lattice vectors parametrizing ${\cal T}$, the T-dual $G^{\prime}$ action mixes
the $T^{2}$ and gauge degrees of freedom. However, if the
${\boldsymbol{\kappa}}_{I}$ are such that $R({\boldsymbol{\kappa}}_{I})={\cal
R}^{J}{}_{I}{\boldsymbol{\kappa}}_{J}$, we are led to the much simpler T-dual
action

$\displaystyle g^{\prime}_{k}=g_{\text{t}}[{\cal
R}^{\prime}]g_{\text{g}}[R]~{}.$ (5.36)

This form of $g^{\prime}_{k}$ now has the exact same geometric interpretation
as $g_{k}$: a $\operatorname{SL}(2,{\mathbb{Z}})$ isometry of the torus by
${\cal R}^{\prime}={\cal R}^{-t}$ accompanied by a global rotation $R$ in the
gauge sector. However, recall that the $G$ action on vertex operators also
involves a phase $U(g_{k},{\boldsymbol{p}})$. In the T-dual description, the
symmetry group $G^{\prime}$ acts with a different phase
$U(g^{\prime}_{k},{\boldsymbol{p}})$.191919Details on the computation of this
$U(g^{\prime}_{k},{\boldsymbol{p}})$ are spelled out in appendix A.4. It might
be that the resulting $G^{\prime}$ action does not have a simple geometric
interpretation but involves, for example, an action on winding modes,
similarly to what we found for CHL quotients in section 4.

###  5.4 Examples

We now examine some examples of quotient geometries that preserve $4$
supercharges.

#### Smooth ${\mathbb{Z}}_{2}$ quotients

Let us first focus on ${\mathbb{Z}}_{2}$ orbifolds. The construction starts
from a K3 base $M$ tuned to the Enriques locus. The Enriques involution
$\sigma_{2}$ acts freely on $M$, and the holomorphic $(2,0)$ form transforms
as $\sigma_{2}^{\ast}\Omega=-\Omega$. We can lift the Enriques involution to a
$G\simeq{\mathbb{Z}}_{2}$ action on the configuration $X_{v}\to M$ using the
$\operatorname{O{}}(\Gamma_{2,18})$ element202020In the above orbifold set up,
the reflection $g_{\text{ref}}$ corresponds to the
$\operatorname{SL}(2,{\mathbb{Z}})$ action
$g_{\text{ref}}=g_{\text{t}}[-1_{2}]$.

$\displaystyle g_{2}=g_{\text{ref}}\,g_{\text{flip}}~{}.$ (5.37)

In other words, the action on $X$ combines the Enriques involution on the base
with a reflection on both $T^{2}$ coordinates and an exchange
$g_{\text{flip}}=g_{\text{g}}[R_{\text{flip}}]$ of the two
$\operatorname{E}_{8}$ factors of the gauge bundle. As explained above, this
${\mathbb{Z}}_{2}$ lift is compatible with the fibration structure of $X_{v}$
if the vector $v$ satisfies the condition (5.33). Here this condition amounts
to $\sigma_{2}^{\ast}v=g_{\text{ref}}\,g_{\text{flip}}v$. We can solve this
requirement in the following way. Recall the Enriques action on the second
integral cohomology of $M$:

$\displaystyle\Gamma_{\text{inv}}$
$\displaystyle=(2\Gamma_{1,1})+(2\Gamma_{8})~{},$
$\displaystyle(\Gamma_{\text{inv}})^{\perp}$
$\displaystyle=\Gamma_{1,1}+(2\Gamma_{1,1})+(2\Gamma_{8})~{},$ (5.38)

where the two sublattices correspond to even and odd classes under the
pullback by $\sigma_{2}$. The condition on $v$ constrains $\upomega^{I}$ and
$\upnu_{I}$ to belong to $(\Gamma_{\text{inv}})^{\perp}$, while
${\boldsymbol{\Uplambda}}$ should split as
${\boldsymbol{\Uplambda}}={\boldsymbol{\Uplambda}}_{(+)}+{\boldsymbol{\Uplambda}}_{(-)}$
where the two factors ${\boldsymbol{\Uplambda}}_{(+)}$ and
${\boldsymbol{\Uplambda}}_{(-)}$ are respectively elements of
$\Gamma_{\text{inv}}$ and $(\Gamma_{\text{inv}})^{\perp}$ and satisfy
$R_{\text{flip}}({\boldsymbol{\Uplambda}}_{(\pm)})=\pm{\boldsymbol{\Uplambda}}_{(\pm)}$.
The action of $\mu_{g_{2}}$ on the moduli

$\displaystyle\mu_{g_{2}}(\tau,\rho,{{\textbf{a}}}_{1},{{\textbf{a}}}_{2})=(\tau,\rho,-R_{\text{flip}}({{\textbf{a}}}_{1}),-R_{\text{flip}}({{\textbf{a}}}_{2}))~{},$
(5.39)

constrains the Wilson line parameters
${{\textbf{a}}}_{I}\in\Gamma_{8}+\Gamma_{8}$ to be antisymmetric under the
exchange of their two $\Gamma_{8}$ components.

We can of course consider ${\mathbb{Z}}_{2}$-symmetric configurations $X_{v}$
that are T-dual to direct product geometries $X_{v^{\prime}}\simeq M\times
T^{2}$, using the $\operatorname{O{}}(\Gamma_{2,18})$ element ${\cal T}$
defined in (5.6). In order for the T-dual configuration to admit a geometric
${\mathbb{Z}}_{2}$ quotient $g^{\prime}_{2}$, the transformation ${\cal T}$
must be parametrized by lattice vectors ${\boldsymbol{\kappa}}_{I}$ such that
$R_{\text{flip}}({\boldsymbol{\kappa}}_{I})=-{\boldsymbol{\kappa}}_{I}$. This
condition is easily satisfied for most configurations, and the T-duality leads
to a trivial product configuration with

$\displaystyle{\boldsymbol{\Uplambda}}^{\prime}={\boldsymbol{\Uplambda}}-{\boldsymbol{\kappa}}_{1}\upomega^{1}-{\boldsymbol{\kappa}}_{2}\upomega^{2}~{}.$
(5.40)

We see that the T-duality introduces abelian instantons in configurations that
are antisymmetric with respect to the exchange of the two
$\operatorname{E}_{8}$ factors. However, this antisymmetry is compensated by
the non-trivial transformation of the classes $\upomega^{I}$ under the
Enriques involution, resulting in
$\sigma_{2}^{\ast}{\boldsymbol{\Uplambda}}^{\prime}=R_{\text{flip}}({\boldsymbol{\Uplambda}}^{\prime})$.
The T-dual theory corresponds, after orbifolding, to a compactification on the
Enriques Calabi–Yau manifold $(M\times T^{2})/{\mathbb{Z}}_{2}$, endowed with
an abelian gauge bundle with non-trivial holonomy turned on for the
${\mathbb{Z}}_{2}$ factor of
$(\operatorname{E}_{8}\times\operatorname{E}_{8})\rtimes{\mathbb{Z}}_{2}$.

When viewed from the $8$-dimensional perspective, this example may at first
appear to be slightly puzzling, since it appears that we are able to put the
$8$-dimensional theory on a non-spin manifold—the Enriques surface—while
preserving supersymmetry. This is reminiscent of the six-dimensional IIB
orientifold compactification recently discussed in [61], and the resolution to
the seeming puzzle is similar in spirit: the $8$–dimensional gravitinos are
charged under a lift of the ${\mathbb{Z}}_{2}$ reflection symmetry inherited
from the torus, and by turning on a non-trivial holonomy for this symmetry
supported by the fundamental group of the Enriques surface, we obtain well-
defined gravitinos and preserve four supercharges in four dimensions.

#### Singular ${\mathbb{Z}}_{3}$ orbifolds

We now turn our attention to ${\mathbb{Z}}_{3}$ N=1 orbifolds. The
construction requires a K3 surface $M$ admitting a non-symplectic automorphism
of order three, with only three isolated fixed points. An example of such K3
manifold is obtained by taking $M$ to be a double cover of a del Pezzo surface
$\text{dP}_{6}$ of degree six, branched over a curve of genus seven. Such a
construction yields a K3 surface, as can be checked from the Euler
characteristic $\chi(M)=2\chi(\text{dP}_{6})-(2-2\times 7)=24$. The surface
$\text{dP}_{6}$ can be represented as a complete intersection of tridegree
$(1,1,1)$ in a projective ambient space
${\mathbb{P}}^{1}\times{\mathbb{P}}^{1}\times{\mathbb{P}}^{1}$. Denoting the
projective coordinates of the $a$-th ${\mathbb{P}}^{1}$ by
$y^{a}=[y_{0}^{a}:y_{1}^{a}]$ and by $x$ the homogeneous coordinate describing
the branched cover, the K3 surface $M$ admits the algebraic description

$\displaystyle p_{1}(y^{1},y^{2},y^{3})$ $\displaystyle=0~{},$ $\displaystyle
x^{2}$ $\displaystyle=p_{2}(y^{1},y^{2},y^{3})~{},$ (5.41)

where $p_{1}$ and $p_{2}$ are polynomials of respective degree $(1,1,1)$ and
$(2,2,2)$ in the ${\mathbb{P}}^{1}$ homogeneous coordinates. The polynomials
can be tuned to the symmetric locus
$p_{1}(y^{2},y^{3},y^{1})=p_{1}(y^{1},y^{2},y^{3})$ and
$p_{2}(y^{2},y^{3},y^{1})=\zeta_{3}p_{2}(y^{1},y^{2},y^{3})$ while keeping the
hypersurface $M$ smooth. On this locus, the Kähler isometry of the projective
space

$\sigma_{3}\cdot[x:y_{0}^{1}:y_{1}^{1}:y_{0}^{2}:y_{1}^{2}:y_{0}^{3}:y_{1}^{3}]=[\zeta_{3}^{2}x:y_{0}^{2}:y_{1}^{2}:y_{0}^{3}:y_{1}^{3}:y_{0}^{1}:y_{1}^{1}]$
(5.42)

descends to $M$, providing a non-symplectic automorphism of order three, with
$\sigma_{3}^{\ast}\Omega=\zeta_{3}\Omega$. The three isolated fixed points of
$\sigma_{3}$ are located at $y^{1}=y^{2}=y^{3}$ and $x=0$. This example is the
one introduced in [28], and is amenable to a torsional linear sigma model
description of the orbifold.

We lift the action of $\sigma_{3}$ to the $T^{2}$ bundle by the
$\operatorname{SL}(2,{\mathbb{Z}})$ action

$\displaystyle{\cal R}=\begin{pmatrix}-1&1\\\ -1&0\end{pmatrix}~{}.$ (5.43)

Note that this $T^{2}$ rotation leaves the three points
$(\theta^{1},\theta^{2})=(0,0),(\tfrac{2\pi}{3},\tfrac{4\pi}{3}),(\tfrac{4\pi}{3},\tfrac{2\pi}{3})$
invariant. The quotient space $X/G$ will consequently inherit nine isolated
${\mathbb{C}}^{3}/{\mathbb{Z}}_{3}$ singularities.

The ${\mathbb{Z}}_{3}$ action on the gauge bundle should also be specified. In
contrast to the ${\mathbb{Z}}_{2}$ outer automorphism of
$\operatorname{E}_{8}\times\operatorname{E}_{8}$, there is no canonical choice
of ${\mathbb{Z}}_{3}$ action on
$\operatorname{E}_{8}\times\operatorname{E}_{8}$, so we need to fix an
embedding ${\mathbb{Z}}_{3}\subset\operatorname{O{}}(\Gamma_{8}+\Gamma_{8})$.
A convenient choice in the following will be to pick two roots
${\boldsymbol{\beta}}_{1}$ and ${\boldsymbol{\beta}}_{2}$ of
$\Gamma_{8}+\Gamma_{8}$ at an angle
${\boldsymbol{\beta}}_{1}\cdot{\boldsymbol{\beta}}_{2}=-1$. The product of
Weyl reflections

$\displaystyle R=R_{{\boldsymbol{\beta}}_{2}}R_{{\boldsymbol{\beta}}_{1}}$
(5.44)

provides an order three lattice isometry, with
$R^{2}=R_{{\boldsymbol{\beta}}_{1}}R_{{\boldsymbol{\beta}}_{2}}$. An advantage
of such construction is that the phase $U(g_{\text{g}}[R],{\boldsymbol{p}})$
associated to this ${\mathbb{Z}}_{3}$ action can be explicitly computed. We
give a choice of representative in appendix A.4.

#### An example with trivial gauge bundle

Consider a simply-connected configuration $X_{v}\to M$ with trivial gauge
bundle, fibered over the K3 surface $M$ introduced above. The bundle is
described by a vector $v$ of the form (5.1) with ${\boldsymbol{\Uplambda}}=0$.
The lift of the ${\mathbb{Z}}_{3}$ action to $X_{v}$ requires
$\sigma_{3}^{\ast}v=g_{3}v$, where $g_{3}$ is the
$\operatorname{O{}}(\Gamma_{2,18})$ element defined in (5.32). This condition
constrains the torus classes $\upomega^{I}_{\text{p}}$ to obey
$\sigma_{3}^{\ast}\upomega^{1}_{\text{p}}=-\upomega^{1}_{\text{p}}+\upomega^{2}_{\text{p}}$
and $\sigma_{3}^{\ast}\upomega^{2}_{\text{p}}=-\upomega^{1}_{\text{p}}$. In
terms of these classes, the vector $v$ must be of the form

$\displaystyle v=\begin{pmatrix}\upomega_{\text{p}}^{1}\\\
\upomega_{\text{p}}^{2}\\\
l_{1}\upomega_{\text{p}}^{1}+l_{2}\upomega_{\text{p}}^{2}\\\
(-l_{1}-l_{2})\upomega_{\text{p}}^{1}+l_{1}\upomega_{\text{p}}^{2}\\\
0\end{pmatrix}~{}.$ (5.45)

The Bianchi identity reads

$\displaystyle-
l_{1}\left(\upomega_{\text{p}}^{1}.\upomega_{\text{p}}^{1}+\upomega_{\text{p}}^{2}.\upomega_{\text{p}}^{2}-\upomega_{\text{p}}^{1}.\upomega_{\text{p}}^{2}\right)=24~{},$
(5.46)

and for a smooth K3 surface $M$ this restricts the integer $l_{1}$ to values
$1\leq l_{1}\leq 4$.

The second integral cohomology of $M$ splits under the action of
$\sigma_{3}^{\ast}$ as [62]

$\displaystyle\Gamma_{\text{inv}}$
$\displaystyle=(3\Gamma_{1,1})+(-3\Gamma_{\mathfrak{e}_{6}}^{\ast})~{},$
$\displaystyle(\Gamma_{\text{inv}})^{\perp}$
$\displaystyle=\Gamma_{1,1}+(3\Gamma_{1,1})+(-\Gamma_{\mathfrak{a}_{2}})^{5}~{}.$
(5.47)

Every $-\Gamma_{\mathfrak{a}_{2}}$ factor in the cohomology sublattice
$(\Gamma_{\text{inv}})^{\perp}$ is generated by two curves $x$, $y$ with
intersection $x.x=y.y=-2$, $x.y=1$, and on which the non-symplectic
automorphism acts as $\sigma_{3}^{\ast}x=-x-y$ and $\sigma_{3}^{\ast}y=x$. We
can thus pick a $(-\Gamma_{\mathfrak{a}_{2}})^{2}$ sublattice of
$(\Gamma_{\text{inv}})^{\perp}$ with generators $x_{1}$, $y_{1}$ and $x_{2}$,
$y_{2}$, and set

$\displaystyle\upomega_{\text{p}}^{1}$ $\displaystyle=x_{1}-x_{2}~{},$
$\displaystyle\upomega_{\text{p}}^{2}$ $\displaystyle=-y_{1}+y_{2}~{}.$ (5.48)

The Bianchi identity is straightforwardly solved by taking $l_{1}=4$. Using a
$B$-field shift parametrized by $m=-l_{2}-4$, we can bring $v$ to the simple
form

$\displaystyle v=g_{\text{b}}[m]\begin{pmatrix}\upomega_{\text{p}}^{1}\\\
\upomega_{\text{p}}^{2}\\\
4\upomega_{\text{p}}^{1}+l_{2}\upomega_{\text{p}}^{2}\\\
-l_{2}\upomega_{\text{p}}^{1}+4\upomega_{\text{p}}^{2}\\\
0\end{pmatrix}=\begin{pmatrix}\upomega_{\text{p}}^{1}\\\
\upomega_{\text{p}}^{2}\\\
4\upomega_{\text{p}}^{1}-4\upomega_{\text{p}}^{2}\\\
4\upomega_{\text{p}}^{2}\\\ 0\end{pmatrix}~{}.$ (5.49)

The above configuration can then be “unwound” to $M\times T^{2}$ using the
transformation ${\cal
T}=g_{\text{i}}g_{\text{s},1}[{\boldsymbol{\kappa}}_{1}]g_{\text{s},2}[{\boldsymbol{\kappa}}_{2}]$,
with two lattice vectors such that
${\boldsymbol{\kappa}}_{1}\cdot{\boldsymbol{\kappa}}_{1}={\boldsymbol{\kappa}}_{2}\cdot{\boldsymbol{\kappa}}_{2}=8$
and ${\boldsymbol{\kappa}}_{1}\cdot{\boldsymbol{\kappa}}_{2}=-4$. We easily
see that these conditions are satisfied by the two $\Gamma_{8}+\Gamma_{8}$
vectors

$\displaystyle{\boldsymbol{\kappa}}_{1}$
$\displaystyle=2{\boldsymbol{\beta}}_{1}~{},$
$\displaystyle{\boldsymbol{\kappa}}_{2}$
$\displaystyle=2{\boldsymbol{\beta}}_{2}~{},$ (5.50)

and this is also consistent with the requirement
$R({\boldsymbol{\kappa}}_{I})={\cal R}^{J}{}_{I}{\boldsymbol{\kappa}}_{I}$ for
our choice of order three rotation
$R=R_{{\boldsymbol{\beta}}_{2}}R_{{\boldsymbol{\beta}}_{1}}$.

The topology of the T-dual configuration is described by the vector

$\displaystyle v^{\prime}={\cal T}v=\begin{pmatrix}0\\\ 0\\\
\upomega_{\text{p}}^{1}\\\ \upomega_{\text{p}}^{2}\\\
-{\boldsymbol{\kappa}}_{1}\upomega_{\text{p}}^{1}-{\boldsymbol{\kappa}}_{2}\upomega_{\text{p}}^{2}\end{pmatrix}~{},$
(5.51)

which represents a product geometry $M\times T^{2}$ equipped with an abelian
bundle
${\boldsymbol{\Uplambda}}^{\prime}=-{\boldsymbol{\kappa}}_{I}\upomega^{I}_{\text{p}}$.
We can observe that this configuration obeys

$\displaystyle\sigma_{3}^{\ast}v^{\prime}=g^{\prime}_{3}v^{\prime}~{},$ (5.52)

with the order three $\operatorname{O{}}(\Gamma_{2,18})$ element
$g^{\prime}_{3}={\cal T}g_{3}{\cal T}^{-1}=g_{\text{t}}[{\cal
R}^{-t}]g_{\text{g}}[R]$.

The $T^{2}$ and Wilson line parameters of the two configurations can also be
tracked through the duality. Starting from a configuration $v$ without Wilson
lines,212121In principle, the condition
$(\tau,\rho,{{\textbf{a}}}_{1},{{\textbf{a}}}_{2})=\mu_{g_{3}}(\tau,\rho,{{\textbf{a}}}_{1},{{\textbf{a}}}_{2})$
allows for non-trivial Wilson lines parameters satisfying
$R({{\textbf{a}}}_{1})=-{{\textbf{a}}}_{1}-{{\textbf{a}}}_{2}$ and
$R({{\textbf{a}}}_{2})={{\textbf{a}}}_{1}$. The following discussion stays
valid when such parameters are turned on. the initial theory is described by
the parameters

$\displaystyle\tau$ $\displaystyle=\zeta_{3}~{},$ $\displaystyle\rho$
$\displaystyle=4(\zeta_{3}+1)~{},$ $\displaystyle{{\textbf{a}}}_{I}=0~{},$
(5.53)

which solve the quantization conditions and are invariant under the action of
$\mu_{g_{3}}$. The T-dual configuration $v^{\prime}$ has parameters

$\displaystyle\tau^{\prime}$ $\displaystyle=\zeta_{3}+1~{},$
$\displaystyle\rho^{\prime}$ $\displaystyle=\tfrac{1}{24}(2\zeta_{3}+1)~{},$
$\displaystyle{{\textbf{a}}}^{\prime}_{1}$
$\displaystyle=-\tfrac{1}{12}(2{\boldsymbol{\kappa}}_{1}+{\boldsymbol{\kappa}}_{2})~{},$
$\displaystyle{{\textbf{a}}}^{\prime}_{2}$
$\displaystyle=-\tfrac{1}{12}({\boldsymbol{\kappa}}_{1}+2{\boldsymbol{\kappa}}_{2})~{}.$
(5.54)

Notice that the quantization conditions on $v^{\prime}$ amount to

$\displaystyle{{\textbf{a}}}^{\prime}_{I}\cdot{\boldsymbol{\Uplambda}}^{\prime}\in
H^{2}(M,{\mathbb{Z}})~{},$ (5.55)

and those conditions are obeyed for the specific choices of
${{\textbf{a}}}^{\prime}_{I}$ and ${\boldsymbol{\Uplambda}}^{\prime}$ of the
T-dual geometry. Moreover, we easily see that the action

$\displaystyle\mu_{g_{3}^{\prime}}^{-1}(\tau^{\prime},\rho^{\prime},{{\textbf{a}}}^{\prime}_{1},{{\textbf{a}}}^{\prime}_{2})=\left(\frac{1}{1-\tau^{\prime}},\rho^{\prime},-R({{\textbf{a}}}^{\prime}_{1})+R({{\textbf{a}}}^{\prime}_{2}),-R({{\textbf{a}}}^{\prime}_{1})\right)$
(5.56)

of the T-dual ${\mathbb{Z}}_{3}$ symmetry on the moduli leaves invariant the
parameters associated to $v^{\prime}$. Finally, let us note that to fully
specify the orbifold of the T-dual worldsheet CFT, it is necessary to compute
the phase factor $U(g^{\prime}_{3},{\boldsymbol{p}})$ that enter into the
action of $g^{\prime}_{3}$ on vertex operators. For the unwound example
described above, the cocycle representatives can be chosen such that
$U(g^{\prime}_{3},{\boldsymbol{p}})=U(g_{3},{\boldsymbol{p}})$. The
${\mathbb{Z}}_{3}$ orbifold can therefore be understood in the T-dual
description as the same geometric quotient as in the original theory. This
feature is not generic: for different configurations, one should expect that
the T-dual orbifold does not always have a geometric interpretation.

Having obtained a T-dual Kähler description of the ${\mathbb{Z}}_{3}$ action,
a question naturally arises: how can we understand the ${\mathbb{Z}}_{3}$
quotient of the T-dual theory? On the unwound side, the orbifold corresponds
to a ${\mathbb{Z}}_{3}$ symmetry of the direct product geometry
$X^{\prime}\simeq M\times T^{2}$ accompanied by an order three action on the
gauge bundle. The action on the six-dimensional Kähler geometry is by a non-
symplectic action on the K3 factor $M$ along with an order three rotation of
the torus. This isometry of $X^{\prime}$ leads to a quotient with nine
${\mathbb{C}}^{3}/{\mathbb{Z}}_{3}$ singular points. The singular
$X^{\prime}/{\mathbb{Z}}_{3}$ geometry admits a smooth Ricci-flat resolution:
the orbifold corresponds to the Borcea–Voisin construction of Calabi–Yau
threefolds [63, 64]. However, the understanding of the resolved geometry
necessitates a description of the gauge bundle. Indeed, on the unwound T-dual
side, the gauge bundle is non-trivial by construction: the T-duality
introduces two abelian instantons fibered over the classes $\upomega^{1}$ and
$\upomega^{2}$. If those curves were to intersect the singular locus of the
order three K3 automorphism, then the resolution of the singularities would
also impact the gauge instantons. However, it is possible to construct
geometries where this is not the case. Consider the K3 surface $M$ defined by
(5.41). This surface inherits three classes $C_{1}$, $C_{2}$, $C_{3}$ from its
ambient projective space
${\mathbb{P}}^{1}\times{\mathbb{P}}^{1}\times{\mathbb{P}}^{1}$, cut out by
hyperplanes of the form ${\mathbb{P}}^{1}\times{\mathbb{P}}^{1}$. The curves
$C_{i}$ have zero self-intersection and intersect each other at two points.
Under the non-symplectic automorphism (5.42), they transform by the
permutation $\sigma^{\ast}_{3}C_{i}=C_{i+1}$. The torsional geometry $X\to M$
can be constructed from two classes

$\displaystyle\upomega^{1}$ $\displaystyle=C_{1}-C_{2}~{},$
$\displaystyle\upomega^{2}$ $\displaystyle=C_{1}-C_{3}~{},$ (5.57)

with the ASD condition on $\upomega^{I}$ requiring the three
${\mathbb{P}}^{1}$ factors to be of the same size. A generic curve
representing the class $C_{i}$ does not intersect the fixed locus of
$\sigma_{3}$. Therefore, we can expect that the Kähler resolution of the
$(M\times T^{2})/{\mathbb{Z}}_{3}$ orbifold geometry will be unobstructed. It
would be interesting to better understand the resolved geometry and its
worldsheet realization. If the resolution modes admit a description in terms
of twisted marginal operators of the unwound $(0,2)$ CFT, then studying the
corresponding modes on the T-dual non-Kähler side could give some insight on
the existence of a non-Kähler $\operatorname{SU}(3)$-structure resolution of
the $X/{\mathbb{Z}}_{3}$ quotient geometry.

###  5.5 Duality and the supersymmetric index

The supersymmetric index of N=2 four-dimensional heterotic vacua was
introduced in [65] for $\text{K3}\times T^{2}$ compactifications and
generalized in [13, 14] to heterotic flux vacua. Our aim is to reformulate the
index in a duality-covariant way and examine in particular the effect of the
unwinding dualities. We refer to the aforementioned publications for
conventions and details.

Let us consider for definiteness a compactification to four dimensions endowed
with the pullback of a rank $r$ HYM bundle ${\cal P}$ on K3 with structure
group in the second $\operatorname{E}_{8}$ factor with vanishing first Chern
class. We will consider later on possible line bundles and Wilson lines in the
first $\operatorname{E}_{8}$ factor only. Since $c_{1}({\cal P})=0$, the
$(0,2)$ SCFT associated with the non-linear sigma model contains a non-
anomalous left-moving $U(1)_{\text{\tiny{L}}}$ symmetry, associated to a
holomorphic current $J(z)$, that will eventually be used for the left-moving
GSO projection. We also denote by $\bar{J}(\bar{z})$ the R-current of the
right-moving N=2 superconformal algebra.

We first define the dressed elliptic genus of the $(0,2)$ two-dimensional SCFT
with target space $T^{2}\hookrightarrow X\stackrel{{\scriptstyle p}}{{\to}}M$
and central charges $(c,\bar{c})=(14+r,9)$ as the following trace in the
Ramond–Ramond sector :

$\displaystyle
Z_{\text{deg}}(\boldsymbol{q},\bar{\boldsymbol{q}};\boldsymbol{y})=\frac{1}{\bar{\eta}^{2}(\bar{\boldsymbol{q}})}\text{Tr}_{\textsc{r}}\left\\{\boldsymbol{y}^{J_{0}}\bar{J}_{0}(-1)^{F}\boldsymbol{q}^{L_{0}-c/24}\bar{\boldsymbol{q}}^{\bar{L}_{0}-\bar{c}/24}\right\\}~{},$
(5.58)

where $J_{0}$ (resp. $\bar{J}_{0}$) is the zero-mode of the
$U(1)_{\text{\tiny{L}}}$ current (resp. the $U(1)_{\text{\tiny{R}}}$ current)
and $(-1)^{F}=\exp i\pi(J_{0}-\bar{J}_{0})$. One can then obtain the new
supersymmetric index proper as:

$\displaystyle
Z_{\text{new}}(\boldsymbol{q},\bar{\boldsymbol{q}})=\frac{\bar{\eta}^{2}(\bar{\boldsymbol{q}})}{2\eta(\boldsymbol{q})}\sum_{\gamma,\delta=0}^{1}\boldsymbol{q}^{\gamma^{2}}\left\\{\left(\frac{\vartheta_{1}(\boldsymbol{q}|\boldsymbol{y})}{\eta(\boldsymbol{q})}\right)^{8-r}Z_{\text{deg}}(\boldsymbol{q},\bar{\boldsymbol{q}};\boldsymbol{y})\right\\}\Bigg{|}_{\boldsymbol{y}=\boldsymbol{q}^{\gamma/2}e^{i\pi\delta}}~{}.$
(5.59)

The dressed elliptic genus of flux N=2 compactifications to four dimensions
was obtained in [13, 14] using a UV completion of the corresponding $(0,2)$
non-linear sigma model as a $(0,2)$ gauged linear sigma model [11] and
supersymmetric localization, building on previous works about the ordinary
elliptic genera [66, 67]. From this result a geometric formulation of the
dressed elliptic genus was derived for any compactification such that the K3
base $M$ is realized as a subvariety of a weighted projective space.

In the absence of Wilson lines, the $\Gamma_{2,10}$ lattice splits into a
$(2,2)$ lattice and the $\operatorname{E}_{8}$ lattice. The former is the
lattice associated with the $T^{2}$ fiber, spanned by

$\displaystyle{\boldsymbol{p}}=\text{w}^{I}{{\boldsymbol{e}}}_{I}+\text{n}_{I}{{\boldsymbol{e}}}^{\ast
I}={\boldsymbol{p}}_{\text{\tiny{L}}}+{\boldsymbol{p}}_{\text{\tiny{R}}}$
(5.60)

in $\mathbb{R}^{2,2}$, where the $\text{n}_{I}$ (resp. the $\text{w}^{I}$) are
the momenta (resp. the winding numbers). The decomposition in left- and right-
moving components is defined through the basis vectors

$\displaystyle{{\boldsymbol{e}}}_{\text{\tiny{L}}}^{1}$
$\displaystyle=\frac{1}{\sqrt{2\tau_{2}\rho_{2}}}\left(\tau_{2}{{\boldsymbol{e}}}_{1}+\rho_{2}{{\boldsymbol{e}}}^{\ast
1}+(\tau_{1}\rho_{2}+\tau_{2}\rho_{1}){{\boldsymbol{e}}}^{\ast 2}\right)~{},$
$\displaystyle{{\boldsymbol{e}}}_{\text{\tiny{L}}}^{2}$
$\displaystyle=\frac{1}{\sqrt{2\tau_{2}\rho_{2}}}\left(-\tau_{1}{{\boldsymbol{e}}}_{1}+{{\boldsymbol{e}}}_{2}-\rho_{1}{{\boldsymbol{e}}}^{\ast
1}+(\tau_{2}\rho_{2}-\tau_{1}\rho_{1}){{\boldsymbol{e}}}^{\ast 2}\right)~{},$
$\displaystyle{{\boldsymbol{e}}}_{\text{\tiny{R}}}^{1}$
$\displaystyle=\frac{1}{\sqrt{2\tau_{2}\rho_{2}}}\left(-\tau_{2}{{\boldsymbol{e}}}_{1}+\rho_{2}{{\boldsymbol{e}}}^{\ast
1}+(\tau_{1}\rho_{2}-\tau_{2}\rho_{1}){{\boldsymbol{e}}}^{\ast 2}\right)~{},$
$\displaystyle{{\boldsymbol{e}}}_{\text{\tiny{R}}}^{2}$
$\displaystyle=\frac{1}{\sqrt{2\tau_{2}\rho_{2}}}\left(\tau_{1}{{\boldsymbol{e}}}_{1}-{{\boldsymbol{e}}}_{2}+\rho_{1}{{\boldsymbol{e}}}^{\ast
1}+(\tau_{2}\rho_{2}+\tau_{1}\rho_{1}){{\boldsymbol{e}}}^{\ast 2}\right)~{}.$
(5.61)

This basis satisfies

$\displaystyle{{\boldsymbol{e}}}_{\text{\tiny{L}}}^{a}\cdot{{\boldsymbol{e}}}_{\text{\tiny{L}}}^{b}$
$\displaystyle=\delta^{ab}~{},$
$\displaystyle{{\boldsymbol{e}}}_{\text{\tiny{R}}}^{a}\cdot{{\boldsymbol{e}}}_{\text{\tiny{R}}}^{b}$
$\displaystyle=-\delta^{ab}~{},$
$\displaystyle{{\boldsymbol{e}}}_{\text{\tiny{L}}}^{a}\cdot{{\boldsymbol{e}}}_{\text{\tiny{R}}}^{b}$
$\displaystyle=0~{},$ (5.62)

and the left and right projections are defined by
${\boldsymbol{p}}_{\text{\tiny{L}}}=({\boldsymbol{p}}\cdot{{\boldsymbol{e}}}_{\text{\tiny{L}}a}){{\boldsymbol{e}}}_{\text{\tiny{L}}}^{a}$
and
${\boldsymbol{p}}_{\text{\tiny{R}}}=-({\boldsymbol{p}}\cdot{{\boldsymbol{e}}}_{\text{\tiny{R}}a}){{\boldsymbol{e}}}_{\text{\tiny{R}}}^{a}$.
As we have seen in subsection 3.4 the moduli of the $T^{2}$ are quantized in
such a way that the $T^{2}$ CFT is rational, or in other words the sublattices
$\Gamma_{\text{\tiny{L}}}=\Gamma_{2,2}\cap\mathbb{R}^{2,0}$ and
$\Gamma_{\text{\tiny{R}}}=\Gamma_{2,2}\cap\mathbb{R}^{0,2}$ are both rank two
lattices. Elements of the lattice $\Gamma_{2,2}$ with
${\boldsymbol{p}}_{\text{\tiny{R}}}=0$ are such that the momenta
$\text{n}_{1}$ and $\text{n}_{2}$ can be expressed in terms of the winding
numbers $\text{w}^{1}$ and $\text{w}^{2}$:

$\displaystyle\text{n}_{1}$
$\displaystyle=\frac{\rho_{2}}{\tau_{2}}(\text{w}^{1}+\tau_{1}\text{w}^{2})-\rho_{1}\text{w}^{2}\
\in\ \mathbb{Z}\,,$ (5.63a) $\displaystyle\text{n}_{2}$
$\displaystyle=\frac{\rho_{2}}{\tau_{2}}(\tau_{1}\text{w}^{1}+|\tau|^{2}\text{w}^{2})+\rho_{1}\text{w}^{1}\
\in\ \mathbb{Z}\,.$ (5.63b)

Note that any lattice vector in $\Gamma_{\text{\tiny{L}}}$ can be expressed
purely in terms of the winding numbers, using (5.63):

$\displaystyle{\boldsymbol{p}}_{\text{\tiny{L}}}=\sqrt{\frac{2\rho_{2}}{\tau_{2}}}\left(\text{w}^{1}{{\boldsymbol{e}}}_{\text{\tiny{L}}}^{1}+\text{w}^{2}(\tau_{1}{{\boldsymbol{e}}}_{\text{\tiny{L}}}^{1}+\tau_{2}{{\boldsymbol{e}}}_{\text{\tiny{L}}}^{2})\right)~{}.$
(5.64)

In the geometrical formulation for the dressed elliptic genus derived in [13]
from the torsional GLSM approach, the coupling between the $\Gamma_{2,2}$
lattice of the two-torus and the anti-self-dual (1,1) forms $\upomega^{1}$ and
$\upomega^{2}$ associated with the curvatures of the torus bundle appeared
naturally in the formula in terms of a formal extension of the winding lattice
as a module over $H^{2}(M,{\mathbb{Z}})$:

$\displaystyle{\boldsymbol{p}}_{\upomega}=\sqrt{\frac{2\rho_{2}}{\tau_{2}}}\left(\upomega^{1}{{\boldsymbol{e}}}_{\text{\tiny{L}}}^{1}+\upomega^{2}(\tau_{1}{{\boldsymbol{e}}}_{\text{\tiny{L}}}^{1}+\tau_{2}{{\boldsymbol{e}}}_{\text{\tiny{L}}}^{2})\right)~{}.$
(5.65)

As noticed above, this element of the winding lattice can be also viewed as an
element of the lattice $\Gamma_{\text{\tiny{L}}}$ of purely left-moving
momenta. Adding the $\Gamma_{8}$ associated with the first
$\operatorname{E}_{8}$ factor, one can then use the same lattice formalism as
in subsection 2.1 and associate to the principal torus bundle the element
$v=\upomega^{I}{{\boldsymbol{e}}}_{I}+\upnu_{I}{{\boldsymbol{e}}}^{\ast I}$ of
$\Gamma_{2,10}\otimes H^{2}(M,{\mathbb{Z}})$, where the classes $\upnu_{1}$
and $\upnu_{2}$ are given by (3.50). We recall that, for the time being, there
are neither Wilson lines nor line bundles in the first $\operatorname{E}_{8}$
factor.

In terms of $v$, one can express the dressed elliptic genus in a manifestly
T-duality covariant presentation. As for the ordinary elliptic genus [68] we
define first the formal power series with bundle coefficients:

$\displaystyle\mathbb{E}_{\boldsymbol{q},\boldsymbol{y}}=\bigotimes_{n=0}^{\infty}\bigwedge\nolimits_{-\boldsymbol{y}\boldsymbol{q}^{n}}{\cal
P}^{*}\otimes\bigwedge\nolimits_{-\boldsymbol{y}^{-1}\boldsymbol{q}^{n}}{\cal
P}\otimes\bigotimes_{n=1}^{\infty}S_{\boldsymbol{q}^{n}}T_{M}^{*}\otimes\bigotimes_{n=1}^{\infty}S_{\boldsymbol{q}^{n}}T_{M}~{},$
(5.66)

where

$\displaystyle\bigwedge\nolimits_{\boldsymbol{t}}{\cal P}$
$\displaystyle=\sum_{k=0}^{\infty}\boldsymbol{t}^{k}\bigwedge\nolimits^{k}{\cal
P}~{},$ $\displaystyle S_{\boldsymbol{t}}T_{M}$
$\displaystyle=\sum_{k=0}^{\infty}\boldsymbol{t}^{k}S^{k}T_{M}~{},$ (5.67)

$\bigwedge\nolimits^{k}$ and $S^{k}$ being respectively the $k$-th exterior
product and the $k$-th symmetric product. One has then

$\displaystyle
Z_{\text{deg}}(\boldsymbol{q},\bar{\boldsymbol{q}};\boldsymbol{y})=\boldsymbol{q}^{\frac{r-2}{12}}\boldsymbol{y}^{r/2}\int_{M}\,\text{ch}\,\big{(}\mathbb{E}_{\boldsymbol{q},\boldsymbol{y}}\big{)}\,\text{td}\big{(}T_{M}\big{)}\sum_{{\boldsymbol{p}}\in\Gamma_{2,10}}\frac{\boldsymbol{q}^{\frac{1}{2}{\boldsymbol{p}}_{\text{\tiny{L}}}^{2}}}{\eta^{2}(\boldsymbol{q})}\frac{\bar{\boldsymbol{q}}^{\frac{1}{2}{\boldsymbol{p}}_{\text{\tiny{R}}}^{2}}}{\bar{\eta}^{2}(\bar{\boldsymbol{q}})}\,e^{-{\boldsymbol{p}}\cdot
v}~{},$ (5.68)

where as in subsection 2.1, $\cdot$ denotes the inner product on
$\Gamma_{2,10}$ preserving the ${\mathbb{R}}^{2,10}$ metric $\eta$.

Using the splitting principle, let $c(T_{M})=(1+\uppsi_{1})(1+\uppsi_{2})$ and
$c({\cal P})=\prod_{a=1}^{r}(1+\upxi_{a})$. One obtains then the final
expression for the dressed elliptic genus, given as a function of the vector
$v\in\Gamma_{2,10}\otimes H^{2}(M,{\mathbb{Z}})$:

$Z_{\text{deg}}(\boldsymbol{q},\bar{\boldsymbol{q}};\boldsymbol{y}|v)=\int_{M}\prod_{a=1}^{r}\frac{i\vartheta_{1}\big{(}\boldsymbol{q}\big{|}\boldsymbol{y}^{-1}e^{\,\upxi_{a}}\big{)}}{\eta(\boldsymbol{q})}\prod_{i=1}^{2}\frac{\eta(\boldsymbol{q})\uppsi_{i}}{i\vartheta_{1}\big{(}\boldsymbol{q}\big{|}e^{\uppsi_{i}}\big{)}}\,\sum_{{\boldsymbol{p}}\in\Gamma_{2,10}}\,\frac{\boldsymbol{q}^{\frac{1}{2}{\boldsymbol{p}}_{\text{\tiny{L}}}^{2}}}{\eta^{2}(\boldsymbol{q})}\frac{\bar{\boldsymbol{q}}^{\frac{1}{2}{\boldsymbol{p}}_{\text{\tiny{R}}}^{2}}}{\bar{\eta}^{2}(\bar{\boldsymbol{q}})}\,e^{-{\boldsymbol{p}}\cdot
v}~{},$ (5.69)

where it is understood that the integral over the base $M$ selects the top
form. This index, although non-holomorphic, transforms under the action of the
(worldsheet) modular group as a Jacobi form of weight -2 and rank $r/2$,
whenever the Bianchi identity (3.40) is satisfied in cohomology [13]. The
dressed elliptic genus (5.69) is also invariant under the action of a space-
time perturbative duality $g\in O(\Gamma_{2,10})$:

$\displaystyle
Z_{\text{deg}}(\boldsymbol{q},\bar{\boldsymbol{q}};\boldsymbol{y}|gv)=Z_{\text{deg}}(\boldsymbol{q},\bar{\boldsymbol{q}};\boldsymbol{y}|v)~{},$
(5.70)

where it is understood that the index
$Z_{\text{deg}}(\boldsymbol{q},\bar{\boldsymbol{q}};\boldsymbol{y}|gv)$ is
computed using the left/right decomposition
${\boldsymbol{p}}={\boldsymbol{p}}_{\text{\tiny{L}}}+{\boldsymbol{p}}_{\text{\tiny{R}}}$
of $\Gamma_{2,18}$ lattice vectors defined by the T-dual moduli.

#### Unwinding the index

The dressed elliptic genus (5.69) has been given in an
$\operatorname{O{}}(\Gamma_{2,10})$ covariant way, allowing to understand its
transformation under topology-changing perturbative dualities. Explicitly, one
would want to map the new supersymmetric index of a torsional compactification
to the new supersymmetric index of a $\text{K3}\times T^{2}$ compactification
with an extra abelian gauge bundle.

The main example discussed in subsection 5.1 involved a torus fibration built
out of two primitive elements of $H^{2}(M,\mathbb{Z})$, corresponding to the
vector

$\displaystyle v=\begin{pmatrix}\upomega_{\text{p}}^{1}\\\
\upomega_{\text{p}}^{2}\\\
l_{11}\upomega_{\text{p}}^{1}+l_{12}\upomega_{\text{p}}^{2}\\\
l_{21}\upomega_{\text{p}}^{1}+l_{22}\upomega_{\text{p}}^{2}\\\
0\end{pmatrix}~{},$ (5.71)

where $\rho=l_{11}\tau-l_{12}$ and $\rho\tau=l_{21}\tau-l_{22}$. Recall that
in the present context $v$ is a vector of $\Gamma_{2,10}$ rather than a vector
of $\Gamma_{2,18}$, but it does not change the logic.

This configuration can be T-dualized by an $\operatorname{O{}}(\Gamma_{2,18})$
transformation ${\cal T}$ of the form (5.6), with
${\boldsymbol{\kappa}}_{1,2}\in\Gamma_{8}$ to a configuration describing a
direct product $\text{K3}\times T^{2}$, endowed with an abelian bundle
characterized by ${\boldsymbol{\Uplambda}}^{\prime}\in\Gamma_{8}\otimes
H^{2}(M,{\mathbb{Z}})$ and quantized Wilson lines in the same
$\operatorname{E}_{8}$, i.e. with parameters ${{\textbf{a}}}_{I}^{\prime}$
such that
${{\textbf{a}}}_{I}^{\prime}\cdot{\boldsymbol{\Uplambda}}^{\prime}\neq 0$. A
further duality—Wilson line integer shift of parameters
${\boldsymbol{\kappa}}^{\prime}_{1,2}\in\Gamma_{8}$—allows to disentangle the
Wilson lines from the abelian gauge bundle, see around (5.15), reaching the
configuration specified by the vector

$\displaystyle v^{\prime\prime}=\begin{pmatrix}0\\\ 0\\\ 0\\\ 0\\\
{\boldsymbol{\Uplambda}}^{\prime\prime}\end{pmatrix}~{},$ (5.72)

with Wilson line parameters ${{\textbf{a}}}^{\prime\prime}_{1,2}$ given by
(5.16) and abelian gauge bundle specified by
${\boldsymbol{\Uplambda}}^{\prime\prime}=-{\boldsymbol{\kappa}}_{1}\upomega^{1}_{\text{p}}-{\boldsymbol{\kappa}}_{2}\upomega^{2}_{\text{p}}$.
For definiteness, and without loss of generality, let us consider that the
non-zero entries of the 8-dimensional vector
$-{\boldsymbol{\kappa}}_{1}\upomega^{1}_{\text{p}}-{\boldsymbol{\kappa}}_{2}\upomega^{2}_{\text{p}}$
are the first $s$ ones, such that the ${{\textbf{a}}}^{\prime\prime}_{1,2}$
Wilson line parameters have non-zero entries only for the remaining $8-s$
components.

A better understanding of the dressed elliptic genus
$Z_{\text{deg}}(\boldsymbol{q},\bar{\boldsymbol{q}};\boldsymbol{y}|v^{\prime\prime})$
associated with the T-dual “unwound” configuration is obtained in the
fermionic description, especially regarding the gauge bundle. The set of 8
chiral bosons with momentum lattice $\Gamma_{8}$ are then traded for a set of
8 left-moving fermions. For a given spin structure $[\alpha,\beta]$ the
contribution to the path integral from the $8-s$ fermions supporting the
Wilson lines of parameters ${{\textbf{a}}}_{I}^{{}^{\prime\prime}}$ is given
by [69]

$Z_{w}\left[\begin{smallmatrix}\alpha\\\
\beta\end{smallmatrix}\right]\left(\boldsymbol{q}\big{|}{{\textbf{a}}}_{I}^{{}^{\prime\prime}}\right)=\prod_{a=s+1}^{8}\frac{1}{\eta(\boldsymbol{q})}\vartheta\left[\begin{smallmatrix}\alpha+2{{\textbf{a}}}_{I}^{{}^{\prime\prime}a}\,\text{w}^{I}\\\
\beta+2{{\textbf{a}}}_{I}^{{}^{\prime\prime}a}\,\text{m}^{I}\end{smallmatrix}\right]\big{(}\boldsymbol{q}\big{|}0\big{)}\,,$
(5.73)

where we have used the orthonormal basis (2.5). In this expression
$\text{w}^{I}$ and $\text{m}^{I}$ are the winding numbers along the worldsheet
space-like and time-like one-cycles respectively. After a Poisson resummation
on $\text{m}^{1}$ and $\text{m}^{2}$, which are traded for $\text{n}_{1}$ and
$\text{n}_{2}$, one obtains indeed left- and right-moving momenta of the
expected form described in subsection 2.1.

As far as the abelian gauge bundle is concerned, the coupling of the
$\Gamma_{2,10}$ lattice vectors with the background vector $v^{\prime\prime}$
given by (5.15) in the dressed elliptic genus (5.69) takes the form

$e^{\,-{\boldsymbol{p}}\cdot
v}=e^{\,{\boldsymbol{p}}\cdot\big{(}\upomega^{1}_{\text{p}}{\boldsymbol{\kappa}}_{1}+\upomega^{2}_{\text{p}}{\boldsymbol{\kappa}}_{2}\big{)}}=\exp{\,\sum_{a=1}^{s}\ell^{a}\big{(}\upomega^{1}_{\text{p}}{\boldsymbol{\kappa}}_{1}^{a}+\upomega^{2}_{\text{p}}{\boldsymbol{\kappa}}_{2}^{a}\big{)}}~{}.$
(5.74)

In the fermionic presentation, it gives a contribution to the dressed elliptic
genus of the form:

$Z_{l}\left[\begin{smallmatrix}\alpha\\\
\beta\end{smallmatrix}\right]\big{(}\boldsymbol{q}\big{|}{\boldsymbol{\Uplambda}}^{\prime}\big{)}=\prod_{a=1}^{s}\frac{1}{\eta(\boldsymbol{q})}\vartheta\left[\begin{smallmatrix}\alpha\\\
\beta\end{smallmatrix}\right]\Big{(}\boldsymbol{q}\Big{|}e^{\,\upomega^{1}_{\text{p}}{\boldsymbol{\kappa}}_{1}^{a}+\upomega^{2}_{\text{p}}{\boldsymbol{\kappa}}_{2}^{a}}\Big{)}\,.$
(5.75)

After summing over the spin structures, these terms will appear in the new
supersymmetric index (5.59) exactly on the same footing as Fermi multiplets
transforming as sections of abelian bundles over the base $M$, see the first
term in the integrand of (5.69). Thus we have demonstrated that, by the chain
of Narain dualities described in subsection 5.1, the new supersymmetric index
of flux compactifications with eight supercharges can be recast in the form of
the new supersymmetric index for $\text{K3}\times T^{2}$ compactifications
with appropriate Wilson lines an abelian gauge bundles.

##  6 N=1 flux vacua in $4$ dimensions

The observations in the previous sections show that four-dimensional heterotic
vacua preserving $8$ supercharges can always be connected to flux-free
$M\times T^{2}$ configurations by Narain T-duality. The N=2 requirements force
the torus classes to be primitive $(1,1)$ forms, so these classes enter on the
same footing as any abelian instanton in the gauge sector. It is then natural
to expect that there exists a duality exchanging torus and gauge data.

Our goal in this section will be to repeat a similar T-duality analysis for
geometries with only $4$ supercharges. As reviewed in section 3, anti-self-
duality of the torus Chern classes is not required for heterotic geometries
with minimal supersymmetry: the classes $\upomega^{I}$ can have non-zero
$(2,0)\oplus(0,2)$ components as long as the complex combination
$\upomega^{1}+\tau\upomega^{2}$ stays orthogonal to $\Omega$. Turning on such
components leads to drastic consequences for the heterotic geometry. In
particular, a crucial modification is the necessity of a formal
$\alpha^{\prime}$ expansion of the internal fields to solve the leading order
equations of motion: the curvature $R_{+}$ of the $H$-twisted connection is no
longer horizontal, and the Bianchi identity can only be solved at first order
in $\alpha^{\prime}$. The same $\alpha^{\prime}$ expansion is necessary for a
derivation of the flux quantization conditions as in [9]. Another difference
that will play an important role in the following is the quantization of
Wilson line parameters. As described in section 3, there are no continuous
deformations of the parameters ${{\textbf{a}}}_{I}$, and it is always possible
to set ${{\textbf{a}}}_{I}=0$ by a large gauge transformation.

###  6.1 Quantization of the K3 periods

Supersymmetry and flux quantization impose strong conditions on the K3 surface
$M$ used to build the heterotic geometry. A particularly simple example
corresponds to configurations for which the torus Chern classes only have
$(2,0)\oplus(0,2)$ components. To ensure supersymmetry, the complexified
$T^{2}$ curvature must be proportional to the holomorphic $2$-form of $M$:

$\displaystyle\gamma\Omega=\upomega^{1}+\tau\upomega^{2}~{},$ (6.1)

where $\gamma$ is a non-zero complex constant. Consequently, supersymmetry
fixes the periods of the K3 manifold. In particular, its Picard number is
$\varrho(M)=20$: in other words, $M$ is an attractive K3 surface.222222We
refer to [70] for a pedagogical description of attractive K3 manifolds. By the
Shioda–Inose theorem [71], this attractive K3 surface is completely
characterized by the $\operatorname{SL}(2,{\mathbb{Z}})$ equivalence class of
the positive-definite even matrix $D$, where we denote by
$D^{IJ}=\upomega^{I}.\upomega^{J}$ the intersection numbers of the torus
classes. In addition, for the definition (6.1) of the holomorphic $2$-form to
be consistent with the condition $\Omega.\Omega=0$, the torus complex
structure is fixed to the value

$\tau=\frac{-D^{12}+i\sqrt{\det D}}{D^{22}}~{}.$ (6.2)

After gauging away the Wilson line parameters, the quantization conditions
(3.45) take the simple form

$\displaystyle\upnu_{2}-\tau\upnu_{1}=\rho(\upomega^{1}+\tau\upomega^{2})~{}.$
(6.3)

For this equation to admit solutions, the torus parameters $(\tau,\rho)$ are
constrained to belong to the same quadratic field, corresponding to the
conditions on the moduli of a rational $T^{2}$ CFT, as in the N=2 case.

Configurations with a purely holomorphic complexified $T^{2}$ curvature are
quite special: in a generic N=1 geometry, the classes $\upomega^{I}$ also have
$(1,1)$ components. When this is the case, the complexified $T^{2}$ class
decomposes as

$\displaystyle\upomega^{1}+\tau\upomega^{2}=\xi+\gamma\Omega~{},$ (6.4)

in terms of a complex primitive $(1,1)$ form $\xi$ and a complex constant
$\gamma$. The quantization conditions (3.45) relate the Hodge duals of the
torus classes to a linear combination of the four integral classes
$\upomega^{I}$ and $\upnu_{I}$. Since the Hodge operator acts with a definite
sign on $\xi$ and on $\Omega$, the quantization conditions can be used to
extract their expansion in integral cohomology:

$\displaystyle\gamma\Omega$
$\displaystyle=\frac{1}{2i\rho_{2}}\left(-\bar{\rho}(\upomega^{1}+\tau\upomega^{2})-\tau\upnu_{1}+\upnu_{2}\right)~{},$
(6.5a) $\displaystyle\xi$
$\displaystyle=\frac{1}{2i\rho_{2}}\left(\rho(\upomega^{1}+\tau\upomega^{2})+\tau\upnu_{1}-\upnu_{2}\right)~{}.$
(6.5b)

In particular, the holomorphic $2$-form is defined by four integral classes,
and the Picard number of the base satisfies $\varrho(M)\geq 18$. Note that K3
surfaces of high Picard rank are quite special in the K3 moduli space. In
particular, it has been conjectured in [31] that an analogue of the
Shioda–Inose theorem exists when $\varrho(M)\geq 18$, and the case
$\varrho(M)=19$ was proved in [32].

The decomposition (6.5) of $\xi$ and $\Omega$ in integral cohomology yields a
solution to the flux quantization conditions. However, not every choice of
integral classes leads to a well-defined K3 geometry. The forms $\xi$ and
$\Omega$ must obey

$\displaystyle\Omega.\Omega$ $\displaystyle=0~{},$ $\displaystyle\Omega.\xi$
$\displaystyle=0~{},$ $\displaystyle\Omega.\bar{\xi}$ $\displaystyle=0~{},$
(6.6)

as well as the inequalities $\Omega.\bar{\Omega}>0$ and $\xi.\bar{\xi}\leq 0$.
These equations fix the value of the torus moduli $(\tau,\rho)$ and constrain
the allowed intersection numbers of the four integral classes. Indeed, the
above equalities can be rewritten as polynomial equations for $\tau$, and the
intersection numbers must be chosen such that these polynomials admit a root
in the upper half-plane. When this holds, the complexified Chern class (6.4)
defines a principal $T^{2}$ bundle over the K3 surface $M$ with quantized
$\tau$ and $\rho$ parameters, which preserves minimal supersymmetry. Note that
the geometry might be subject to additional constraints due to smoothness
requirements for the K3 surface. Let us also emphasize that, while the complex
constant $\gamma$ superficially looks like a continuous parameter that can be
used to deform off the N=2 locus where the torus Chern classes are ASD, this
is not the case. As already pointed out in [30], such deformations of a N=2
geometry induce complex structure deformations of $X$ that are incompatible
with the supersymmetry conditions on the flux. In the present analysis, we see
another crucial difference: flux quantization fixes the $\operatorname{SU}(2)$
structure of the K3 base to a specific integral locus which cannot be reached
by a smooth deformation of a generic N=2 geometry.

###  6.2 Obstructions to unwinding

The N=1 flux geometries admit an action of the T-duality group
$\operatorname{O{}}(\Gamma_{2,18})$. However, we will see that T-duality
transformations act within this class, and these configurations cannot be
connected by duality to a direct product $M\times T^{2}$. This is, of course,
not surprising: any T-dual theory should preserve exactly $4$ supercharges,
whereas a compactification on $M\times T^{2}$ preserves twice more
supersymmetry, unless orbifolded.

To study the $\operatorname{O{}}(\Gamma_{2,18})$ action in detail, we can
split the $2$-forms specifying the fibration structure of $X$ into their self-
dual and anti-self-dual parts. For the Chern classes of the $T^{2}$ fibration,
we can extract from (6.4) the self-dual $(2,0)\oplus(0,2)$ components:

$\displaystyle\upomega^{1}_{+}$
$\displaystyle=\frac{i\bar{\tau}}{2\tau_{2}}\gamma\Omega-\frac{i\tau}{2\tau_{2}}\bar{\gamma}\bar{\Omega}~{},$
$\displaystyle\upomega^{2}_{+}$
$\displaystyle=-\frac{i}{2\tau_{2}}\gamma\Omega+\frac{i}{2\tau_{2}}\bar{\gamma}\bar{\Omega}~{}.$
(6.7)

Abelian gauge instantons are characterized by the vector of classes
${\boldsymbol{\Uplambda}}=[\frac{1}{2\pi}{\cal
F}^{s}{\boldsymbol{\sigma}}_{s}]$ and, up to a large gauge transformation, the
curvatures ${\cal F}^{s}$ must be $(1,1)$ primitive forms. The vector
${\boldsymbol{\Uplambda}}$ can be split as

$\displaystyle{\boldsymbol{\Uplambda}}={\boldsymbol{\Uplambda}}_{0}-{{\textbf{a}}}_{I}\upomega^{I}~{},$
(6.8)

where ${\boldsymbol{\Uplambda}}_{0}$ encodes anti-self-dual classes
corresponding to a HYM connection pulled back from the base $M$. The Wilson
line parameters ${{\textbf{a}}}_{I}$ are quantized and could be shifted away
by a large gauge transformation, however we keep them in the present
discussion as they are generically produced by T-duality. When the parameters
${{\textbf{a}}}_{I}$ are non-trivial, ${\boldsymbol{\Uplambda}}$ has non-
vanishing $(2,0)\oplus(0,2)$ components. Turning on those self-dual components
introduces new terms in the expansion (6.5) of $\Omega$ in integral
cohomology. The self-dual components of the classes $\upnu_{I}$ can be
obtained from this expansion.

To summarize, supersymmetry imposes strong constraints on the self-dual part
$v_{+}$ of $v$, which has to be of the form

$\displaystyle v_{+}=\frac{i}{2\tau_{2}}\begin{pmatrix}\bar{\tau}\\\ -1\\\
\rho+\tfrac{1}{2}{{\textbf{a}}}_{1}\cdot({{\textbf{a}}}_{2}-\bar{\tau}{{\textbf{a}}}_{1})\\\
\rho\bar{\tau}+\tfrac{1}{2}{{\textbf{a}}}_{2}\cdot({{\textbf{a}}}_{2}-\bar{\tau}{{\textbf{a}}}_{1})\\\
{{\textbf{a}}}_{2}-\bar{\tau}{{\textbf{a}}}_{1}\end{pmatrix}\gamma\Omega+\text{c.c.}~{}.$
(6.9)

Note that $v_{+}$ is completely determined by the moduli and the complex
constant $\gamma$. In addition, it is not too hard to see that the first four
components of $v_{+}$ are non-zero, no matter what values the Wilson line
parameters take. This is a first sign that the presence of $(2,0)\oplus(0,2)$
components for $v$ obstructs unwinding dualities: the analysis of section 5
relied on dualizing $v$ using Wilson line shifts to set the classes
$\upnu_{1}$ and $\upnu_{2}$ to zero, and this is no longer possible.

To make this argument more precise, we need more information on the T-duality
orbit of $v_{+}$. Let us introduce the quantity

$\displaystyle
h(\tau,\rho,{{\textbf{a}}}_{1},{{\textbf{a}}}_{2})=\frac{i}{2\tau_{2}}\begin{pmatrix}\bar{\tau}\\\
-1\\\
\rho+\tfrac{1}{2}{{\textbf{a}}}_{1}\cdot({{\textbf{a}}}_{2}-\bar{\tau}{{\textbf{a}}}_{1})\\\
\rho\bar{\tau}+\tfrac{1}{2}{{\textbf{a}}}_{2}\cdot({{\textbf{a}}}_{2}-\bar{\tau}{{\textbf{a}}}_{1})\\\
{{\textbf{a}}}_{2}-\bar{\tau}{{\textbf{a}}}_{1}\end{pmatrix}$ (6.10)

that appears in the self-dual part (6.9) of $v$. While this $h$ is written as
a vector, it is not a proper $\operatorname{O{}}(2,18,{\mathbb{R}})$ multiplet
but has a more intricate transformation law under T-duality. The vector $v$
admits the decomposition

$\displaystyle v=v_{(1,1)}+h\gamma\Omega+\bar{h}\bar{\gamma}\bar{\Omega}~{},$
(6.11)

and its contribution to the Bianchi identity can be written

$-\tfrac{1}{2}v\bullet v=-\tfrac{1}{2}v_{(1,1)}\bullet
v_{(1,1)}+\frac{\rho_{2}}{\tau_{2}}|\gamma|^{2}\Omega.\bar{\Omega}~{},$ (6.12)

using $h\cdot h=0$ and $\bar{h}\cdot h=-\rho_{2}/\tau_{2}$. Both $h$ and
$\gamma$ transform under T-duality. Their general transformation law can be
derived from the action of $\operatorname{O{}}(\Gamma_{2,18})$ generators.
There is essentially one possibility for a duality
$g\in\operatorname{O{}}(\Gamma_{2,18})$ to preserve the form (6.11) of $v$:
$g$ must act on $h$ as

$\displaystyle
gh(\tau,\rho,{{\textbf{a}}}_{1},{{\textbf{a}}}_{2})=f_{g}(\tau,\rho,{{\textbf{a}}}_{1},{{\textbf{a}}}_{2})h(\mu_{g}^{-1}(\tau,\rho,{{\textbf{a}}}_{1},{{\textbf{a}}}_{2}))~{},$
(6.13)

with some complex moduli-dependent constant
$f_{g}(\tau,\rho,{{\textbf{a}}}_{1},{{\textbf{a}}}_{2})$. When this is the
case, the corresponding T-dual configuration $X_{v^{\prime}}=X_{gv}$ respects
the decomposition (6.11), with new parameters
$(\tau^{\prime},\rho^{\prime},{{\textbf{a}}}^{\prime}_{1},{{\textbf{a}}}^{\prime}_{2})=\mu_{g}^{-1}(\tau,\rho,{{\textbf{a}}}_{1},{{\textbf{a}}}_{2})$
and a complex constant
$\gamma^{\prime}=f_{g}(\tau,\rho,{{\textbf{a}}}_{1},{{\textbf{a}}}_{2})\gamma$.
We can easily see that the $\operatorname{O{}}(\Gamma_{2,18})$ generators
$g_{\text{t}}[{\cal R}]$, $g_{\text{b}}[m]$,
$g_{\text{s},I}[{\boldsymbol{\kappa}}]$ and $g_{\text{g}}[R]$ yield a
transformation rule of this sort.232323The $\operatorname{SL}(2,{\mathbb{Z}})$
transformations $g_{\text{t}}[{\cal R}]$ have a non-trivial factor
$f_{g_{\text{t}}[{\cal R}]}=({\cal R}^{2}{}_{2}-{\cal R}^{2}{}_{1}\tau)^{-1}$,
while the other generators have $f_{g}=1$. However, the generators
$g_{\text{i},I}$ and $g_{\text{ref},I}$ satisfy a different identity:

$\displaystyle
gh(\tau,\rho,{{\textbf{a}}}_{1},{{\textbf{a}}}_{2})=\overline{f_{g}(\tau,\rho,{{\textbf{a}}}_{1},{{\textbf{a}}}_{2})h(\mu_{g}^{-1}(\tau,\rho,{{\textbf{a}}}_{1},{{\textbf{a}}}_{2}))}~{}.$
(6.14)

For those T-duality transformations, the dual geometries are characterized by
a vector $v^{\prime}=gv$ which splits in self-dual and anti-self-dual parts as

$\displaystyle
v^{\prime}=v^{\prime}_{(1,1)}+h^{\prime}\bar{\gamma}^{\prime}\bar{\Omega}+\bar{h}^{\prime}\gamma^{\prime}\Omega~{},$
(6.15)

where
$h^{\prime}=h(\mu_{g}^{-1}(\tau,\rho,{{\textbf{a}}}_{1},{{\textbf{a}}}_{2}))$
and
$\gamma^{\prime}=f_{g}(\tau,\rho,{{\textbf{a}}}_{1},{{\textbf{a}}}_{2})\gamma$.
At first, this T-dual geometry seems incompatible with supersymmetry: it
explicitly breaks the form (6.11) required by the supersymmetry equations. In
particular, the complexified curvature of the $T^{2}$ fibration of
$X_{g^{\prime}}$ is given by

$\displaystyle\upomega^{\prime 1}+\tau^{\prime}\upomega^{\prime
2}=\xi^{\prime}+\bar{\gamma}^{\prime}\bar{\Omega}~{},$ (6.16)

and has $(1,1)$ and $(0,2)$ components, which is incompatible with the
conditions in (3.19). This apparent inconsistency is resolved by looking at
the action of the transformation $g$ on the holomorphic $(1,0)$ form $\Theta$,
which transforms as

$\displaystyle
g~{}:~{}\Theta~{}\to~{}\Theta^{\prime}=f_{g}(\tau,\rho,{{\textbf{a}}}_{1},{{\textbf{a}}}_{2})\,\bar{\Theta}~{}.$
(6.17)

In particular, any T-duality under which $h$ transforms by (6.14) must act
non-trivially on the $\operatorname{SU}(3)$ structure. The holomorphic
$3$-form $\Omega_{X}$ transforms as

$\displaystyle
g~{}:~{}\Omega_{X}~{}\to~{}\Omega_{X^{\prime}}=e^{\text{i}\alpha^{\prime}}e^{2\phi}\sqrt{\frac{\alpha^{\prime}\rho^{\prime}_{2}}{\tau^{\prime}_{2}}}\;\Omega\wedge\bar{\Theta}~{},$
(6.18)

where the overall phase factor is defined by
$e^{\text{i}\alpha^{\prime}}\sqrt{\rho^{\prime}_{2}/\tau^{\prime}_{2}}=f_{g}(\tau,\rho,{{\textbf{a}}}_{1},{{\textbf{a}}}_{2})\sqrt{\rho_{2}/\tau_{2}}$
and could be absorbed in a redefinition of $\Omega_{X^{\prime}}$. In the
T-dual picture, $\Theta$ is now a $(0,1)$ form, and supersymmetry requires

$\displaystyle J\wedge\boldsymbol{F}$ $\displaystyle=0~{},$
$\displaystyle\bar{\Omega}\wedge\boldsymbol{F}$ $\displaystyle=0~{},$ (6.19)

instead of (3.19). The complexified curvature $\boldsymbol{F}$ is constrained
to have $(1,1)$ and $(0,2)$ components only, and this is the case for the
T-dual geometry.

Examining the action of T-duality generators in detail, we find that every
generator obeys one of the two conditions (6.13) and (6.14). Consequently, the
same holds for a generic $\operatorname{O{}}(\Gamma_{2,18})$ element. As a
result, all T-dual configurations of an N=1 geometry are non-trivially
fibered: their complexified first Chern class has a non-trivial component
proportional to $\Omega$ or to $\bar{\Omega}$. As expected, we find that
Narain T-duality cannot undo the torus fibration of this class of backgrounds.
Torsional geometries with minimal supersymmetry are disconnected from flux-
free $M\times T^{2}$ compactifications.

##  7 Outlook

In this work we examined a large class of heterotic flux vacua distinguished
by the topologically non-Kähler target space as the underlying geometry. In
vacua that preserve $8$ supercharges we argued that all such vacua can be
deformed to a locus in the moduli space where a T-duality relates the vacuum
to a more familiar compactification on a topologically Kähler space. By taking
quotients of these dual pairs we uncovered surprising relations between Kähler
and non-Kähler compactifications, and studying the implications of these
relations may well provide some general lessons for heterotic flux vacua, and
perhaps flux vacua in general.

For instance, the ${\mathbb{Z}}_{3}$ quotient flux geometry described in
section (5.3) is singular. While the existence of a torsional linear sigma
model realization suggests that this is indeed a well-defined CFT, it is not
clear whether the orbifold singularities can be resolved by a marginal
deformation. The existence of the conventional Kähler T-dual geometry may shed
light on the nature of these singularities. If there is indeed a mechanism to
resolve these non-Kähler singularities, it may well generalize to other non-
Kähler flux geometries.

As we mentioned in the introduction, there are flux configurations that
preserve $8$ supercharges and are not covered by our analysis. These are
geometries introduced in [10], where the base of the torus fibration is
allowed to degenerate in such a way that the local geometry is of the form
$({\mathbb{C}}^{2}\times T^{2})/\Gamma$, where ${\mathbb{C}}^{2}/\Gamma$ is a
standard K3 orbifold singularity, but there is an added shift on the torus
fiber such that the action on the total space is free.242424The simplest
example is a ${\mathbb{Z}}_{2}$ action on ${\mathbb{C}}^{2}\times S^{1}$ which
acts on the local coordinates as
$(z_{1},z_{2},\theta)\to(-z_{1},-z_{2},\theta+\pi)$. Since our analysis is
ultimately based on compactifying an $8$-dimensional heterotic string theory
on a K3 manifold $M$, it does not immediately extend to the case when $M$ has
orbifold singularities.

Nevertheless, we can still ask what is the action of T-duality on these
configurations—after all the total space still has two commuting isometries. A
possible resolution comes from [35], where just such singularities were
studied in toroidal ${\mathbb{Z}}_{n}$ orbifolds of type II string, and it was
argued that the T-dual of the local singularity of this form is a singular
geometry ${\mathbb{C}}^{2}/{\mathbb{Z}}_{n}\times S^{1}$ equipped with a flat
but topologically non-trivial $B$-field gerbe. It would be interesting to see
if such structures arise in the T-duals of the geometries studied in [10] and
to precisely identify the singular Kähler geometries and the gerbe structures.

Another clear line of investigation is to return to the M/F-theoretic origins
of heterotic flux vacua [6]. For instance, we might consider the duals of
$3$-dimensional heterotic flux vacua realized as M-theory compactification on
K3$\times$K3 = $M\times M_{f}$, where we think of $M$ as the base K3 geometry,
while identifying M-theory compactified on $M_{f}$ with the heterotic string
on $T^{3}$. The data describing the abelian gauge bundle and $T^{3}$ fibration
on the heterotic side of the duality is encoded in the choice of the M-theory
$G$-flux. Taking $M_{f}$ to be elliptically fibered we can identify the
corresponding $\Gamma_{2,18}=\Gamma_{2,2}+\Gamma_{8}+\Gamma_{8}$ lattice in
$H^{2}(M_{f},{\mathbb{Z}})$ with the $T^{2}$ Narain lattice of the heterotic
theory. Then, as in [6], we have a clear picture of the significance of
various components of $G\in H^{2}(M,{\mathbb{Z}})\times\Gamma_{2,18}$:

  1. 1.

activating components in the negative-definite part of $\Gamma_{2,2}$ will
lead to backgrounds with $4$ supercharges;

  2. 2.

a component in the positive-definite part of $\Gamma_{2,2}$ is dual to a non-
trivial $T^{2}$ fibration;

  3. 3.

a component in the $\Gamma_{8}+\Gamma_{8}$ lattice corresponds to an abelian
instanton.

Finally, T-duality on the heterotic side can be understood as automorphisms of
$M_{f}$ that act on the $\Gamma_{2,18}$ lattice. This clear set-up seems
ideally suited to investigating the action of various quotient constructions
across string duality with reduced supersymmetry, and it should be useful to
investigate which heterotic orbifold actions dualize to “non-geometric”
actions on the M-theory side along the lines of [72].

We can also consider adding non-perturbative ingredients to the discussion:
for example, it seems reasonably clear that we can introduce $5$-branes in the
$8$–dimensional compactification while preserving supersymmetry. How does this
addition affect the effective theory of the flux vacua?

These and other questions indicate that while heterotic flux vacua may indeed
be “unwound” to more familiar configurations, they provide a useful duality
frame for a subset of the string landscape, and we suspect there are general
lessons to be learned that may be applied to backgrounds where we do not have
as much control.

## Appendix

##  Appendix A Cocycle details

In this appendix we give the explicit form for the cocycle factor
$\varepsilon({\boldsymbol{p}}_{1},{\boldsymbol{p}}_{2})$ used in our
computations, as well as details of the phase calculations for the generators.

###  A.1 Form of the cocycle

As discussed in [39, 38, 37], this is far from unique, but physical quantities
are independent of the choice.

For each $\Gamma_{8}$ factor, we define an antisymmetric bilinear form

$\displaystyle\Omega({\boldsymbol{\ell}}_{1},{\boldsymbol{\ell}}_{2})=\frac{1}{2}\sum_{i>j}\left(\ell_{1}^{i}\ell_{2}^{j}-\ell_{2}^{i}\ell_{1}^{j}\right){\boldsymbol{\alpha}}_{i}\cdot{\boldsymbol{\alpha}}_{j}~{},$
(A.1)

where the ${\boldsymbol{\alpha}}_{i}$ denote the simple roots. Decomposing

$\displaystyle\Gamma_{d,d+16}=\Gamma_{d,d}+\Gamma_{8}+\Gamma_{8}~{},$ (A.2)

we then take

$\displaystyle\varepsilon({\boldsymbol{p}}_{1},{\boldsymbol{p}}_{2})=\exp\left[\frac{i\pi}{2}\left(\text{n}_{1i}\text{w}^{i}_{2}-\text{w}_{1}^{i}\text{n}_{2i}\right)+i\pi\Omega({\boldsymbol{\ell}}_{1},{\boldsymbol{\ell}}_{2})+i\pi\Omega({\boldsymbol{\ell}}^{\prime}_{1},{\boldsymbol{\ell}}^{\prime}_{2})\right]$
(A.3)

It is a nice exercise to check that this satisfies the conditions

$\displaystyle\varepsilon({\boldsymbol{p}}_{2},{\boldsymbol{p}}_{1})$
$\displaystyle=e^{i\pi{\boldsymbol{p}}_{1}\cdot{\boldsymbol{p}}_{2}}\varepsilon({\boldsymbol{p}}_{1},{\boldsymbol{p}}_{2})~{},$
$\displaystyle\varepsilon({\boldsymbol{p}}_{1},{\boldsymbol{p}}_{3})\varepsilon({\boldsymbol{p}}_{2},{\boldsymbol{p}}_{3})$
$\displaystyle=\varepsilon({\boldsymbol{p}}_{1}+{\boldsymbol{p}}_{2},{\boldsymbol{p}}_{3})~{},$
(A.4)

and

$\displaystyle\varepsilon({\boldsymbol{p}}_{1},{\boldsymbol{p}}_{2})\varepsilon({\boldsymbol{p}}_{1}+{\boldsymbol{p}}_{2},{\boldsymbol{p}}_{3})$
$\displaystyle=\varepsilon({\boldsymbol{p}}_{1},{\boldsymbol{p}}_{2}+{\boldsymbol{p}}_{3})\varepsilon({\boldsymbol{p}}_{2},{\boldsymbol{p}}_{3})~{}.$
(A.5)

The first two conditions are necessary so that the operators commute and the
OPE closes. The last condition follows from associativity of the OPE.

###  A.2 Phases for generators of $\operatorname{O{}}(\Gamma_{1,17})$

In this appendix we provide some details for the phases in the actions on the
Narain CFT quoted in the text. We fix the form

$\displaystyle{\boldsymbol{p}}$
$\displaystyle=\text{w}{{\boldsymbol{e}}}+\text{n}{{\boldsymbol{e}}}^{\ast}+{{\boldsymbol{L}}}$
(A.6)

as the initial form of the lattice vector.

We begin with the factorized duality $g_{\text{i}}$, for which

$\displaystyle\varphi_{g_{\text{i}}}({\boldsymbol{p}})$
$\displaystyle=\text{n}{{\boldsymbol{e}}}+\text{w}{{\boldsymbol{e}}}^{\ast}+{{\boldsymbol{L}}}~{},$
(A.7)

and therefore

$\displaystyle\frac{\varepsilon(\varphi_{g_{\text{i}}}({\boldsymbol{p}}_{1}),\varphi_{g_{\text{i}}}({\boldsymbol{p}}_{2}))}{\varepsilon({\boldsymbol{p}}_{1},{\boldsymbol{p}}_{2})}=e^{i\pi(\text{n}_{1}\text{w}_{2}+\text{w}_{1}\text{n}_{2})}~{}.$
(A.8)

On the other hand, with $U(g_{\text{i}},{\boldsymbol{p}})=e^{i\pi\text{nw}}$,
we calculate

$\displaystyle\frac{U(g_{\text{i}},{\boldsymbol{p}}_{1}+{\boldsymbol{p}}_{2})}{U(g_{\text{i}},{\boldsymbol{p}}_{1})U(g_{\text{i}},{\boldsymbol{p}}_{2})}=e^{i\pi(\text{n}_{1}\text{w}_{2}+\text{w}_{1}\text{n}_{2})}~{},$
(A.9)

and we can also check that
$U(g_{\text{i}},\varphi_{g_{\text{i}}}({\boldsymbol{p}}))U(g_{\text{i}},{\boldsymbol{p}})=1$.

Next, for the reflection $g_{\text{ref}}$

$\displaystyle\varphi_{g_{\text{ref}}}({\boldsymbol{p}})$
$\displaystyle=-\text{n}{{\boldsymbol{e}}}-\text{w}{{\boldsymbol{e}}}^{\ast}+{{\boldsymbol{L}}}~{},$
(A.10)
$\displaystyle\frac{\varepsilon(\varphi_{g_{\text{ref}}}({\boldsymbol{p}}_{1}),\varphi_{g_{\text{r}}}({\boldsymbol{p}}_{2}))}{\varepsilon({\boldsymbol{p}}_{1},{\boldsymbol{p}}_{2})}=1~{},$
(A.11)

so we can set $U(g_{\text{ref}},{\boldsymbol{p}})=1$.

The Wilson line shift is more complicated. Using the explicit form of the
cocycle and the action

$\displaystyle{\boldsymbol{p}}^{\prime}=\varphi_{g_{\text{s}}}[{\boldsymbol{\kappa}}]({\boldsymbol{p}})=\text{w}{{\boldsymbol{e}}}+\left(\text{n}+{\boldsymbol{\kappa}}\cdot{{\boldsymbol{L}}}-{\textstyle\frac{1}{2}}{\boldsymbol{\kappa}}\cdot{\boldsymbol{\kappa}}\right){{\boldsymbol{e}}}^{\ast}+{{\boldsymbol{L}}}-{\boldsymbol{\kappa}}\text{w}$
(A.12)

we find

$\displaystyle\frac{\varepsilon({\boldsymbol{p}}^{\prime}_{1},{\boldsymbol{p}}^{\prime}_{2})}{\varepsilon({\boldsymbol{p}}_{1},{\boldsymbol{p}}_{2})}=\exp\left[i\pi\left(\text{w}_{1}S_{{\boldsymbol{\kappa}}}({{\boldsymbol{\Lambda}}}_{2})+\text{w}_{2}S_{{\boldsymbol{\kappa}}}({{\boldsymbol{\Lambda}}}_{1})\right)\right]~{},$
(A.13)

where

$\displaystyle
S_{{\boldsymbol{\kappa}}}({{\boldsymbol{\Lambda}}})=-\sum_{i}\boldsymbol{\kappa}^{i}{{\boldsymbol{L}}}^{i}-\sum_{i>j}\boldsymbol{\kappa}^{i}{{\boldsymbol{L}}}^{j}{\boldsymbol{\alpha}}_{i}\cdot{\boldsymbol{\alpha}}_{j}=\Omega({\boldsymbol{\kappa}},{{\boldsymbol{L}}})-{\textstyle\frac{1}{2}}{\boldsymbol{\kappa}}\cdot{{\boldsymbol{L}}}~{},$
(A.14)

where $i,j\in\\{1,\ldots,16\\}$, and we slightly abused our previous notation
by combining the simple roots ${\boldsymbol{\alpha}}$ and
${\boldsymbol{\alpha}}^{\prime}$ into a single set of $16$ vectors. We compare
this to the ratio of phases obtained with

$\displaystyle
U(g_{\text{s}}[{\boldsymbol{\kappa}}],{\boldsymbol{p}})=e^{i\pi(\text{w}+1)S_{{\boldsymbol{\kappa}}}({{\boldsymbol{L}}})}~{},$
(A.15)

which leads to

$\displaystyle\frac{U(g_{\text{s}}[{\boldsymbol{\kappa}}],{\boldsymbol{p}}_{1}+{\boldsymbol{p}}_{2})}{U(g_{\text{s}}[{\boldsymbol{\kappa}}],{\boldsymbol{p}}_{1})U(g_{\text{s}}[{\boldsymbol{\kappa}}],{\boldsymbol{p}}_{2})}=\exp\left[i\pi\left(\text{w}_{1}S_{{\boldsymbol{\kappa}}}({{\boldsymbol{L}}}_{2})+\text{w}_{2}S_{{\boldsymbol{\kappa}}}({{\boldsymbol{L}}}_{1})\right)\right]~{}.$
(A.16)

We also have the composition

$\displaystyle
U(g_{\text{s}}[{\boldsymbol{\kappa}}_{2}],\varphi_{g_{\text{s}}[{\boldsymbol{\kappa}}_{1}]}({\boldsymbol{p}}))U(g_{\text{s}}[{\boldsymbol{\kappa}}_{1}],{\boldsymbol{p}})$
$\displaystyle=U(g_{\text{s}}[\boldsymbol{\kappa}_{2}+\boldsymbol{\kappa}_{1}],{\boldsymbol{p}})e^{-i\pi\text{w}(\text{w}+1)S_{{\boldsymbol{\kappa}}_{2}}({\boldsymbol{\kappa}}_{1})}$
$\displaystyle=U(g_{\text{s}}[\boldsymbol{\kappa}_{2}+\boldsymbol{\kappa}_{1}],{\boldsymbol{p}})~{},$
(A.17)

where the last equality follows because
$S_{{\boldsymbol{\kappa}}_{2}}({\boldsymbol{\kappa}}_{1})\in{\mathbb{Z}}$.

Next we consider the phases associated to a rotation of
$\Gamma_{8}+\Gamma_{8}$. Since any such rotation can be obtained as a product
of Weyl reflections (see, for example [73]), we can restrict attention to
$R_{{\boldsymbol{\beta}}}$, where ${\boldsymbol{\beta}}$ is a root, and the
action on the simple roots is

$\displaystyle
R_{{\boldsymbol{\beta}}}({\boldsymbol{\alpha}})={\boldsymbol{\alpha}}-({\boldsymbol{\beta}}\cdot{\boldsymbol{\alpha}}){\boldsymbol{\beta}}~{}.$
(A.18)

Denoting this reflection by $g_{\text{g}}[{\boldsymbol{\beta}}]$, and setting
${\boldsymbol{p}}^{\prime}=\varphi_{g_{\text{g}}[{\boldsymbol{\beta}}]}({\boldsymbol{p}})$,
we then compute

$\displaystyle\frac{\varepsilon({\boldsymbol{p}}^{\prime}_{1},{\boldsymbol{p}}^{\prime}_{2})}{\varepsilon({\boldsymbol{p}}_{1},{\boldsymbol{p}}_{2})}=\exp\left[2\pi
i\left(S_{{\boldsymbol{\beta}}}({{\boldsymbol{L}}}_{1})\Omega({\boldsymbol{\beta}},{{\boldsymbol{L}}}_{2})+S_{{\boldsymbol{\beta}}}({{\boldsymbol{L}}}_{2})\Omega({\boldsymbol{\beta}},{{\boldsymbol{L}}}_{1})\right)\right]~{}.$
(A.19)

We set

$\displaystyle U(g_{\text{g}}[{\boldsymbol{\beta}}],{\boldsymbol{p}})=e^{2\pi
iS_{{\boldsymbol{\beta}}}({{\boldsymbol{L}}})\Omega({\boldsymbol{\beta}},{{\boldsymbol{L}}})+i\pi\Omega({\boldsymbol{\beta}},{{\boldsymbol{L}}})}~{},$
(A.20)

and this satisfies

$\displaystyle\frac{\varepsilon({\boldsymbol{p}}^{\prime}_{1},{\boldsymbol{p}}^{\prime}_{2})}{\varepsilon({\boldsymbol{p}}_{1},{\boldsymbol{p}}_{2})}$
$\displaystyle=\frac{U(g_{\text{g}}[{\boldsymbol{\beta}}],{\boldsymbol{p}}_{1}+{\boldsymbol{p}}_{2})}{U(g_{\text{g}}[{\boldsymbol{\beta}}],{\boldsymbol{p}}_{1})U(g_{\text{g}}[{\boldsymbol{\beta}}],{\boldsymbol{p}}_{2})}~{},$
(A.21)

as well as
$U(g_{\text{g}}[{\boldsymbol{\beta}}],{\boldsymbol{p}}^{\prime})U(g_{\text{g}}[{\boldsymbol{\beta}}],{\boldsymbol{p}})=1$.
The induced map on the moduli is

$\displaystyle\mu_{g_{\text{g}}[{\boldsymbol{\beta}}]}(r,{{\textbf{a}}})=(r,{{\textbf{a}}}-({{\textbf{a}}}\cdot{\boldsymbol{\beta}}){\boldsymbol{\beta}})~{}.$
(A.22)

###  A.3 Phases for generators of $\operatorname{O{}}(\Gamma_{2,18})$

In the following section, we collect the relevant details of the action of
$\operatorname{O{}}(\Gamma_{2,18})$ on vertex operators and on the associated
moduli. The $T^{2}$ parameters are encoded in two constants $\tau$ and $\rho$
valued in the upper half-plane.

Factorized dualities act with a phase

$\displaystyle U(g_{\text{i},1},{\boldsymbol{p}})$
$\displaystyle=e^{i\pi\text{n}_{1}\text{w}^{1}}~{},$ $\displaystyle
U(g_{\text{i},2},{\boldsymbol{p}})$
$\displaystyle=e^{i\pi\text{n}_{2}\text{w}^{2}}~{}.$ (A.23)

Their action on moduli is

$\displaystyle\mu_{g_{\text{i},1}}(\tau,\rho,{{\textbf{a}}}_{1},{{\textbf{a}}}_{2})=$
$\displaystyle\left(\rho+\tfrac{1}{2}{{\textbf{a}}}_{1}\cdot({{\textbf{a}}}_{2}-\bar{\tau}{{\textbf{a}}}_{1}),\frac{\rho_{2}\tau+\tfrac{1}{2}\tau_{2}{{\textbf{a}}}_{1}\cdot{{\textbf{a}}}_{2}}{\rho_{2}+\tfrac{1}{2}\tau_{2}{{\textbf{a}}}_{1}\cdot{{\textbf{a}}}_{1}},\frac{-\tau_{2}}{\rho_{2}+\tfrac{1}{2}\tau_{2}{{\textbf{a}}}_{1}\cdot{{\textbf{a}}}_{1}}{{\textbf{a}}}_{1},-\frac{\tau_{1}\rho_{2}+\tau_{2}\rho_{1}+\tfrac{1}{2}\tau_{2}{{\textbf{a}}}_{1}\cdot{{\textbf{a}}}_{2}}{\rho_{2}+\tfrac{1}{2}\tau_{2}{{\textbf{a}}}_{1}\cdot{{\textbf{a}}}_{1}}{{\textbf{a}}}_{1}+{{\textbf{a}}}_{2}\right)~{},$
$\displaystyle\mu_{g_{\text{i},2}}(\tau,\rho,{{\textbf{a}}}_{1},{{\textbf{a}}}_{2})=$
$\displaystyle\left(\frac{-\bar{\tau}}{\rho\bar{\tau}+\tfrac{1}{2}{{\textbf{a}}}_{2}\cdot({{\textbf{a}}}_{2}-\bar{\tau}{{\textbf{a}}}_{1})},\frac{-\rho_{2}\bar{\tau}-\tfrac{1}{2}\tau_{2}{{\textbf{a}}}_{1}\cdot{{\textbf{a}}}_{2}}{\rho_{2}|\tau|^{2}+\tfrac{1}{2}\tau_{2}{{\textbf{a}}}_{2}\cdot{{\textbf{a}}}_{2}},{{\textbf{a}}}_{1}-\frac{\tau_{1}\rho_{2}-\tau_{2}\rho_{1}+\tfrac{1}{2}\tau_{2}{{\textbf{a}}}_{1}\cdot{{\textbf{a}}}_{2}}{\rho_{2}|\tau|^{2}+\tfrac{1}{2}\tau_{2}{{\textbf{a}}}_{2}\cdot{{\textbf{a}}}_{2}}{{\textbf{a}}}_{2},\frac{-\tau_{2}}{\rho_{2}|\tau|^{2}+\tfrac{1}{2}\tau_{2}{{\textbf{a}}}_{2}\cdot{{\textbf{a}}}_{2}}{{\textbf{a}}}_{2}\right)~{}.$
(A.24)

Let us also quote the action $\mu_{g_{\text{i}}}$ of the product
$g_{\text{i}}=g_{\text{i},1}g_{\text{i},2}$ of the two factorized dualities:

$\displaystyle\mu_{g_{\text{i}}}(\tau,\rho,{{\textbf{a}}}_{1},{{\textbf{a}}}_{2})=$
$\displaystyle\Bigg{(}\frac{-\bar{\rho}-\tfrac{1}{2}{{\textbf{a}}}_{1}\cdot({{\textbf{a}}}_{2}-\tau{{\textbf{a}}}_{1})}{\tau\bar{\rho}+\tfrac{1}{2}{{\textbf{a}}}_{2}\cdot({{\textbf{a}}}_{2}-\tau{{\textbf{a}}}_{1})},~{}\frac{-\tau_{2}\bar{\rho}}{\tau_{2}|\rho|^{2}+\frac{1}{2}\rho_{2}|{{\textbf{a}}}_{2}-\tau{{\textbf{a}}}_{1}|^{2}+\tfrac{1}{4}\tau_{2}({{\textbf{a}}}_{1}^{2}{{\textbf{a}}}_{2}^{2}-({{\textbf{a}}}_{1}\cdot{{\textbf{a}}}_{2})^{2})},$
$\displaystyle\frac{-(\rho_{2}|\tau|^{2}+\tfrac{1}{2}\tau_{2}{{\textbf{a}}}_{2}^{2}){{\textbf{a}}}_{1}+(\tau_{1}\rho_{2}-\tau_{2}\rho_{1}+\tfrac{1}{2}\tau_{2}{{\textbf{a}}}_{1}\cdot{{\textbf{a}}}_{2}){{\textbf{a}}}_{2}}{\tau_{2}|\rho|^{2}+\frac{1}{2}\rho_{2}|{{\textbf{a}}}_{2}-\tau{{\textbf{a}}}_{1}|^{2}+\tfrac{1}{4}\tau_{2}({{\textbf{a}}}_{1}^{2}{{\textbf{a}}}_{2}^{2}-({{\textbf{a}}}_{1}\cdot{{\textbf{a}}}_{2})^{2})},~{}\frac{(\tau_{1}\rho_{2}+\tau_{2}\rho_{1}+\frac{1}{2}\tau_{2}{{\textbf{a}}}_{1}\cdot{{\textbf{a}}}_{2}){{\textbf{a}}}_{1}-(\rho_{2}+\tfrac{1}{2}\tau_{2}{{\textbf{a}}}_{1}^{2}){{\textbf{a}}}_{2}}{\tau_{2}|\rho|^{2}+\frac{1}{2}\rho_{2}|{{\textbf{a}}}_{2}-\tau{{\textbf{a}}}_{1}|^{2}+\tfrac{1}{4}\tau_{2}({{\textbf{a}}}_{1}^{2}{{\textbf{a}}}_{2}^{2}-({{\textbf{a}}}_{1}\cdot{{\textbf{a}}}_{2})^{2})}\Bigg{)}~{},$
(A.25)

where we denoted for brevity
${{\textbf{a}}}^{2}={{\textbf{a}}}\cdot{{\textbf{a}}}$ and
$|{{\textbf{a}}}|^{2}={{\textbf{a}}}\cdot\bar{{{\textbf{a}}}}$.

For the $\operatorname{SL}(2,{\mathbb{Z}})$ transformations
$g_{\text{t}}[{\cal R}]$ corresponding to $T^{2}$ isometries, the cocycle
condition is compatible with $U(g_{\text{t}}[{\cal R}],{\boldsymbol{p}})=1$,
and the moduli action takes the following form:

$\displaystyle\mu_{g_{\text{t}}}[{\cal
R}](\tau,\rho,{{\textbf{a}}}_{I})=\left(\frac{{\cal R}^{1}{}_{2}+{\cal
R}^{2}{}_{2}\tau}{{\cal R}^{1}{}_{1}+{\cal R}^{2}{}_{1}\tau},\rho,{\cal
R}^{J}{}_{I}{{\textbf{a}}}_{J}\right)~{},$ (A.26)

with the usual $\operatorname{PSL}(2,{\mathbb{Z}})$ action on $\tau$. We can
also set $U(g_{\text{ref},I},{\boldsymbol{p}})=1$ for the $T^{2}$ reflections.
The action of $\mu_{g_{\text{ref},I}}$ reads

$\displaystyle\mu_{g_{\text{ref},1}}(\tau,\rho,{{\textbf{a}}}_{1},{{\textbf{a}}}_{2})$
$\displaystyle=\left(-\bar{\tau},-\bar{\rho},-{{\textbf{a}}}_{1},{{\textbf{a}}}_{2}\right)~{},$
$\displaystyle\mu_{g_{\text{ref},2}}(\tau,\rho,{{\textbf{a}}}_{1},{{\textbf{a}}}_{2})$
$\displaystyle=\left(-\bar{\tau},-\bar{\rho},{{\textbf{a}}}_{1},-{{\textbf{a}}}_{2}\right)~{},$
(A.27)

with a complex conjugation of the $T^{2}$ parameters.

Both $B$-field shifts and Wilson line shifts require a non-trivial phase. We
take

$\displaystyle U(g_{\text{b}}[m],{\boldsymbol{p}})$ $\displaystyle=e^{i\pi
m\text{w}^{1}\text{w}^{2}}~{},$ $\displaystyle
U(g_{\text{s}}[{\boldsymbol{\kappa}}_{I}],{\boldsymbol{p}})$
$\displaystyle=e^{i\pi\text{w}^{1}S_{{\boldsymbol{\kappa}}_{1}}({{\boldsymbol{L}}}+\frac{1}{2}{\boldsymbol{\kappa}}_{1})+i\pi\text{w}^{2}S_{{\boldsymbol{\kappa}}_{2}}({{\boldsymbol{L}}}+\frac{1}{2}{\boldsymbol{\kappa}}_{2})+\frac{i\pi}{2}\text{w}^{1}\text{w}^{2}S_{{\boldsymbol{\kappa}}_{1}}({\boldsymbol{\kappa}}_{2})-S_{{\boldsymbol{\kappa}}_{2}}({\boldsymbol{\kappa}}_{1}))}~{}.$
(A.28)

The corresponding transformation law of the moduli is

$\displaystyle\mu_{g_{\text{b}}}[m](\tau,\rho,{{\textbf{a}}}_{1},{{\textbf{a}}}_{2})$
$\displaystyle=(\tau,\rho+m,{{\textbf{a}}}_{1},{{\textbf{a}}}_{2})~{},$
$\displaystyle\mu_{g_{\text{s}}}[{\boldsymbol{\kappa}}_{I}](\tau,\rho,{{\textbf{a}}}_{1},{{\textbf{a}}}_{2})$
$\displaystyle=\left(\tau,\rho+\tfrac{1}{2}\epsilon^{IJ}{{\textbf{a}}}_{I}\cdot{\boldsymbol{\kappa}}_{J},{{\textbf{a}}}_{1}-{\boldsymbol{\kappa}}_{1},{{\textbf{a}}}_{2}-{\boldsymbol{\kappa}}_{2}\right)~{}.$
(A.29)

Finally, the action of $\Gamma_{8}+\Gamma_{8}$ rotations is identical to the
one already discussed for $\operatorname{O{}}(\Gamma_{1,17})$ dualities.

###  A.4 Phases in the T-dual description

In this appendix, we make some comments on the action of the T-dual group
$G^{\prime}={\cal T}G{\cal T}^{-1}$ on the vertex operators ${\cal
V}_{\boldsymbol{p}}$ , given an action of the group $G$. Every element
$g^{\prime}$ of $G^{\prime}$ is obtained by conjugating some $g\in G$ by the
transformation ${\cal T}$. Since ${\cal T}$ has an action $\varphi_{\cal T}$
on the $\Gamma_{d,d+16}$ lattice, we can obtain the lattice action of
$g^{\prime}={\cal T}g{\cal T}^{-1}$ by

$\displaystyle\varphi_{g^{\prime}}=\varphi_{\cal T}\varphi_{g}\varphi_{\cal
T}^{-1}~{}.$ (A.30)

Whenever $\varphi_{\cal T}$ acts non-trivially on $\Gamma_{d,d+16}$, we must
also define a phase $U({\cal T},{\boldsymbol{p}})$ in order for the action of
${\cal T}$ on vertex operators to be compatible with the operator product
expansion. The phase associated to the inverse transformation ${\cal T}^{-1}$
is

$\displaystyle U({\cal T}^{-1}\\!,{\boldsymbol{p}})=U({\cal T},\varphi_{{\cal
T}^{-1}}({\boldsymbol{p}}))^{-1}~{},$ (A.31)

which ensures that the product law ${\cal T}^{-1}\circ({\cal T}\circ{\cal
V}_{\boldsymbol{p}})={\cal V}_{\boldsymbol{p}}$ is satisfied. Every element
$g$ of the group $G$ also has a corresponding phase $U(g,{\boldsymbol{p}})$.
This phase obeys the cocycle condition (2.17) and the group law (2.15), since
we assume a well-defined action of $G$.

Consider now an element $g^{\prime}={\cal T}g{\cal T}^{-1}$ of $G^{\prime}$.
We define the phase $U(g^{\prime},{\boldsymbol{p}})$ by

$\displaystyle U(g^{\prime},{\boldsymbol{p}})=U({\cal T},\varphi_{g{\cal
T}^{-1}}({\boldsymbol{p}}))~{}U(g,\varphi_{{\cal
T}^{-1}}({\boldsymbol{p}}))U({\cal T}^{-1},{\boldsymbol{p}})~{}.$ (A.32)

We can easily check that this phase obeys the cocycle condition (2.17).
Moreover, for two elements $g^{\prime}_{1}$ and $g^{\prime}_{2}$ of
$G^{\prime}$, we can check that
$U(g^{\prime}_{2},\varphi_{g^{\prime}_{1}}({\boldsymbol{p}}))U(g^{\prime}_{1},{\boldsymbol{p}})$
and $U(g^{\prime}_{2}g^{\prime}_{1},{\boldsymbol{p}})$ agree. In other words,
the product law (2.15) for the phases associated to $G$ implies the product
law for the phases associated to $G^{\prime}$. We emphasize that this
construction does not necessitate an embedding $G\subseteq H$ of the group $G$
in a bigger group $H$, with ${\cal T}\in H$. The phases
$U(g^{\prime},{\boldsymbol{p}})$ define a consistent $G^{\prime}$ action
without having to specify such embedding in $H$ or the corresponding phases
$U(h,{\boldsymbol{p}})$ for every $h\in H$.

As a particular case, notice that if an element $g\in G$ has a trivial lattice
action $\varphi_{g}({\boldsymbol{p}})={\boldsymbol{p}}$, then the conjugated
element $g^{\prime}$ also has
$\varphi_{g^{\prime}}({\boldsymbol{p}})={\boldsymbol{p}}$ and the
corresponding phase, defined in (A.32), takes the simple form

$\displaystyle U(g^{\prime},{\boldsymbol{p}})=U(g,\varphi_{{\cal
T}}^{-1}({\boldsymbol{p}}))~{}.$ (A.33)

Let us also describe in more details the case of a cyclic group
$G\simeq{\mathbb{Z}}_{k}$. The action of $G$ is fully specified by the action
of a generator $g$, with $\varphi_{g}^{k}({\boldsymbol{p}})={\boldsymbol{p}}$
and

$\displaystyle U(g,\varphi_{g}^{k-1}({\boldsymbol{p}}))\dots
U(g,\varphi_{g}^{2}({\boldsymbol{p}}))U(g,\varphi_{g}({\boldsymbol{p}}))U(g,{\boldsymbol{p}})=1~{}.$
(A.34)

As an example, consider the product of Weyl reflections
$R=R_{{\boldsymbol{\beta}}_{2}}R_{{\boldsymbol{\beta}}_{1}}$ introduced in
section 5, where the two roots ${\boldsymbol{\beta}}_{1}$ and
${\boldsymbol{\beta}}_{2}$ are at a $\frac{2\pi}{3}$ angle, i.e.
${\boldsymbol{\beta}}_{1}\cdot{\boldsymbol{\beta}}_{2}=-1$. This
transformation generates a ${\mathbb{Z}}_{3}$ subgroup of
$\operatorname{O{}}(\Gamma_{8}+\Gamma_{8})$. Since
${\boldsymbol{\beta}}_{1}\cdot{\boldsymbol{\beta}}_{2}=-S_{{\boldsymbol{\beta}}_{1}}({\boldsymbol{\beta}}_{2})-S_{{\boldsymbol{\beta}}_{2}}({\boldsymbol{\beta}}_{1})$,
we can—without loss of generality—order the two roots such that
$S_{{\boldsymbol{\beta}}_{1}}({\boldsymbol{\beta}}_{2})$ is even and
$S_{{\boldsymbol{\beta}}_{2}}({\boldsymbol{\beta}}_{1})$ is odd. We define the
following cocycle252525Here we slightly abuse notation by writing
$U(R,{\boldsymbol{p}})$ instead of $U(g_{\text{g}}[R],{\boldsymbol{p}})$ and
$\varphi_{R}$ instead of $\varphi_{g_{\text{g}}[R]}$. associated to $R$:

$\displaystyle U(R,{\boldsymbol{p}})$
$\displaystyle=e^{i\pi\left(S_{{\boldsymbol{\beta}}_{1}}({{\boldsymbol{L}}}){\boldsymbol{\beta}}_{1}\cdot{{\boldsymbol{L}}}+S_{{\boldsymbol{\beta}}_{2}}(R_{{\boldsymbol{\beta}}_{1}}({{\boldsymbol{L}}})){\boldsymbol{\beta}}_{2}\cdot
R_{{\boldsymbol{\beta}}_{1}}({{\boldsymbol{L}}})\right)}$
$\displaystyle=e^{i\pi\left(S_{{\boldsymbol{\beta}}_{1}}({{\boldsymbol{L}}}){\boldsymbol{\beta}}_{1}\cdot{{\boldsymbol{L}}}+S_{{\boldsymbol{\beta}}_{2}}({{\boldsymbol{L}}})({\boldsymbol{\beta}}_{1}+{\boldsymbol{\beta}}_{2})\cdot{{\boldsymbol{L}}}+({\boldsymbol{\beta}}_{1}\cdot{{\boldsymbol{L}}})({\boldsymbol{\beta}}_{1}+{\boldsymbol{\beta}}_{2})\cdot{{\boldsymbol{L}}}\right)}~{}.$
(A.35)

We can easily check that $U(R,{\boldsymbol{p}})$ satisfies the cocycle
property (2.17). Moreover, we can compute

$\displaystyle U(R,\varphi_{R}({\boldsymbol{p}}))$
$\displaystyle=e^{i\pi\left(S_{{\boldsymbol{\beta}}_{1}}({{\boldsymbol{L}}}){\boldsymbol{\beta}}_{2}\cdot{{\boldsymbol{L}}}+S_{{\boldsymbol{\beta}}_{2}}({{\boldsymbol{L}}}){\boldsymbol{\beta}}_{1}\cdot{{\boldsymbol{L}}}+({\boldsymbol{\beta}}_{1}\cdot{{\boldsymbol{L}}}){\boldsymbol{\beta}}_{2}\cdot{{\boldsymbol{L}}}\right)}~{},$
$\displaystyle U(R,\varphi_{R^{2}}({\boldsymbol{p}}))$
$\displaystyle=e^{i\pi\left(S_{{\boldsymbol{\beta}}_{1}}({{\boldsymbol{L}}})({\boldsymbol{\beta}}_{1}+{\boldsymbol{\beta}}_{2})\cdot{{\boldsymbol{L}}}+S_{{\boldsymbol{\beta}}_{2}}({{\boldsymbol{L}}}){\boldsymbol{\beta}}_{2}\cdot{{\boldsymbol{L}}}+({\boldsymbol{\beta}}_{1}\cdot{{\boldsymbol{L}}}){\boldsymbol{\beta}}_{1}\cdot{{\boldsymbol{L}}}\right)}~{},$
(A.36)

and we see that $U(R,{\boldsymbol{p}})$ satisfies the ${\mathbb{Z}}_{3}$
condition

$\displaystyle
U(R,{\boldsymbol{p}})U(R,\varphi_{R}({\boldsymbol{p}}))U(R,\varphi_{R^{2}}({\boldsymbol{p}}))=1~{}.$
(A.37)

As a result, any action obtained by conjugation of this ${\mathbb{Z}}_{3}$
group by some transformation ${\cal T}$ will satisfy the same
${\mathbb{Z}}_{3}$ product law. For the unwinding T-duality ${\cal
T}=g_{\text{i}}g_{\text{s},1}[2{\boldsymbol{\beta}}_{1}]g_{\text{s},2}[2{\boldsymbol{\beta}}_{2}]$
discussed in section 5, the cocycle condition for ${\cal T}$ is solved by the
choice $U({\cal T},{\boldsymbol{p}})=e^{i\pi\text{w}^{I}\text{n}_{I}}$.
Computing the explicit form of the orbifold phase using (A.32) yields $U({\cal
T}g_{3}{\cal T}^{-1},{\boldsymbol{p}})=U(g_{3},{\boldsymbol{p}})$.

##  Appendix B Some details on the cohomology of $X$

Because the base $M$ is simply connected, it is easy to use the Leray–Serre
spectral sequence to calculate the cohomology of $X$.262626Since $M$ is simply
connected, we can avoid the complications of having to work with local
coefficients. See, for example, chapter 5 of [41] for an introduction. This
spectral sequence has $E_{2}^{p,q}=H^{p}(M,H^{q}(F))$ as its second stage,
where $F=T^{2}$ denotes the torus fiber, and the spectral sequence converges
to $\oplus_{p+q=k}E^{p,q}_{\infty}=H^{k}(X)$. In our case it will degenerate
when we take cohomology at the second page.

The universal coefficients theorem determines the cohomology groups
$H^{p}(M,H^{q}(F))$. In general [74] these are given by the exact sequence

${0}$${\text{Ext}(H_{p-1}(M),H^{q}(F))}$${H^{p}(M,H^{q}(F))}$${\text{Hom}(H_{p}(M),H^{q}(F))}$${0}$
(B.1)

But, this simplifies because $\text{Ext}(H,G)=0$ whenever $H$ is a freely
generated. Thus, we find

$\displaystyle H^{p}(M,H^{q}(F))=\begin{cases}H^{p}(M)\\\ H^{p}(M)\times
H^{p}(M)\\\ H^{p}(M)~{},\end{cases}$ (B.2)

and the second page of the spectral sequence has the form

${H^{0}(M)}$${0}$${H^{2}(M)}$${0}$${H^{4}(M)}$${H^{0}(M,H^{1}(F))}$${0}$${H^{2}(M,H^{1}(F))}$${0}$${H^{4}(M,H^{1}(F))}$${H^{0}(M)}$${0}$${H^{2}(M)}$${0}$${H^{4}(M)}$$\scriptstyle{d_{22}}$$\scriptstyle{d_{24}}$$\scriptstyle{d_{21}}$$\scriptstyle{d_{23}}$
(B.3)

We indicated the $d_{2}$ map, and it is clear that $d_{3}=0$, so that
$E^{p,q}_{3}=E^{p,q}_{\infty}$. All of the maps are fixed in terms of the cup
product and the classes $\upomega^{I}$:

${H^{0}(M,H^{1}(F))}$${H^{2}(M)}$${(x_{1},x_{2})}$${(x_{1}\upomega^{1}+x_{2}\upomega^{2})~{},}$$\scriptstyle{d_{21}}$$\scriptstyle{d_{21}}$
(B.4)
${H^{0}(M)}$${H^{2}(M,H^{1}(F))}$${H^{4}(M)}$${(x)}$${(x\upomega^{1},x\upomega^{2})}$${(C_{1},C_{2})}$${(C_{1}\cup\upomega^{2}-C_{2}\cup\upomega^{1})~{},}$$\scriptstyle{d_{22}}$$\scriptstyle{d_{23}}$$\scriptstyle{d_{22}}$$\scriptstyle{d_{23}}$
(B.5)
${H^{2}(M)}$${H^{4}(M,H^{1}(F))}$${(C)}$${(C\cup\upomega^{1},C\cup\upomega^{2})~{}.}$$\scriptstyle{d_{24}}$$\scriptstyle{d_{24}}$
(B.6)

So, we find $H^{0}(X)={\mathbb{Z}}$ and $H^{6}(X)={\mathbb{Z}}$, while

$\displaystyle H^{1}(X)$ $\displaystyle=\ker d_{21}~{},$ $\displaystyle
H^{2}(X)$ $\displaystyle=\ker d_{22}\oplus
H^{2}(M)/\operatorname{im}d_{21}~{},$ $\displaystyle H^{3}(X)$
$\displaystyle=\ker d_{23}/\operatorname{im}d_{22}~{},$ $\displaystyle
H^{4}(X)$ $\displaystyle=H^{4}(M)/\operatorname{im}d_{23}\oplus\ker
d_{24}~{},$ $\displaystyle H^{5}(X)$ $\displaystyle=\left(H^{4}(M)\oplus
H^{4}(M)\right)/\operatorname{im}d_{24}~{}.$ (B.7)

To study the groups in detail, we identify $H^{4}(M)\simeq{\mathbb{Z}}$, so
that $C\cup\upomega^{1}=C.\upomega^{1}\in{\mathbb{Z}}$, and then we use the
basis $\\{\upomega_{\text{p}}^{1},\upomega_{\text{p}}^{2}\,\ldots,\\}$ and its
dual basis.

We have three distinct situations to consider. First, there is the trivial
fibration with $m_{1}=0$. In this case the cohomology is determined by the
Künneth formula, and of course every group is free. In what follows we will
leave this case out and work with $m_{1}>0$.

  1. 1.

$H^{1}(X)$. There are two non-trivial possibilities.

    1. (a)

$m_{1}>0$, $m_{2}=0$. In this case just one circle is fibered non-trivially
and $H^{1}(X)={\mathbb{Z}}$. We may as well factor out the trivial circle from
$X=Y\times S^{1}$ and work with $Y$ directly.

    2. (b)

$m_{1}>0$, $m_{2}>0$. Both circles are non-trivially fibered, and
$H^{1}(X)=0$.

  2. 2.

$H^{2}(X)$. If a circle is non-trivially fibered, $\ker d_{22}=0$, so we just
need to consider the quotient $H^{2}(M)/\operatorname{im}d_{21}$.

    1. (a)

When $m_{1}>0$ and $m_{2}=0$, then
$\operatorname{im}d_{21}=\text{Span}\\{m_{1}\upomega_{\text{p}}^{1}\\}$, and
therefore $H^{2}(X)={\mathbb{Z}}^{21}\times{\mathbb{Z}}_{m_{1}}$.

    2. (b)

When $m_{1}>0$ and $m_{2}=0$, then
$\operatorname{im}d_{21}=\text{Span}\\{m_{1}\upomega_{\text{p}}^{1},m_{1}m_{2}\upomega_{\text{p}}^{1}\\}$,
and therefore
$H^{2}(X)={\mathbb{Z}}^{20}\times{\mathbb{Z}}_{m_{1}}\times{\mathbb{Z}}_{m_{1}m_{2}}$.

Notice that in either case the torsion subgroup reproduces the torsion
subgroup of $H_{1}(X)=\pi_{1}(X)/{[\pi_{1}(X),\pi_{1}(X)]}$.

  3. 3.

$H^{3}(X)$. To study this group it is convenient to expand $C_{2}$ in the dual
basis:

$\displaystyle
C_{2}=a\upomega_{\text{p}1}^{\ast}+b\upomega_{\text{p}2}^{\ast}+\widehat{C}_{2}~{}.$
(B.8)

Then

$\displaystyle
d_{23}(C_{1},C_{2})=C_{1}.\upomega^{2}-C_{2}.\upomega^{1}=m_{1}\left(m_{2}C_{1}.\upomega^{2}_{\text{p}}-\widehat{C}_{2}.\upomega_{\text{p}}^{1}-a\right)=m_{1}\left(m_{2}C_{1}.\upomega^{2}_{\text{p}}-a\right)~{},$
(B.9)

where we used $\widehat{C}_{2}.\upomega_{\text{p}}^{1}=0$. We see therefore,
that $a$ is determined, and $\ker
d_{23}={\mathbb{Z}}^{22}\times{\mathbb{Z}}^{21}$:

$\displaystyle\ker
d_{23}=\\{(C_{1},(m_{2}C_{1}.\upomega^{2}_{\text{p}})\upomega^{\ast}_{\text{p}1}+b\upomega^{\ast}_{\text{p}2}+\widehat{C}_{2})\\}~{}.$
(B.10)

The quotient by $\operatorname{im}d_{22}$ gives rise to possible torsion in
$H^{3}(X)$. This is because

$\displaystyle\operatorname{im}d_{22}=\text{Span}\\{(m_{1}\upomega^{1}_{\text{p}},m_{1}m_{2}\upomega^{2}_{\text{p}})\\}~{},$
(B.11)

and therefore the element
$(\upomega^{1}_{\text{p}},m_{2}\upomega^{2}_{\text{p}})$ generates a
${\mathbb{Z}}_{m_{1}}$ subgroup in $H^{3}(X)$. This is the only source of
torsion, so we conclude that whenever $m_{1}>0$

$\displaystyle H^{3}(X)={\mathbb{Z}}^{2\times
22-2}\times{\mathbb{Z}}_{m_{1}}~{}.$ (B.12)

Note that when $m_{2}=0$, so that $X=Y\times S^{1}$, then the Künneth formula
determines

$\displaystyle H^{2}(X)$ $\displaystyle=H^{2}(Y)~{},$ $\displaystyle H^{3}(X)$
$\displaystyle=H^{3}(Y)\times H^{2}(Y)~{},$ (B.13)

so that

$\displaystyle H^{3}(Y)={\mathbb{Z}}^{21}~{}.$ (B.14)

  4. 4.

$H^{4}(X)$. If $m_{1}>0$, then $\operatorname{im}d_{23}=m_{1}{\mathbb{Z}}$:

$\displaystyle d_{23}(0,-\upomega^{\ast}_{\text{p}1})=m_{1}~{}.$ (B.15)

Thus,

$\displaystyle H^{4}(M)/\operatorname{im}d_{23}={\mathbb{Z}}_{m_{1}}~{}.$
(B.16)

The free part of the group comes from $\ker d_{24}$, and this depends on
whether $m_{2}=0$ or not:

    1. (a)

$m_{1}>0,m_{2}>0$. Then $\ker
d_{24}=\text{Span}\\{\widehat{C}\\}={\mathbb{Z}}^{20}$.

    2. (b)

$m_{1}>0,m_{2}=0$. Then $\ker
d_{24}=\text{Span}\\{\upomega_{\text{p}2}^{\ast},\widehat{C}\\}={\mathbb{Z}}^{21}$.

  5. 5.

$H^{5}(X)$. Here we obtain

$\displaystyle\operatorname{im}d_{24}=\text{Span}\\{(m_{1},0),(0,m_{1}m_{2})\\}~{}.$
(B.17)

So, the free part is the isomorphic to the free part of $H^{1}(X)$, while the
torsion part is the Pontryagin dual to the torsion in $H^{2}(X)$.

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