac{1}{\eta\overline{\eta}}}\sum_{n,w}e^{2\pi
i{\textstyle\frac{aw}{N}}}{\textbf{q}}^{h_{n,w}}{\overline{{\textbf{q}}}}^{\overline{h}_{n,w}}~{},$
(4.48)

but now the form of the $b$-dependence under the $x$ integral shows that more
generally including $b$ just leads to the shift $n\to n+b/N$ in the weights
$h_{n,w}$ and $\overline{h}_{n,w}$. But this is precisely the partition
function of the T-dual of the shift orbifold of the circle! Since the
remaining degrees of freedom, i.e. the fermions and the bosons of the
${{\mathbb{T}}}^{d}$ theory, are treated in the same way in both the $X_{5}$
and ${{\mathbb{T}}}^{d}/{\mathbb{Z}}_{N}\times S^{1}$ orbifolds, the gerbe
data and the associated phase correctly reproduce the T-dual of the
${\mathbb{Z}}_{N}$ quotient that leads to the smooth $X_{5}$ geometries.

It would be interesting to generalize this treatment to the other pure
holonomy equivariant gerbes that appear in the T-dualities, i.e. the ones with
$B\neq 0$. On one hand, it would provide a detailed test of the dualities
constructed via the correspondence space analysis. On the other hand, such a
generalization may shed light on the precise meaning of (4.41) in
compactifications that go beyond the simple toroidal examples considered here.
We leave such a study for future work.

#### The circle and “non-geometry”

Much like the inexhaustible electron [56], the compact boson is a seemingly
perpetual source of insights and lessons, and we end this section by
specializing our analysis to it.

The compact boson CFT admits various orbifold actions, and some of these, such
as the shift orbifold which acts by the phase $e^{2\pi in/N}$ or the
reflection orbifold, have a straightforward geometric interpretation: for
example, the former is a quotient that leads to a circle of radius $r/N$. On
the other hand, there are also orbifolds that seemingly do not have such a
straightforward interpretation, such as the ${\mathbb{Z}}_{N}$ orbifold that
acts by the phase $e^{2\pi iw/N}$. If we are willing to include the gerbe data
as part of our geometric information, then we can assign a geometric meaning
to this phase: it corresponds to turning on a gerbe on $S^{1}$!

Applying this logic to the compact boson CFT is surely an indication that one
has spent too much time with gerbes, but the example has an important lesson
for more elaborate theories: the gerbe data should be included in any
discussion of “non-geometric” features of string compactification.

###  4.5 Frozen singularities and a dual perspective

The geometry ${{\mathbb{T}}}^{4}/G\times S^{1}$ is familiar to every string
theorist as a limit of compactification on $\text{K3}\times S^{1}$. The
resulting spacetime physics depends on the way in which the limit is taken.
For example, if we focus on type II compactification, then the choice of
$B$-field determines whether the limiting theory will lead to additional
massless gauge degrees of freedom (in IIA) or tensionless strings (in IIB)
[57, 36, 58]. These arise at points in the moduli space where the CFT breaks
down, and string non-perturbative effects are crucial. On the other hand,
there is a choice of $B$-field such that the limit leads precisely to the
orbifold CFT for ${{\mathbb{T}}}^{4}/G$. In each of these cases it is possible
to perturb the moduli to a more generic point and arrive at a smooth K3
geometry with some small cycles. From the point of view of the orbifold CFT
these deformations correspond to massless spacetime fields that arise from the
twisted sectors.

As we have seen, in $5$–dimensional compactification there is a new
possibility: by turning on a flat vertical gerbe on the $\text{K3}\times
S^{1}$ geometry, we obtain a theory without moduli associated to any of the
blow-up modes. The disappearance of the moduli is easy to understand in the
T-dual picture, since the dual geometry is smooth, and every twisted sector is
massive. In the original formulation the singularity is frozen by the holonomy
of the gerbe.

We can understand this as follows in terms of the local structure of the
geometry.212121In this discussion we stick to the equivariant gerbe with
$B=0$. The equivariant gerbe implies that the orbifold geometry
${{\mathbb{T}}}^{4}/G\times S^{1}$ carries non-trivial gerbe holonomy: a curve
$\widetilde{\gamma}$ in ${{\mathbb{T}}}^{4}$ with $\widetilde{\gamma}(0)=x$
and $\widetilde{\gamma}(1)=\varphi_{a}(x)$ projects to a closed curve
$\gamma$, and the cycle $\gamma\times S^{1}$ then carries a gerbe holonomy