# Flat F-theory and friends

Peng Cheng  Ilarion V. Melnikov  Ruben Minasian

###### Abstract

We discuss F-theory backgrounds associated to flat torus bundles over Ricci-
flat manifolds. In this setting the F-theory background can be understood as a
IIB orientifold with a large radius limit described by a supersymmetric
compactification of IIB supergravity on a smooth, Ricci flat, but in general
non-spin geometry. When compactified on an additional circle these backgrounds
are T-dual to IIA compactifications on smooth non-orientable manifolds with a
$\text{Pin}^{-}$ structure.

##  1 Introduction

Given a type IIB string theory compactified on a manifold
$Y\times{\mathbb{R}}^{1,d-1}$ with a discrete symmetry action $G$, we can
orientifold the construction to obtain a compactification
$Y\times{\mathbb{R}}^{1,d-1}/H$, where $H=G^{\prime}\cup G\Pi$, and $\Pi$
denotes a worldsheet parity operation [1, 2, 3]. Typically the $G$-action will
have a non-empty fixed locus on $Y$, and this locus will mark the locations of
orientifold planes. The orientifold planes source a tadpole that can be
canceled by adding D-branes on top of the orientifold planes, and the
resulting theory can preserve some reduced amount of supersymmetry. As shown
in the classic work [4, 5], such constructions can often be identified with
certain limits of related F-theory compactifications.

The introduction of open string degrees of freedom is a central feature of
this construction that leads to much of the beauty and intricate structure of
the resulting theory. In this note we point out an interesting corner of the
orientifold landscape: when $G$ acts freely on $Y$ no orientifold planes are
required, so that the resulting theories can be understood without additional
D-branes or open strings. Furthermore, in the large radius limit the effective
theory is described as compactification of IIB supergravity on $X=Y/G$—a
smooth space that need not admit a spin structure. These compactifications
also have an F-theory interpretation in terms of an elliptically fibered
Calabi-Yau manifold $\pi:Z\to X$, where the fibration corresponds to a flat
$\operatorname{SL}(2,{\mathbb{Z}})$ bundle, so that the F-theory elliptic
fiber is locally constant.

While well-known to experts since the inception of F-theory—see for example
sections 6.3 and 6.4 of [6], there are a number of reasons why these examples
deserve more attention. First, they provide very simple examples of
supersymmetric string compactification on non-spin manifolds in IIB or non-
orientable manifolds in IIA, where a crucial role is played by the lift of a
bosonic duality group, such as $\operatorname{SL}(2,{\mathbb{Z}})$ of the ten-
dimensional IIB string theory, to the action on the full theory, including the
spacetime fermion fields [7, 8, 9]. Second, they illustrate the importance of
global topological features, such as certain non-trivial torsion classes in
cohomology, and allow us to probe these features directly on the string
worldsheet. Third, they provide additional examples of perturbative string
compactifications with reduced supersymmetry in various dimensions. Finally,
their simple structure makes these vacua ideal for detailed explorations of
string duality.

In the rest of this note we first illustrate the idea with a well-known
example based on [10]. We then describe a number of related examples in
diverse dimensions, point out a relationship to compactification of the IIA
string on non-orientable manifolds, and we conclude with a brief outlook.
There is a small geometric appendix.

## Acknowledgements

PC is partially supported by the DFG Excellence Strategy EXC-2094 390783311.
IVM’s work is supported in part by the Humboldt Research Award and the Jean
d’Alembert Program at the University of Paris–Saclay, as well as the
Educational Leave program at James Madison University. RM is partially
supported by ERC grants 772408-Stringlandscape and 787320-QBH Structure, as
well as the Humboldt Foundation. We thank F. Bonetti, I. Brunner, M. Del
Zotto, A. Font, J. Gray, and I. Saberi for useful discussions. This work
initiated during a stay at the Albert Einstein Institute (Max Planck Institute
for Gravitational Physics), and we are grateful to the AEI for its generous
hospitality.

##  2 The Enriques orientifold

###  2.1 The Enriques geometry as a quotient

The starting point for our discussion is compactification of IIB string theory
on a smooth K3 surface $Y$ . The massless spectrum is well-known [11, 12, 13]:
it is (2,0) supergravity with $21$ chiral tensor multiplets and a moduli space
${\cal
M}_{5,21}\simeq\operatorname{Gr}(5,21)/\operatorname{O{}}(\Gamma_{5,21})$,
where $\operatorname{Gr}(n_{1},n_{2})$ is the Grassmannian

$\displaystyle\operatorname{Gr}(n_{1},n_{2})=\operatorname{O{}}(n_{1},n_{2})/\operatorname{O{}}(n_{1})\times\operatorname{O{}}(n_{2})~{},$
(2.1)

and $\operatorname{O{}}(\Gamma_{n_{1},n_{2}})$ is the group of isometries of
the integral lattice $\Gamma_{n_{1},n_{2}}$ with signature $(n_{1},n_{2})$.

Working in the RNS formulation, the string-perturbative regime is described by
a worldsheet theory with (4,4) superconformal symmetry and central charges
$c_{\text{\tiny{L}}}=c_{\text{\tiny{R}}}=6$, and if we further take a large
radius limit, then the worldsheet theory has a description as a non-linear
sigma model with target space $Y$. This SCFT has moduli space ${\cal
M}_{4,20}$ defined in an analogous fashion to ${\cal M}_{5,21}$ [14, 15]. In
the large radius limit we can decompose ${\cal M}_{4,20}$ into the moduli
space of Einstein volume $1$ metrics $g$ on K3—this is isomorphic to ${\cal
M}_{3,19}$, a choice of closed B-field $B\in
H^{2}(Y,{\mathbb{R}})/H^{2}(Y,{\mathbb{Z}})$, and the K3 volume $V$ [16]. $Y$
is a hyper-Kähler manifold with holonomy
$\operatorname{SU}(2)=\operatorname{Sp}(1)$ and consequently has $3$
covariantly constant forms $j\in H^{1,1}(Y,{\mathbb{C}})$, $\omega\in
H^{2,0}(Y,{\mathbb{C}})$, and $\overline{\omega}\in H^{0,2}(Y,{\mathbb{C}})$
which obey

$\displaystyle j\wedge j$
$\displaystyle={\textstyle\frac{1}{2}}\omega\wedge\overline{\omega}=\operatorname{dVol}_{g}(Y)~{},$
$\displaystyle j\wedge\omega$
$\displaystyle=\omega\wedge\omega=\overline{\omega}\wedge\overline{\omega}=0~{}.$
(2.2)

It is useful to think of the holonomy group of $Y$ in terms of the reduction
of the holonomy of $Y$ from the generic
$\operatorname{SO}(4)=\left\\{\operatorname{SU}(2)_{1}\times\operatorname{SU}(2)_{2}\right\\}/{\mathbb{Z}}_{2}$.
Here the quotient is by the diagonal central element, and we can identify the
holonomy action with $\operatorname{SU}(2)_{2}$. The forms
$j,\omega,\overline{\omega}$ are singlets with respect to
$\operatorname{SU}(2)_{2}$ but transform in ${{\boldsymbol{3}}}$ of
$\operatorname{SU}(2)_{1}$, which describes the hyper-Kähler rotations. We can
think of $Y$ as a Calabi-Yau manifold with trivial canonical bundle $K_{Y}$
and $h^{1}({\cal O}_{Y})=0$.

There is a locus in the metric moduli space ${\cal M}_{E}\subset{\cal
M}_{3,19}$, where $Y$ admits a freely-acting Kähler isometry group
$G_{E}\simeq{\mathbb{Z}}_{2}$ [17], and its generator $U_{E}$ acts on the
triplet of forms as

$\displaystyle U_{E}\cdot j$ $\displaystyle=j~{},$ $\displaystyle
U_{E}\cdot\omega$ $\displaystyle=-\omega~{},$ $\displaystyle
U_{E}\cdot\overline{\omega}$ $\displaystyle=-\overline{\omega}~{}.$ (2.3)

This free action is known as the Enriques involution, and the quotient
$X=Y/G_{E}$ is an Enriques surface— a smooth compact Kähler Ricci-flat surface
with a torsion canonical bundle: $K_{X}^{\otimes 2}\simeq{\cal O}_{X}$, but
$K_{X}\neq{\cal O}_{X}$. Recall that a complex manifold $X$ is spin if and
only if $c_{1}(K_{X})=0\mod 2$ (see, e.g. [18]); thus, the Enriques surface
$X$ is not spin. We can see this explicitly in the quotient construction, as
the action of $U_{E}$ on the hyper-Kähler forms and on the covariantly
constant spinors is by the element $i\sigma_{3}\in\operatorname{SU}(2)_{1}$.
Letting $J^{3}$ and $J^{\pm}$ be the generators of $\operatorname{SU}(2)_{1}$
with the non-zero commutators

$\displaystyle{[J^{3},J^{\pm}]}$ $\displaystyle=\pm 2J^{\pm}~{},$
$\displaystyle{[J^{\pm},J^{\mp}]}$ $\displaystyle=\pm J^{3}~{},$ (2.4)

the action can also be expressed as $e^{i{\textstyle\frac{\pi}{2}}J^{3}}$. It
is no coincidence that the quotient also enlarges the holonomy group: we have
$\pi_{1}(X)={\mathbb{Z}}_{2}$, and the quotient metric $g_{X}$ has holonomy
group $\text{Hol}(X,g_{X})=\operatorname{SU}(2)\rtimes{\mathbb{Z}}_{2}$. The
first factor is the null-homotopic holonomy inherited from $Y$, while the
second factor is associated to the holonomy around a loop $\gamma$ that
generates $\pi_{1}(X)\simeq{\mathbb{Z}}_{2}$.111The semi-direct product in the
holonomy group is discussed in the appendix.

The last geometric point we will need is the $G_{E}$ action on the lattice
$H^{2}(Y,{\mathbb{Z}})=\Gamma_{3,19}$ with respect to the Enriques action [17,
10]. Decomposing

$\displaystyle\Gamma_{3,19}=(\Gamma_{0,8}+\Gamma_{1,1})+(\Gamma_{0,8}+\Gamma_{1,1})+\Gamma_{1,1}~{},$
(2.5)

the generator acts as

$\displaystyle U_{E}$ $\displaystyle=\begin{pmatrix}0&{\mathbbm{1}}_{10}&0\\\
{\mathbbm{1}}_{10}&0&0\\\ 0&0&-{\mathbbm{1}}_{2}\end{pmatrix}~{},$ (2.6)

and consequently the spaces of dual and anti-self-dual forms decompose into
even and odd components as

$\displaystyle H^{2}_{+}(Y,{\mathbb{R}})$
$\displaystyle={\mathbb{R}}_{\text{e}}\oplus\left({\mathbb{R}}^{2}\right)_{\text{o}}~{},$
$\displaystyle H^{2}_{-}(Y,{\mathbb{R}})$
$\displaystyle=\left({\mathbb{R}}^{9}\right)_{\text{e}}\oplus\left({\mathbb{R}}^{10}\right)_{\text{o}}~{}.$
(2.7)

###  2.2 The Enriques symmetry in the SCFT

Denote by ${\cal S}[Y]$ the superconformal theory associated to the non-linear
sigma model on $Y$ equipped with Einstein metric $G=\sqrt{V}g$, where $g$ is
an Einstein metric with volume $1$, and a closed $B$-field $B$. Choosing
$g\in{\cal M}_{E}$ and setting $B=0$, we expect ${\cal S}[Y]$ to admit a
global ${\mathbb{Z}}_{2}$ symmetry, which we will also call $G_{E}$.222It is
not necessary to set $B=0$. The most general choice is to choose $B$ so that
the difference $g_{E}\cdot B-B$ is contained in the image of
$H^{2}(Y,{\mathbb{Z}})$ inside $H^{2}(Y,{\mathbb{R}})$. Its generator is a
unitary operator which we will also denote $U_{E}$. Although we will use some
features of the large radius limit, we will phrase our discussion as much as
possible in terms of the SCFT properties to emphasize that in principle we
have a construction that works at generic points on the Enriques locus. This
includes solvable points, such as $T^{4}/{\mathbb{Z}}_{n}$ orbifolds and
Gepner points.

In the NS-NS sector the action of the generator $U_{E}$ is a combination of
two operations. The first, which we call $x_{E}$, is a ${\mathbb{Z}}_{2}$
action on the (4,4) primary fields of the SCFT, while the second is a rotation
generated by the global charges of the
$\widehat{\operatorname{\mathfrak{su}}}(2)_{\text{\tiny{L}}}\oplus\widehat{\operatorname{\mathfrak{su}}}(2)_{\text{\tiny{R}}}$
Kac-Moody algebra contained in the superconformal algebra:

$\displaystyle\exp\left[{\textstyle\frac{i\pi}{2}}J^{3}_{\text{\tiny{L}}0}+{\textstyle\frac{i\pi}{2}}J^{3}_{\text{\tiny{R}}0}\right]~{}.$
(2.8)

The symmetry generator is then

$\displaystyle U_{E}$
$\displaystyle=x_{E}\exp\left[{\textstyle\frac{i\pi}{2}}J^{3}_{\text{\tiny{L}}0}+{\textstyle\frac{i\pi}{2}}J^{3}_{\text{\tiny{R}}0}\right]~{}.$
(2.9)

It is not hard to see that $U_{E}^{2}={\mathbbm{1}}$ in the NS-NS sector. The
only subtlety is in the factor
$\exp\left[{\textstyle\frac{i\pi}{2}}J^{3}_{\text{\tiny{L}}0}+{\textstyle\frac{i\pi}{2}}J^{3}_{\text{\tiny{R}}0}\right]$.
To see that it squares to identity note that the primary fields in the NS-NS
sector transform either in $({{\boldsymbol{1}}},{{\boldsymbol{1}}})$ or
$({{\boldsymbol{2}}},{{\boldsymbol{2}}})$ of
$\widehat{\operatorname{\mathfrak{su}}}(2)_{\text{\tiny{L}}}\oplus\widehat{\operatorname{\mathfrak{su}}}(2)_{\text{\tiny{R}}}$.333
At $c=6$ unitarity only allows the representations ${{\boldsymbol{1}}}$ and
${{\boldsymbol{2}}}$ to appear, and by considering a limiting case such as
$Y=T^{4}/{\mathbb{Z}}_{2}$, where the theory is solvable, we see that in fact
the only representations possible are
$({{\boldsymbol{1}}},{{\boldsymbol{1}}})$ or
$({{\boldsymbol{2}}},{{\boldsymbol{2}}})$. Unitarity constraints on
representations of the N=4 superconformal algebra are explored in detail in
[19]. In the latter case the action on the (4,4) primary field is determined
by specifying the action of $x_{E}$ on, say, the state of charge $+1$ with
respect to $J^{3}_{\text{\tiny{L}}0}$ and $-1$ with respect to
$J^{3}_{\text{\tiny{R}}0}$.

For example we can consider the operators that contribute to the massless
spacetime spectrum. In addition to the identity operator these are (4,4)
primary operators with $h_{\text{\tiny{L}}}=h_{\text{\tiny{R}}}=1/2$,
transforming in $({{\boldsymbol{2}}},{{\boldsymbol{2}}})$ of
$\operatorname{SU}(2)_{\text{\tiny{L}}}\times\operatorname{SU}(2)_{\text{\tiny{R}}}$.
Picking a subgroup
$\operatorname{U{}}(1)_{\text{\tiny{L}}}\times\operatorname{U{}}(1)_{\text{\tiny{R}}}\subset\operatorname{SU}(2)_{\text{\tiny{L}}}\times\operatorname{SU}(2)_{\text{\tiny{R}}}$
with corresponding Kac-Moody currents
$J^{3}_{\text{\tiny{L}},\text{\tiny{R}}}$ as above, we find that these
operators transform in a quartet consisting of $N=2$ (anti)-chiral primary
operators on the left and the right:

${{\cal O}^{\text{ca}}}$${{\cal O}^{\text{cc}}~{}}$${{\cal
O}^{\text{aa}}}$${{\cal
O}^{\text{ac}}}$$\scriptstyle{J^{+}_{\text{\tiny{R}}}}$$\scriptstyle{J^{-}_{\text{\tiny{L}}}}$$\scriptstyle{J^{-}_{\text{\tiny{L}}}}$$\scriptstyle{J^{+}_{\text{\tiny{R}}}}$
(2.10)

Here ${\cal O}^{\text{cc}}$ is an operator that is chiral-primary both on the
left and right. We can be very explicit when the SCFT is a non-linear sigma
model in a large radius limit, i.e. $g$ is at a smooth point in ${\cal M}_{E}$
and $\sqrt{V}\gg\alpha^{\prime}$. In this case taking complex coordinates for
the bosonic target space fields $Z^{i}$, $i=1,2$, with their left- and right-
moving superpartners Weyl fermions $\Psi^{i}_{\text{\tiny{L}}}$ and
$\Psi^{i}_{\text{\tiny{R}}}$, as well as their target-space complex
conjugates, we find that there are two types quartets contributing to the
massless spectrum:

  1. 1.

the self-dual quartet with components

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}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{10.22813pt}{-25.64719pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-7.23549pt}{-21.92775pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{$\scriptstyle{J^{++}_{\text{\tiny{R}}}}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}~{},$ (2.11)

  2. 2.

the $19$ anti-self-dual quartets, each with components

$\leavevmode\hbox to108.64pt{\vbox
to56.39pt{\pgfpicture\makeatletter\hbox{\hskip
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20.05354pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
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A}^{\text{ca}}_{i\overline{\jmath}}\Psi^{i}_{\text{\tiny{L}}}\overline{\Psi}^{\overline{\jmath}}_{\text{\tiny{R}}}}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}}}&\hskip
20.05354pt\hfil&\hfil\hskip 46.26515pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
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}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{${{\cal
A}^{\text{cc}}_{ij}\Psi^{i}_{\text{\tiny{L}}}\Psi^{j}_{\text{\tiny{R}}}~{}}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}&\hskip
22.26518pt\hfil\cr\vskip 18.00005pt\cr\hfil\hskip
19.85558pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-15.55003pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{${{\cal
A}^{\text{aa}}_{\overline{\imath}\overline{\jmath}}\overline{\Psi}^{\overline{\imath}}_{\text{\tiny{L}}}\overline{\Psi}^{\overline{\jmath}}_{\text{\tiny{R}}}}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}&\hskip
19.85558pt\hfil&\hfil\hskip 44.40222pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
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A}^{\text{ac}}_{\overline{\imath}j}\overline{\Psi}^{\overline{\imath}}_{\text{\tiny{L}}}\Psi^{j}_{\text{\tiny{R}}}}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}&\hskip
20.40225pt\hfil\cr}}}\pgfsys@invoke{\lxSVG@closescope
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{}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}{}{}{{{}{}}}{}{{}}{}{}{}{{{}{}}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{
}{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{-13.61162pt}{11.2878pt}\pgfsys@lineto{8.78838pt}{11.2878pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{8.98836pt}{11.2878pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-7.60019pt}{15.00723pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
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}\hbox{$\scriptstyle{J^{++}_{\text{\tiny{R}}}}$}
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\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{ {}{}{}}{}{ {}{}{}}
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{}{}}{}{}{{}{}}}}}{{}}{}{}{}{}{}{{{}{}}}{}{{}}{}{}{}{{{}{}}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{
}{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{-34.26517pt}{2.25697pt}\pgfsys@lineto{-34.26517pt}{-15.44864pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.0}{-1.0}{1.0}{0.0}{-34.26517pt}{-15.64862pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-31.9124pt}{-8.9347pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{$\scriptstyle{J^{--}_{\text{\tiny{L}}}}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{ {}{}{}}{}{ {}{}{}}
{{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}{}{}{{{}{}}}{}{{}}{}{}{}{{{}{}}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{
}{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{32.05353pt}{1.59253pt}\pgfsys@lineto{32.05353pt}{-14.80754pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.0}{-1.0}{1.0}{0.0}{32.05353pt}{-15.00752pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{34.4063pt}{-8.94637pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{$\scriptstyle{J^{--}_{\text{\tiny{L}}}}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{ {}{}{}}{}{ {}{}{}}
{{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}{}{}{{{}{}}}{}{{}}{}{}{}{{{}{}}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{
}{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{-13.80959pt}{-25.64719pt}\pgfsys@lineto{10.6513pt}{-25.64719pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{10.85129pt}{-25.64719pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-6.76772pt}{-21.92775pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{$\scriptstyle{J^{++}_{\text{\tiny{R}}}}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}~{},$ (2.12)

where ${\cal A}^{\text{ca}}={\cal
A}^{\text{ca}}_{i\overline{\jmath}}dZ^{i}\wedge
d\overline{Z}^{\overline{\jmath}}$ is a harmonic anti-self-dual form444Anti-
self-duality of ${\cal A}^{\text{ca}}$ is equivalent to $j\wedge{\cal
A}^{\text{ca}}=0$., and

$\displaystyle{\cal A}^{\text{cc}}_{ij}$ $\displaystyle=-{\cal
A}^{\text{ac}}_{i\overline{k}}g^{\overline{k}k}\omega_{kj}~{},$
$\displaystyle{\cal A}^{\text{aa}}_{\overline{\imath}\overline{\jmath}}$
$\displaystyle=-\overline{\omega}_{\overline{\imath}\overline{k}}g^{\overline{k}k}{\cal
A}^{\text{ac}}_{k\overline{\jmath}}~{},$ $\displaystyle{\cal
A}^{\text{ac}}_{\overline{\imath}j}$
$\displaystyle=\overline{\omega}_{\overline{\imath}\overline{m}}g^{\overline{m}m}{\cal
A}_{m\overline{k}}g^{\overline{k}k}\omega_{kj}~{}.$ (2.13)

It is not hard to show that anti-self-duality of ${\cal A}^{\text{ca}}$ is
equivalent to ${\cal A}^{\text{cc}}_{ij}={\cal A}^{\text{cc}}_{ji}$.

Since we know the action of $x_{E}$ on the cohomology of $Y$, we can now
decompose these quartets according to the (2,2) superconformal algebra that
commutes with the action of $G_{E}$. The self-dual quartet decomposes as

$\displaystyle\text{SD
quartet}=(\text{c},\text{c})_{\text{o}}\oplus(\text{a},\text{a})_{\text{o}}\oplus(\text{c},\text{a})_{\text{e}}\oplus(\text{a},\text{c})_{\text{e}}~{}.$
(2.14)

An anti-self-dual quartet has different decompositions, depending on whether
$x_{E}\cdot{\cal A}^{\text{ca}}=\pm{\cal A}^{\text{ca}}$:

$\displaystyle\text{ASD
quartet}^{+}=(\text{c},\text{c})_{\text{o}}\oplus(\text{a},\text{a})_{\text{o}}\oplus(\text{c},\text{a})_{\text{e}}\oplus(\text{a},\text{c})_{\text{e}}~{},$
$\displaystyle\text{ASD
quartet}^{-}=(\text{c},\text{c})_{\text{e}}\oplus(\text{a},\text{a})_{\text{e}}\oplus(\text{c},\text{a})_{\text{o}}\oplus(\text{a},\text{c})_{\text{o}}~{}.$

With this set up we can define an orbifold ${\cal S}[Y]/G_{E}$. The resulting
theory will only have (2,2) supersymmetry because the currents
$J^{\pm}_{\text{\tiny{L}}}$ and $J^{\pm}_{\text{\tiny{R}}}$ will be projected
out. It will have non-trivial (c,c) and (a,c) rings (and their conjugates),
each containing the identity operator, $10$ operators with dimension
$h_{\text{\tiny{L}}}=h_{\text{\tiny{R}}}=1/2$, and one operator of top
dimension $h_{\text{\tiny{L}}}=h_{\text{\tiny{R}}}=1$ represented by
$J_{\text{\tiny{L}}}^{+}J_{\text{\tiny{R}}}^{+}$ in the (c,c) ring and
$J_{\text{\tiny{L}}}^{-}J_{\text{\tiny{R}}}^{+}$ in the (a,c) ring. Crucially
for our purposes the projection will also remove the spectral flow operators
that we would usually use to construct the chiral Ramond sectors and define a
chiral GSO projection. This is the worldsheet analogue of the spacetime
statement that the Enriques orbifold projects out the covariantly constant
spinors.

###  2.3 Some RNS details

In order to obtain supersymmetric vacua based on the Enriques orbifold we will
need to enlarge the action of $G_{E}$ to act on the entire world-sheet theory,
including the Minkowski degrees of freedom and the superconformal ghosts. To
describe the structure we will use the covariant formulation of the RNS
string, and we will not need too many details beyond the usual textbook
treatment [20].

The first additional ingredient we will need is a description of the Ramond
sectors associated to the ${\cal S}[Y]$ SCFT. These are easily constructed by
spectral flow once we pick a (2,2) subalgebra of the (4,4) algebra. In our
case there is a preferred choice of a (2,2) sub-algebra — the sub-algebra that
commutes with the action of $U_{E}$. Then, by the usual construction [21], we
obtain an isomorphism between the Ramond ground states of the theory and the
chiral primary operators in the NS sector. Since the theory has chiral
spectral flow, we can do this separately on the left and right. For example,
left spectral flow of the identity operator and $J^{+}_{\text{\tiny{L}}}$
leads to spin fields $\Sigma^{\pm}_{\text{\tiny{L}}}$ which have the OPE

$\displaystyle\Sigma_{\text{\tiny{L}}}^{A}(z_{1})\Sigma_{\text{\tiny{L}}}^{B}(z_{2})$
$\displaystyle\sim\frac{{\epsilon}^{AB}}{\sqrt{z_{12}}}+\frac{1}{2}\sqrt{z_{12}}\delta^{A,B}J_{\text{\tiny{L}}}^{B}~{}.$
(2.16)

Moreover these are primary with respect to
$\widehat{\operatorname{\mathfrak{su}}}(2)_{\text{\tiny{L}}}$, transforming in
a doublet. These universal spin fields will play a role in constructing
spacetime supercharges and a universal tensor multiplet in spacetime. On the
other hand, chiral primary fields with $h_{\text{\tiny{L}}}=1$ lead to left-
moving Ramond ground states that are singlets with respect to
$\widehat{\operatorname{\mathfrak{su}}}(2)_{\text{\tiny{L}}}$. More generally,
every NS-NS field has an image in the R-NS sector determined by the left-
moving spectral flow, and we have the same construction on the right. Finally,
the internal theory has chiral fermion number operators that are represented
by

$\displaystyle e^{i\pi F_{\text{\tiny{L}}}^{\text{int}}}$
$\displaystyle=e^{i\pi J^{3}_{\text{\tiny{L}}0}}~{},$ $\displaystyle e^{i\pi
F_{\text{\tiny{R}}}^{\text{int}}}$ $\displaystyle=e^{i\pi
J^{3}_{\text{\tiny{R}}0}}~{}.$ (2.17)

Next we describe the Minkowski degrees of freedom: these are bosons $X^{\mu}$
for ${\mathbb{R}}^{1,5}$, and their Majorana-Weyl superpartners ${\cal
X}^{\mu}_{\text{\tiny{L}}}$, ${\cal X}^{\mu}_{\text{\tiny{R}}}$. The currents
$\Lambda^{\mu\nu}=:{\cal X}_{L}^{\mu}{\cal X}_{L}^{\nu}$ generate a Lorentz
current algebra $\widehat{\operatorname{\mathfrak{so}}}(1,5)_{1}$, and we can
construct spin fields ${\cal S}_{\text{\tiny{L}}a}$ that transform in the
reducible ${{\boldsymbol{8}}}$ representation of the associated
$\operatorname{Spin}(1,5)$. Conformal invariance, together with the spacetime
Lorentz symmetry constrain the OPEs of the spin fields:

$\displaystyle{\cal X}^{\mu}(z_{1}){\cal S}_{a}(z_{2})$
$\displaystyle\sim\frac{1}{\sqrt{2z_{12}}}\Gamma^{\mu}_{ab}{\cal
S}_{b}(z_{2})~{},$ $\displaystyle{\cal S}_{a}(z_{1}){\cal S}_{b}(z_{2})$
$\displaystyle\sim\frac{{\cal C}_{ab}}{z_{12}^{3/4}}+\frac{({\cal
C}\Gamma^{\mu})_{ab}}{\sqrt{2}z_{12}^{1/4}}{\cal X}_{\mu}(z_{2})~{}.$ (2.18)

Here the $(\Gamma^{\mu})_{ab}$ are the generators of the non-trivial
irreducible representation of $\text{Cl}(1,5)$, and ${\cal C}$ is the
symmetric charge conjugation matrix that anticommutes with the chirality
matrix $\Gamma_{6}=-\Gamma^{0}\Gamma^{1}\cdots\Gamma^{5}$. The action of the
fermion number operator $e^{i\pi F^{\text{min}}_{\text{\tiny{L}}}}$ on the
spin fields can be determined in a number of ways, for example by bosonizing
the fermions ${\cal X}^{\mu}_{\text{\tiny{L}}}$. The result is

$\displaystyle e^{i\pi F^{\text{min}}_{\text{\tiny{L}}}}{\cal
S}_{a}(z)e^{-i\pi
F^{\text{min}}_{\text{\tiny{L}}}}=(e^{-i\pi\Gamma_{6}/2})_{ab}{\cal
S}_{b}(z)~{}.$ (2.19)

In other words, the spin fields ${\cal S}^{\pm}$ satisfying $\Gamma_{6}{\cal
S}^{\pm}=\pm{\cal S}^{\pm}$ carry fermion number $\mp i$. It will be
convenient for us to decompose the reducible ${{\boldsymbol{8}}}$
representation of $\operatorname{\mathfrak{spin}}(1,5)$ into its irreducible
Weyl components. In this representation

$\displaystyle\Gamma_{6}$ $\displaystyle=\begin{pmatrix}{\mathbbm{1}}_{4}&0\\\
0&\\-{\mathbbm{1}}_{4}\end{pmatrix}~{},$ $\displaystyle{\cal C}$
$\displaystyle=\begin{pmatrix}0&{\mathbbm{1}}_{4}\\\
{\mathbbm{1}}_{4}&0\end{pmatrix}~{},$ $\displaystyle{\cal C}\Gamma^{\mu}$
$\displaystyle=\begin{pmatrix}\sigma^{\mu}&0\\\
0&\widetilde{\sigma}^{\mu}\end{pmatrix}~{},$ (2.20)

with $\sigma$ and $\widetilde{\sigma}$ anti-symmetric matrices.

Finally, we need a little bit of information on the ghost sector. The relevant
ghost sector operators are constructed from the fields that bosonize the (1,1)
$\beta_{{\text{\tiny{L}}}}\gamma_{{\text{\tiny{L}}}}$ and
$\beta_{\text{\tiny{R}}}\gamma_{\text{\tiny{R}}}$ currents as
$\partial\varphi_{\text{\tiny{L}}}$,
$\bar{\partial}\varphi_{\text{\tiny{R}}}$. We will need the operators
$e^{-s\varphi_{\text{\tiny{L}}}}(z)$ and
$e^{-s\varphi_{\text{\tiny{R}}}}(\overline{z})$, which have conformal
dimensions $(s-s^{2}/2,0)$ and $(0,s-s^{2}/2)$ and fermion numbers

$\displaystyle e^{i\pi
F^{\text{gh}}_{\text{\tiny{L}}}}e^{-s\varphi_{\text{\tiny{L}}}}e^{-i\pi
F^{\text{gh}}_{\text{\tiny{L}}}}$ $\displaystyle=e^{-i\pi
s}e^{-s\varphi_{\text{\tiny{L}}}}~{},$ $\displaystyle e^{i\pi
F^{\text{gh}}_{\text{\tiny{R}}}}e^{-s\varphi_{\text{\tiny{R}}}}e^{-i\pi
F^{\text{gh}}_{\text{\tiny{R}}}}$ $\displaystyle=e^{-i\pi
s}e^{-s\varphi_{\text{\tiny{R}}}}~{}.$ (2.21)

With these ingredients in hand, we can construct the R-NS, NS-R, and R-R
sectors and impose the IIB GSO projection

$\displaystyle e^{i\pi F^{\text{tot}}_{\text{\tiny{L}}}}$
$\displaystyle=1~{},$ $\displaystyle e^{i\pi
F^{\text{tot}}_{\text{\tiny{R}}}}$ $\displaystyle=1~{}.$ (2.22)

In what follows we will focus on two aspects of the full SCFT with GSO
projection: the massless spectrum and the worldsheet realization of the
spacetime supersymmetry algebra. We begin our discussion with the latter,
following the classic treatment [22, 23]: by stripping the gravitino vertex
operators in the (-1/2,0) and (0,-1/2) pictures we obtain chiral worldsheet
currents

$\displaystyle{\cal J}^{A}_{\text{\tiny{L}}\alpha}$
$\displaystyle=e^{-\varphi_{\text{\tiny{L}}}/2}\Sigma^{A}_{\text{\tiny{L}}}{\cal
S}^{+}_{\text{\tiny{L}}\alpha}~{},$ $\displaystyle{\cal
J}^{A}_{\text{\tiny{R}}\alpha}$
$\displaystyle=e^{-\varphi_{\text{\tiny{R}}}/2}\Sigma^{A}_{\text{\tiny{R}}}{\cal
S}^{+}_{\text{\tiny{R}}\alpha}~{}.$ (2.23)

Using the preceding analysis we calculate

$\displaystyle{\cal J}^{A}_{\text{\tiny{L}}\alpha}(z_{1}){\cal
J}^{B}_{\text{\tiny{L}}\beta}(z_{2})\sim\frac{1}{z_{12}}{\epsilon}^{AB}\sigma^{\mu}_{\beta\alpha}e^{-\varphi_{\text{\tiny{L}}}}{\cal
X}_{\text{\tiny{L}}\mu}(z_{2})~{},$ (2.24)

where we recognize ${\cal P}_{\mu}(z_{2})=e^{-\varphi_{\text{\tiny{L}}}}{\cal
X}_{\mu}(z_{2})$ as the conserved current associated to spacetime translations
in the (-1,0) picture. This means the conserved charges ${\cal
Q}^{A}_{\text{\tiny{L}}\alpha}=\oint{\textstyle\frac{dz}{2\pi i}}{\cal
J}^{A}_{\text{\tiny{L}}\alpha}$ and
$P_{\text{\tiny{L}}\mu}=\oint{\textstyle\frac{dz}{2\pi i}}{\cal
P}_{\text{\tiny{L}}\mu}$ and their right-moving counter-parts obey the
supersymmetry algebra555Notice that the odd pole in the ${\cal J}_{L}$–${\cal
J}_{L}$ OPE, together with anti-symmetry of $\sigma$ implies that the ${\cal
Q}^{A}_{\text{\tiny{L}}\alpha}$ are anticommuting; moreover the operators
$P_{\text{\tiny{L}}\mu}$ and $P_{\text{\tiny{R}}\mu}$ both represent the
spacetime translation current, expressed in different pictures.

$\displaystyle{\\{{\cal Q}^{A}_{\text{\tiny{L}}\alpha},{\cal
Q}^{B}_{\text{\tiny{L}}\beta}\\}}$
$\displaystyle={\epsilon}^{AB}\sigma^{\mu}_{\beta\alpha}P_{\text{\tiny{L}}\mu}~{},$
$\displaystyle{\\{{\cal Q}^{A}_{\text{\tiny{R}}\alpha},{\cal
Q}^{B}_{\text{\tiny{R}}\beta}\\}}$
$\displaystyle={\epsilon}^{AB}\sigma^{\mu}_{\beta\alpha}P_{\text{\tiny{R}}\mu}.$
(2.25)

This is exactly the supersymmetry algebra of (2,0) supergravity in six
dimensions.

Having introduced the Ramond sectors, we also note that the full worldsheet
theory enjoys additional symmetries generated by the spacetime fermion number
operators $e^{i\pi{{{\boldsymbol{F}}}}_{\text{\tiny{L}}}}$ and
$e^{i\pi{{{\boldsymbol{F}}}}_{\text{\tiny{R}}}}$. The latter assigns charge
$-1$ to NS-R and R-R fields and leaves the other sectors invariant, and the
former has an analogous action on the R-NS and R-R fields. We obtain the total
six-dimensional spacetime fermion number by combining these, and it is then
not hard to verify

$\displaystyle
e^{i\pi{{{\boldsymbol{F}}}}}=e^{i\pi{{{\boldsymbol{F}}}}_{\text{\tiny{L}}}}e^{i\pi{{{\boldsymbol{F}}}}_{\text{\tiny{R}}}}=e^{i\pi
J^{3}_{\text{\tiny{L}}0}+i\pi J^{3}_{\text{\tiny{R}}0}}~{}.$ (2.26)

###  2.4 Worldsheet parity and a supersymmetric Enriques action

The Enriques action in the NS-NS sector has a lift to the R-NS and NS-R
sectors, and because the internal spin fields transform in doublets of the
$\operatorname{SU}(2)_{\text{\tiny{L}}}\times\operatorname{SU}(2)_{\text{\tiny{R}}}$
it is easy to see that the spacetime supercharges are charged with respect to
$U_{E}$, and, moreover, $U_{E}^{2}\neq{\mathbbm{1}}$ on these sectors.
However, there is a simple modification of the quotient that will preserve
spacetime supersymmetry through an orientifold construction (see [1, 3] for
thorough reviews).

We can choose moduli of the internal theory, for example by setting $B=0$, so
that in addition to the $U_{E}$ action described above, the IIB worldsheet
theory has a parity symmetry which acts on the worldsheet coordinates as
$z\mapsto\overline{z}$, and there is a unitary Hermitian operator $\Pi$ that
takes each Virasoro primary field to a Virasoro primary field of the same
weight and opposite spin: ${\cal O}(z,\overline{z})\mapsto\Pi{\cal
O}(z,\overline{z})\Pi=\overline{{\cal O}}(\overline{z},z)$, and the action
preserves the OPE and correlation functions. The modified Enriques action is
then taken to be

$\displaystyle\widetilde{U}_{E}=U_{E}e^{i\pi{{{\boldsymbol{F}}}}_{\text{\tiny{L}}}}\Pi~{}.$
(2.27)

Using (2.26) it is then easy to see that $\widetilde{U}_{E}^{2}={\mathbbm{1}}$
on all sectors. Furthermore $\widetilde{U}_{E}$ now preserves a number of
fundamental structures including a diagonal Virasoro algebra with $c=12$
generated by the modes
$\widehat{L}_{m}=L_{\text{\tiny{L}}m}+L_{\text{\tiny{R}}m}$, and a level $2$
“even” Kac-Moody algebra $\widehat{\operatorname{\mathfrak{su}}}(2)_{2}$
generated by the modes

$\displaystyle\widehat{J}_{m}^{3}$
$\displaystyle=J^{3}_{\text{\tiny{L}}}+J^{3}_{\text{\tiny{R}}}~{},$
$\displaystyle\widehat{J}^{\pm}_{m}$
$\displaystyle=J^{\pm}_{\text{\tiny{L}}m}-J^{\pm}_{\text{\tiny{R}}m}~{}.$
(2.28)

The spacetime supersymmetry currents now transform as

$\displaystyle\widetilde{U}_{E}{\cal
J}^{\pm}_{\text{\tiny{L}}}(z)\widetilde{U}_{E}$ $\displaystyle=\pm i{\cal
J}^{\pm}_{\text{\tiny{R}}\alpha}(z)~{},$ $\displaystyle\widetilde{U}_{E}{\cal
J}^{\pm}_{\text{\tiny{R}}}(\overline{z})\widetilde{U}_{E}$ $\displaystyle=\mp
i{\cal J}^{\pm}_{\text{\tiny{L}}\alpha}(\overline{z})~{},$ (2.29)

so that if we define the linear combinations of the charges

$\displaystyle{\cal Q}^{+}_{\text{e}}$ $\displaystyle={\cal
Q}^{+}_{\text{\tiny{L}}}+i{\cal Q}^{+}_{\text{\tiny{R}}}~{},$
$\displaystyle{\cal Q}^{+}_{\text{o}}$ $\displaystyle={\cal
Q}^{+}_{\text{\tiny{L}}}-i{\cal Q}^{+}_{\text{\tiny{R}}}~{},$
$\displaystyle{\cal Q}^{-}_{\text{e}}$ $\displaystyle={\cal
Q}^{-}_{\text{\tiny{L}}}-i{\cal Q}^{-}_{\text{\tiny{R}}}~{},$
$\displaystyle{\cal Q}^{-}_{\text{o}}$ $\displaystyle={\cal
Q}^{-}_{\text{\tiny{L}}}+i{\cal Q}^{-}_{\text{\tiny{R}}}~{},$ (2.30)

then

$\displaystyle\widetilde{U}_{E}{\cal Q}^{A}_{\text{e}}\widetilde{U}_{E}$
$\displaystyle={\cal Q}^{A}_{\text{e}}~{},$
$\displaystyle\widetilde{U}_{E}{\cal Q}^{A}_{\text{o}}\widetilde{U}_{E}$
$\displaystyle=-{\cal Q}^{A}_{\text{o}}~{},$ (2.31)

and each pair transforms as a doublet with respect to the
$\operatorname{SU}(2)_{\text{e}}$ symmetry generated by the “even” Kac-Moody
algebra.

Using the set up of the previous section it is now easy to characterize the
massless spacetime spectrum of IIB on K3 in terms of the $\widetilde{U}_{E}$
action. We can summarize the results in terms of (c,a) fields of the internal
theory and the resulting spacetime multiplets:

  1. 1.

vacuum:

$\displaystyle\\{\text{(2,0) gravity}\\}$ $\displaystyle=\\{\text{(1,0)
gravity}\\}_{\text{e}}\oplus\\{\text{(1,0) gravitinos}\\}_{\text{o}}$
$\displaystyle\\{\text{(2,0) tensor}\\}$ $\displaystyle=\\{\text{(1,0)
tensor}\\}_{\text{o}}\oplus\\{\text{(1,0) hyper}\\}_{\text{e}}~{}.$ (2.32)

  2. 2.

${\cal O}^{\text{ca}}[g]$:

$\displaystyle\\{\text{(2,0) tensor}\\}$ $\displaystyle=\\{\text{(1,0)
tensor}\\}_{\text{o}}\oplus\\{\text{(1,0) hyper}\\}_{\text{e}}~{}.$ (2.33)

  3. 3.

${\cal O}^{\text{ca}}[{\cal A}]$, ${\cal A}$ anti-self-dual and $g_{E}$-even
($9$ of these):

$\displaystyle\\{\text{(2,0) tensor}\\}$ $\displaystyle=\\{\text{(1,0)
tensor}\\}_{\text{e}}\oplus\\{\text{(1,0) hyper}\\}_{\text{o}}~{}.$ (2.34)

  4. 4.

${\cal O}^{\text{ca}}[{\cal A}]$, ${\cal A}$ anti-self-dual and $g_{E}$-odd
($10$ of these):

$\displaystyle\\{\text{(2,0) tensor}\\}$ $\displaystyle=\\{\text{(1,0)
tensor}\\}_{\text{o}}\oplus\\{\text{(1,0) hyper}\\}_{\text{e}}~{}.$ (2.35)

###  2.5 An orientifold without open strings and its supergravity limit

We now use the $\widetilde{U}_{E}$ action on the IIB string worldsheet to
define a ${\mathbb{Z}}_{2}$ orientifold. The $\widetilde{U}_{E}$–even modes
include the spacetime supercharges ${\cal Q}_{\text{e}}^{A}$ transforming in a
doublet of $\operatorname{SU}(2)_{\text{e}}$, and taking the projection in the
untwisted sector leads to a massless spectrum consisting of (1,0) supergravity
coupled to $9$ tensor multiplets and $12$ hypermultiplets—exactly the massless
spectrum of F-theory compactification on $Z=(Y\times T^{2})/{\mathbb{Z}}_{2}$
[6, 24]—the Enriques Calabi-Yau $3$-fold familiar from the Borcea-Voisin
construction of mirror pairs [25, 26] and introduced in the context of string
duality in [10]. The same spectrum arises in the closed string sector of the
orientifold introduced in [27] (there it is also accompanied by open string
modes and a non-trivial gauge symmetry of rank at least $8$), and the same
spectrum is reproduced in the ${\mathbb{Z}}_{4}$ orientifold of $T^{4}$
studied, among other models, in [28]. Because our theory has neither
orientifold planes or ordinary orbifold singularities, by contrast with both
of these constructions, the orientifold we discuss is a consistent closed
unoriented string (as is the model in [28]), but it is also entirely smooth,
and thus has a supergravity description as a supersymmetric compactification
of the IIB string on the Enriques surface $X$, a non-spin manifold.

To understand how it is possible to have a supersymmetric supergravity
compactification on $X$, we recall that the fermions in IIB supergravity are
not really spinors due to the non-trivial action of the duality group [7].
Though it is often left unstated, this point is at the heart of orientifold
and F-theory constructions. The former have been discussed in great generality
in [29, 30]. Our worldsheet construction is an important special case of that
discussion, in that the background preserves supersymmetry and also allows a
complementary spacetime point of view, which we develop next.

The duality group of IIB has been recently discussed in [8, 9, 31], and the
basic point relevant to us is that the bosonic duality symmetry
$\operatorname{SL}(2,{\mathbb{Z}})$ is enlarged to
$\text{Pin}^{+}\operatorname{GL}(2,{\mathbb{Z}})$ when the action on the
fermions is taken into account. Considering the Lorentz symmetry
$\operatorname{SO}(1,9)$ together with the duality group, we find the
corresponding faithful action on the fermions by a group we term
${\operatorname{Spin}_{\text{d}}}(1,9)$, which fits into the exact sequence

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The duality group $\text{Pin}^{+}\operatorname{GL}(2,{\mathbb{Z}})$ includes
the spacetime fermion number $e^{i\pi{{{\boldsymbol{F}}}}}$, and the map $f$
sends the ${\mathbb{Z}}_{2}$ generator to
$(-{\mathbbm{1}}_{16},e^{i\pi{{{\boldsymbol{F}}}}})\in\operatorname{Spin}(1,9)\times\text{Pin}^{+}\operatorname{GL}(2,{\mathbb{Z}})$.
The possibilities for IIB compactification are then clearly enlarged if we
allow to patch fields by ${\operatorname{Spin}_{\text{d}}}(1,9)$ rather than
$\operatorname{Spin}(1,9)$ transition functions.

A lift of a generic $\operatorname{SL}(2,{\mathbb{Z}})$ element leads to
transformations of the fermions by factors that depend on the axio-dilaton
profile. However, if we focus on perturbative backgrounds where the axio-
dilaton remains constant and unconstrained, these factors drop out of the
transformations, and we can restrict attention to the subgroup of perturbative
duality symmetries ${\cal
G}_{\text{per}}\subset\text{Pin}^{+}\operatorname{GL}(2,{\mathbb{Z}})$
generated by two generators
$r=e^{i\pi{{{\boldsymbol{F}}}}_{\text{\tiny{L}}}}\Pi$ and $s=\Pi$ [3, 9]:

$\displaystyle{\cal G}_{\text{per}}\simeq D_{8}=\langle
r,s~{}|~{}r^{4}=s^{2}=(rs)^{2}=1\rangle~{}.$ (2.37)

The analogue of the bosonic duality group $\operatorname{SL}(2,{\mathbb{Z}})$
is now ${\mathbb{Z}}_{2}\times{\mathbb{Z}}_{2}$ generated by $r$ and $s$
action on the bosonic fields, while ${\cal G}_{\text{per}}$ is the lift that
also includes the action on the fermions. We then have

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}\pgfsys<EMAIL_ADDRESS>(2.38)

Working on a general ten-dimensional oriented spacetime $X$ then requires that
the fermions transform in sections of a vector bundle ${\cal F}\otimes T_{X}$,
where the transition functions for ${\cal F}$ are valued in
${\operatorname{Spin}^{\prime}_{\text{d}}}(1,9)$. If we take a good cover
$\\{{\cal U}_{\alpha}\\}_{\alpha\in I}$ for $X$, then over every patch we have

$\displaystyle\left.{\cal F}\right|_{{\cal U}_{\alpha}}={\cal
U}_{\alpha}\times{\mathbb{R}}^{16}\otimes{\mathbb{R}}^{2}~{}.$ (2.39)

A section $S$ of ${\cal F}$ is then locally represented by $S_{\alpha}$, a
real $16\times 2$ matrix, and on ${\cal U}_{\alpha}\cap{\cal
U}_{\beta}\neq\emptyset$ the section patches by

$\displaystyle S_{\alpha}$
$\displaystyle=M_{\alpha\beta}S_{\beta}A_{\beta\alpha}~{},$ (2.40)

where the transition functions take value in
${\operatorname{Spin}^{\prime}_{\text{d}}}(1,9)$:

$\displaystyle(M_{\alpha\beta},A_{\beta\alpha})\in{\operatorname{Spin}^{\prime}_{\text{d}}}(1,9)~{}.$
(2.41)

If we do not consider the non-trivial equivalence relation, we simply think of
$M_{\alpha\beta}\in\operatorname{Spin}(1,9)$ and of $A_{\beta\alpha}\in{\cal
G}_{\text{p}}$. The transition functions must satisfy the usual condition on a
triple-overlap ${\cal U}_{\alpha}\cap{\cal U}_{\beta}\cap{\cal
U}_{\gamma}\neq\emptyset$

$\displaystyle(M_{\alpha\beta}M_{\beta\gamma}M_{\gamma\alpha},A_{\alpha\gamma}A_{\gamma\beta}A_{\beta\alpha})=\text{id}\in{\operatorname{Spin}^{\prime}_{\text{d}}}(1,9)~{}.$
(2.42)

A simple way to satisfy this is to have each factor be identity separately, so
that ${\cal F}={\cal S}_{X}\otimes{\cal R}$, where ${\cal S}_{X}$ is a spin
bundle ${\cal S}_{X}$ over $X$, while ${\cal R}$ is a rank $2$ vector bundle
over $X$ with structure group ${\cal G}_{\text{p}}$. The equivalence relation
$(M,A)\sim(-M,-A)$ encoded in (2.38) allows for a more general possibility:

$\displaystyle M_{\alpha\beta}M_{\beta\gamma}M_{\gamma\alpha}$
$\displaystyle=\pm{\mathbbm{1}}_{16}~{},$ $\displaystyle
A_{\alpha\gamma}A_{\gamma\beta}A_{\beta\alpha}$
$\displaystyle=\pm{\mathbbm{1}}_{2}~{}.$ (2.43)

The lower sign allows for $X$ to be non-spin.

Suppose now that the spacetime is of the form ${\mathbb{R}}^{1,5}\times Y$,
where $Y$ is a four-dimensional compact oriented manifold. In this case we
obtain a decomposition

$\displaystyle{\cal F}$ $\displaystyle={\cal F}_{+}\oplus{\cal F}_{-}~{},$
(2.44)

corresponding to the decomposition

$\displaystyle\operatorname{\mathfrak{spin}}(1,9)$
$\displaystyle\supset\operatorname{\mathfrak{spin}}(1,5)\oplus\operatorname{\mathfrak{su}}(2)\oplus\operatorname{\mathfrak{su}}(2)~{},$
$\displaystyle{{\boldsymbol{16}}}$
$\displaystyle=({{\boldsymbol{4}}},{{\boldsymbol{1}}},{{\boldsymbol{2}}})\oplus({{\boldsymbol{4}}}^{\prime},{{\boldsymbol{2}}},{{\boldsymbol{1}}})~{}.$
(2.45)

A section ${\cal S}$ of ${\cal F}_{+}$ has components ${\cal S}_{aI}^{A}$,
with $a=1,\ldots,8$, $I=1,2$ denoting the
$\operatorname{\mathfrak{spin}}(1,5)$ and $\operatorname{\mathfrak{su}}(2)$
indices, while $A=1,2$ is the ${\cal G}_{\text{p}}$ index. Reality of the
underlying 10-dimensional field then implies

$\displaystyle{\cal S}_{aI}^{A}={\cal C}_{ab}{\epsilon}_{IJ}\left({\cal
S}_{bJ}^{A}\right)^{\ast}~{},$ (2.46)

where ${\cal C}$ is the $6$-dimensional charge conjugation matrix, and
${\epsilon}_{IJ}$ is the invariant tensor of $\operatorname{SU}(2)$.

Since the transition functions are trivial in the ${\mathbb{R}}^{1,5}$
directions, the existence of fermions in ${\cal F}_{+}$ boils down to the
existence of a chiral ${\operatorname{Spin}^{\prime}_{\text{d}}}(4)$ structure
on $Y$. Abusing notation slightly, we denote the corresponding bundle again by
${\cal F}_{+}$, now with section locally having the form ${\cal S}^{A}_{I}$, a
$2\times 2$ complex-valued matrix. If $Y$ is taken to be a K3 manifold, then
clearly such a structure exists when the holonomy $\operatorname{SU}(2)$ is
taken to be valued in the second factor, and we can take ${\cal S}$ to be a
constant $2\times 2$ matrix.

If we tune the moduli of $Y$ to the Enriques locus, then there is a Kähler
isometry $\phi:Y\to Y$ such that $\phi$ is fixed-point free and
$\phi\circ\phi=\text{id}$, and its action on tensors is by pull-back:
$U_{E}\cdot T=\phi^{\ast}T$. The action on the frame bundle with local
orthonormal basis $\boldsymbol{e}_{a}$ is simply
$U_{E}\cdot\boldsymbol{e}_{a}(x)=M_{a}^{b}(x)\boldsymbol{e}_{b}(x)$, where
$M_{a}^{b}\in\operatorname{SO}(4)$, and in terms of a holomorphic basis

$\displaystyle\boldsymbol{p}_{1}$
$\displaystyle={\textstyle\frac{1}{\sqrt{2}}}\left(\boldsymbol{e_{1}}+i\boldsymbol{e}_{4}\right)~{},$
$\displaystyle\boldsymbol{p}_{2}$
$\displaystyle={\textstyle\frac{1}{\sqrt{2}}}\left(\boldsymbol{e_{2}}+i\boldsymbol{e}_{3}\right)~{},$
$\displaystyle\boldsymbol{P}=\begin{pmatrix}\boldsymbol{p}_{1}&\boldsymbol{p}_{2}\\\
-\overline{\boldsymbol{p}}_{2}&\overline{\boldsymbol{p}}_{1}\end{pmatrix}~{},$
(2.47)

the $\operatorname{SO}(4)$ action by elements
$g=(M_{1},M_{2})\in\\{\operatorname{SU}(2)\times\operatorname{SU}(2)\\}/{\mathbb{Z}}_{2}$
is

$\displaystyle g\cdot\boldsymbol{P}=M_{1}\boldsymbol{P}M_{2}~{}.$ (2.48)

It is easy to see that elements of the form $({\mathbbm{1}},M_{2})$ preserve
the hyper-Kähler forms

$\displaystyle j$
$\displaystyle=i\boldsymbol{p}_{1}\wedge\overline{\boldsymbol{p}}_{1}+i\boldsymbol{p}_{2}\wedge\overline{\boldsymbol{p}}_{2}~{},$
$\displaystyle\omega$
$\displaystyle=2\boldsymbol{p}_{1}\wedge\boldsymbol{p}_{2}~{},$
$\displaystyle\overline{\omega}$
$\displaystyle=2\overline{\boldsymbol{p}}_{1}\wedge\overline{\boldsymbol{p}}_{2}~{},$
(2.49)

while those of the form $(M_{1},{\mathbbm{1}})$ rotate them as a triplet. The
Enriques action on the frame and the covariantly constant spinors is then by
the element $U_{E}=(i\sigma_{3},i\sigma_{3})$,666 Here $\sigma_{3}$ is the
usual Pauli matrix $\sigma_{3}=\begin{pmatrix}1&0\\\ 0&-1\end{pmatrix}$.:

$\displaystyle U_{E}\cdot\boldsymbol{P}$
$\displaystyle=\begin{pmatrix}-\boldsymbol{p}_{1}&\boldsymbol{p}_{2}\\\
-\overline{\boldsymbol{p}}_{2}&-\overline{\boldsymbol{p}}_{1}\end{pmatrix}~{},$
$\displaystyle U_{E}\cdot{\cal S}$ $\displaystyle=i\sigma_{3}{\cal S}~{}.$
(2.50)

Taking the quotient $Y/G_{E}=X$ leads to the Enriques surface as the
compactification geometry, but we see that it does not inherit any of the
spinors. Notice also that when extended to act on the spin bundle the action
is no longer order $2$.

On the other hand, we observe that the background on $Y$ has another
${\mathbb{Z}}_{2}$ symmetry, with generator

$\displaystyle\widetilde{U}_{E}=(g,e^{i\pi{{{\boldsymbol{F}}}}_{\text{\tiny{L}}}}\Pi)\in\operatorname{SO}(4)\times{\cal
G}_{\text{p}}~{}.$ (2.51)

This action has a lift to ${\operatorname{Spin}^{\prime}_{\text{d}}}(4)$ and
acts on ${\cal S}$ by

$\displaystyle\widetilde{U}_{E}\cdot{\cal S}$ $\displaystyle=i\sigma_{3}{\cal
S}i\sigma_{2}=-\sigma_{3}{\cal S}\sigma_{2}~{}.$ (2.52)

Now the action squares to identity, and there are invariant fermions that are
solutions to

$\displaystyle{\cal S}$ $\displaystyle=-\sigma_{3}{\cal S}\sigma_{2}~{}.$
(2.53)

In this way we obtain an explicit geometric understanding of supersymmetry
preservation on a non-spin background geometry.

##  3 Further examples

The conclusion of the previous section is that it is possible to compactify
IIB string theory on the Enriques surface by using
${\operatorname{Spin}^{\prime}_{\text{d}}}$ fermions. Moreover, it is possible
to analyze the resulting theory both from supergravity and from the
worldsheet, and the latter does not require any non-perturbative degrees of
freedom. In this section we will discuss a few other examples of these sorts
of “flat F-theory” backgrounds.

###  3.1 Toroidal generalizations

The first idea one might have is to replace the K3 geometry with the simpler
choice of $T^{4}$. The analogue of the Enriques involution on periodic
coordinates $x_{i}$, $x_{i}\sim x_{i}+1$ is then simply

$\displaystyle
U_{E}\cdot(x_{1},x_{2},x_{3},x_{4})=\left(x_{1}+{\textstyle\frac{1}{2}},-x_{2},-x_{3},x_{4}\right)~{},$
(3.1)

and we see that the space is of the form $X_{3}\times S^{1}$, where $X_{3}$ is
a compact flat Riemannian manifold with holonomy ${\mathbb{Z}}_{2}$. Dropping
the spectator circle, we see that there is a $5$-dimensional flat F-theory
background, $(T^{3}\times T^{2})/{\mathbb{Z}}_{2}$, combining a Bieberbach
action on the first factor with a holomorphic involution on the second factor.
This $7$-dimensional background was recently discussed in [32], where it is
also pointed out that it has non-perturbative relatives, where the
${\mathbb{Z}}_{2}$ action is replaced by one of
${\mathbb{Z}}_{3},{\mathbb{Z}}_{4},{\mathbb{Z}}_{6}$. While those quotients
still do not have orientifold planes, the IIB axio-dilaton is now fixed to
special values, and, moreover, the action on the worldsheet spectrum
invariably mixes NS-NS with R-R fields.777In this case the non-trivial
“modular” factors of the form $(c\tau+d)/(c\overline{\tau}+d)$ that appear in
the transformations of the fermions—see, for example, the discussion in
section 3.1 of [9]—must be included to obtain the desired invariant fermions.

While the latter feature makes it difficult to understand the world-sheet
theory in the RNS formalism, the former signals that a perturbative worldsheet
treatment will be at best formal because the dilaton is fixed.

It is also possible to build similar lower-dimensional compactifications. For
instance, consider $Z=((T^{2})^{4}\times S^{1})/{\mathbb{Z}}_{2}$, where the
action of the generator is

$\displaystyle
g\cdot(z_{0},z_{1},z_{2},z_{3};x)=\left(-z_{0},-z_{1},-z_{2},-z_{3};x+{\textstyle\frac{1}{2}}\right)~{}.$
(3.2)

$Z$ has holonomy
${\mathbb{Z}}_{2}\subset\operatorname{Sp}(1)\times\operatorname{Sp}(1)$, and
it is a flat F-theory background with base the $7$-dimensional non-spin
manifold $X=(T^{6}\times S^{1})/{\mathbb{Z}}_{2}$ obtained by dropping the
$z_{0}$ coordinate. We then have the corresponding perturbative IIB
orientifold of $T^{7}\times{\mathbb{R}}^{1,2}$ that preserves $8$ supercharges
in three dimensions.

There is no need to restrict to a quotient by a single cyclic group. Consider
again $T^{9}$ and the action of
$G={\mathbb{Z}}_{2}\times{\mathbb{Z}}_{2}\times{\mathbb{Z}}_{2}$ with
generators acting on the holomorphic coordinates as

$\displaystyle g_{1}\cdot(z_{0},z_{1},z_{2},z_{3};x)$
$\displaystyle=\left(-z_{0},-z_{1},z_{2}+{\textstyle\frac{1}{2}},z_{3};x\right)~{},$
$\displaystyle g_{2}\cdot(z_{0},z_{1},z_{2},z_{3};x)$
$\displaystyle=\left(-z_{0},z_{1},-z_{2},z_{3}+{\textstyle\frac{1}{2}};x\right)~{},$
$\displaystyle g_{3}\cdot(z_{0},z_{1},z_{2},z_{3};x)$
$\displaystyle=\left(-z_{0},z_{1}+{\textstyle\frac{1}{2}},z_{2},-z_{3};x+{\textstyle\frac{1}{2}}\right)~{}.$
(3.3)

The action is free on $T^{9}$, as well as on the $T^{7}$ spanned by the
coordinates $(z_{1},z_{2},z_{3};x)$. The only holomorphic form preserved is
$dz_{0}\wedge dz_{1}\wedge dz_{2}\wedge dz_{3}$, so this is an F-theory
background preserving $4$ supercharges in $3$ dimensions, and it has a
perturbative orientifold description with a large radius limit as a IIB
supergravity compactification on the non-spin manifold $T^{7}/G$.

###  3.2 Flat F-theory with curved target space

It is possible to generalize the discussion to curved geometries. The first
idea one might have is to replace the Enriques surface with a quotient of a
higher-dimensional Calabi-Yau manifold with holonomy $\operatorname{SU}(n)$ by
a freely-acting Kähler isometry that projects out the holomorphic $n$-form.
This is not possible for $n=3$: the Euler characteristic of the structure
sheaf of the Calabi-Yau $3$-fold is zero, and the quotient necessarily
preserves this vanishing. It then follows that the holomorphic $3$-form will
be preserved.888We thank James Gray for explaining this to us. When $n$ is
even this restriction does not apply, and it may be possible to find a
$4$-fold with an appropriate free ${\mathbb{Z}}_{2}$ action. However, even for
$n=2$ the constraint is non-trivial: for example it immediately shows that a
freely-acting Kähler isometry of a K3 surface that projects out the
holomorphic form must have order $2$.

Alternatively, we can allow for a non-free action on the Calabi-Yau $n$-fold
but compensate this by a shift on an additional circle. For example, consider
IIB string theory compactified on $Y\times S^{1}$, where $Y$ is a K3 surface
realized as a smooth quartic hypersurface in ${\mathbb{P}}^{3}$. Denoting the
projective coordinates by $[y_{0}:y_{1}:y_{3}]$, we take the hypersurface as
the vanishing locus of

$\displaystyle P=y_{0}^{3}y_{1}+y_{1}^{4}+y_{2}^{4}+y_{3}^{4}~{}.$ (3.4)

$Y$ has a large group of Kähler isometries inherited from isometries of the
ambient ${\mathbb{P}}^{3}$, including a ${\mathbb{Z}}_{4}$ subgroup generated
by $g_{4}$, and ${\mathbb{Z}}_{6}$ subgroup generated by $g_{6}$, with

$\displaystyle g_{4}\cdot[y_{0}:y_{1}:y_{2}:y_{3}]$
$\displaystyle=[y_{0}:y_{1}:y_{2}:\zeta_{4}y_{3}]~{},$ $\displaystyle
g_{6}\cdot[y_{0}:y_{1}:y_{2}:y_{3}]$
$\displaystyle=[-\zeta_{3}y_{0}:-y_{1}:-y_{2}:y_{3}]~{},$ (3.5)

where $\zeta_{n}=e^{2\pi i/n}$ denotes the $n$-th primitive root of unity. We
recall—see, e.g. [33]—that in this case the holomorphic (2,0) form has a
residue presentation as

$\displaystyle\omega$
$\displaystyle=\operatorname{Res}\frac{1}{P}\sum_{j=0}^{3}(-1)^{j}y_{j}dy_{0}\wedge\cdots\wedge\widehat{dy_{j}}\wedge\cdots\wedge
dy_{d-1}~{},$ (3.6)

which makes it easy to read off the action of the generators on $\omega$:

$\displaystyle g_{4}\cdot\omega$ $\displaystyle=\zeta_{4}\omega~{},$
$\displaystyle g_{6}\cdot\omega=\zeta_{6}\omega~{}.$ (3.7)

There are different ways in which we can use $Y$ for a string
compactification. For example, we could use the Borcea-Voisin construction and
its generalizations to build Calabi-Yau $3$-folds of the form $Z=(Y\times
T^{2})/G$, with $G={\mathbb{Z}}_{n}$ and $n\in\\{2,3,4,6\\}$. This will
typically require a resolution of singularities.999A review of the original
construction, and further references can be found in [33], while more recent
work on the $n>2$ case, as well as more general automorphisms of non-prime
order includes [34, 35]. Alternatively, we define a modified action on
$Y\times S^{1}$ with generators

$\displaystyle\widetilde{g}_{n}=\left(g_{n},x\mapsto
x+{\textstyle\frac{1}{n}}\right)~{}.$ (3.8)

The quotient by the ${\mathbb{Z}}_{n}$ action generated by $\widetilde{g}_{n}$
is a smooth non-spin manifold $X$ with a Kähler contact structure. But, now
just as above we can construct a IIB orientifold of $Y\times S^{1}$, which
will be a purely closed string theory and will preserve $8$ supercharges in
$5$ dimensions and can also be viewed as a IIB compactification on $X$. The
corresponding F-theory background is the quotient of $Z=(T^{2}\times Y\times
S^{1})/{\mathbb{Z}}_{n}$, where the first factor is the elliptic fiber.

We can replace $Y$ by a Calabi-Yau $3$-fold and obtain an analogous
construction of IIB compactification on $X$—a smooth non-spin $7$-dimensional
manifold with a Kähler contact structure—which nevertheless preserves $4$
supercharges in three dimensions.

Going down further to “compactification” on a Calabi-Yau $4$-fold, we can
obtain superficially similar results. For instance there is a flat F-theory
background $Z=(Y_{1}\times Y_{2}\times T^{2})/G$, where each $Y_{i}$ is a K3
admitting an Enriques involution $\phi_{i}$. Denoting the reflection on
$T^{2}$ by $\rho$, we consider the $G={\mathbb{Z}}_{2}\times{\mathbb{Z}}_{2}$
action on the total space generated by

$\displaystyle g_{1}$ $\displaystyle=(\phi_{1},{\mathbbm{1}},\rho)~{},$
$\displaystyle g_{2}$ $\displaystyle=({\mathbbm{1}},\phi_{2},\rho)~{}.$ (3.9)

The action is free and projects out the holomorphic $(1,0)$, $(2,0)$ and
$(3,0)$ forms, but keeps the $(5,0)$ form, so that $Z$ is a smooth
elliptically fibered Calabi-Yau $5$-fold. The holonomy and fundamental group
of the $5$-fold $Z$ can be worked out in a similar fashion to the Enriques
Calabi-Yau reviewed in the appendix below. The result is that
$\pi_{1}(Z)={\mathbb{Z}}^{2}\rtimes G$, and the holonomy is
$\Gamma=(\operatorname{SU}(2)\rtimes{\mathbb{Z}}_{2})^{2}\subset\operatorname{SU}(5)$.
Since $\Gamma$ is not contained in $\operatorname{SU}(4)$, we expect this to
preserve (2,0) supersymmetry in two dimensions. Further examples of this sort
can be constructed by quotients of hyper-Kähler manifolds with holonomy
$\operatorname{Sp}(2)$, such as the Enriques manifolds discussed in [36, 37].

##  4 Unoriented IIA compactification

The introduction of an additional circle direction allows us to also consider
a T-dual type IIA picture. While the considerations apply to all of the
examples described above, in this section we discuss the basic case of IIA
compactified $Y\times S^{1}$, where $Y$ is a K3 manifold admitting the
Enriques involution. Taking an orientifold by worldsheet parity is now
problematic, since the IIA GSO projection is chiral in the Ramond sector.
However, we also have T-duality of the additional circle with an action
$T_{\text{\tiny{L}}}$, which acts on the internal left-moving NS fields of the
circle as

$\displaystyle T_{\text{\tiny{L}}}:X^{5}_{\text{\tiny{L}}}$
$\displaystyle\mapsto-X^{5}_{\text{\tiny{L}}}~{},$ $\displaystyle
T_{\text{\tiny{L}}}:{\cal X}^{5}_{\text{\tiny{L}}}$
$\displaystyle\mapsto-{\cal X}^{5}_{\text{\tiny{L}}}~{},$ (4.1)

while leaving the other NS fields invariant and maps the IIA string on
$Y\times S^{1}$, where $S^{1}$ has radius $R$, to the IIB string on $Y\times
S^{1}$ with the T-dual radius $\alpha^{\prime}/R$. This allows us to define a
symmetry of the IIA string on $Y\times S^{1}$ (note that
$T_{\text{\tiny{L}}}^{2}=\text{id}$):

$\displaystyle\widetilde{U}_{\text{IIA}}=T_{\text{\tiny{L}}}^{-1}\widetilde{U}_{E}T_{\text{\tiny{L}}}=T_{\text{\tiny{L}}}U_{E}e^{i\pi{{{\boldsymbol{F}}}}_{\text{\tiny{L}}}}\Pi
T_{\text{\tiny{L}}}~{},$ (4.2)

and this leads to a consistent orientifold of the IIA string: using $\Pi
T_{\text{\tiny{L}}}=T_{\text{\tiny{R}}}\Pi$, we find

$\displaystyle\widetilde{U}_{\text{IIA}}=R_{5}U_{E}e^{i\pi{{{\boldsymbol{F}}}}_{\text{\tiny{L}}}}\Pi~{},$
(4.3)

where $R_{5}=T_{\text{\tiny{L}}}T_{\text{\tiny{R}}}$ is the spacetime
reflection symmetry acting on the “5”-th circle direction. The action
$R_{5}U_{E}$ is free on $Y\times S^{1}$, but this time it projects out the
volume form! Consequently, the spacetime interpretation is now a
compactification of IIA supergravity on the unoriented manifold
$\widetilde{X}=(Y\times S^{1})/{\mathbb{Z}}_{2}$, where the generator of
${\mathbb{Z}}_{2}$ is $(\phi,\rho)$, where $\phi$ is the Enriques involution
on the K3 factor and $\rho$ is the reflection symmetry of the single circle.
As in our IIB discussion, the novelty is not so much in the loss of the
orientation in the quotient—after all that is inherent in IIA orientifolds, as
was pointed out long ago [38]; instead, it is that the compactification has a
completely string-perturbative description with a supergravity limit, and it
is T-dual to the IIB compactification on the Enriques surface discussed
above.101010Similar arguments should apply to our other examples upon further
compactification on a circle, but for the remainder of this section we will
stick to $\widetilde{X}$.

###  4.1 A check of the massless spectrum

The $5$-dimensional massless spectrum provides a simple check of the proposed
T-dual pair. This can be carried out directly in spacetime and in general
requires the machinery of twisted cohomology, or, more precisely, cohomology
with local coefficients—see, for example [39] for a pedagogical introduction.
In our case the situation is much simpler, because we can work on the covering
space and just project onto invariant states.

To see how this works we start with the massless spectrum of IIB supergravity
compactified on the K3 surface $Y$ on the Enriques locus in the moduli space.
Returning to the cohomology of $Y$, we have as in (2.7) the non-vanishing
groups

$\displaystyle H^{0}(Y,{\mathbb{R}})$
$\displaystyle=\left({\mathbb{R}}\right)_{\text{e}}~{},$ $\displaystyle
H^{2}(Y,{\mathbb{R}})$
$\displaystyle=\left({\mathbb{R}}^{10}\right)_{\text{e}}\oplus\left({\mathbb{R}}^{12}\right)_{\text{o}}~{},$
$\displaystyle H^{4}(Y,{\mathbb{R}})$
$\displaystyle=\left({\mathbb{R}}\right)_{\text{e}}~{},$ (4.4)

while the action of the gauge symmetry $\Pi
e^{i\pi{{{\boldsymbol{F}}}}_{\text{\tiny{L}}}}$ on the ten-dimensional bosonic
fields is as follows:

field $\displaystyle g$ $\displaystyle\phi$ $\displaystyle B$ $\displaystyle
C_{0}$ $\displaystyle C_{2}$ $\displaystyle C_{4}^{+}$ $\displaystyle
e^{i\pi{{{\boldsymbol{F}}}}_{\text{\tiny{L}}}}\Pi~{}$ $\displaystyle+$
$\displaystyle+$ $\displaystyle-$ $\displaystyle+$ $\displaystyle-$
$\displaystyle+$

To describe the massless bosonic spectrum it then suffices to reduce these
fields while keeping the invariant forms and to supplement that counting with
deformations of $g$ on the Enriques locus. We see that the reduction of
$C_{4}^{+}$ yields $10$ tensor fields and a scalar, while $B$ and $C_{2}$ each
lead to $12$ scalar fields. Including the axio-dilaton and the $30$ real
scalars from deformations of $g$ we obtain the six-dimensional massless
bosonic content of the (1,0) supergravity multiplet, $9$ tensor multiplets and
$12$ hypermultiplets.

We obtain the $5$-dimensional spectrum by reducing the theory further on
$S^{1}$. Upon reduction each tensor multiplet gives to a $5$-dimensional
vector multiplet, each hypermultiplet descends to a $5$-dimensional
hypermultiplet, and the (1,0) supergravity multiplet reduces to the minimal
$5$-dimensional supergravity multiplet and a vector multiplet, so that the
massless bosonic content consists of the metric, $11$ vectors, and $58$ real
scalars. The latter include the $31$ real scalars corresponding to
deformations of the Enriques metric and the radius of the additional circle.

The same methods are easily applied to the massless spectrum of IIA string on
the unoriented manifold $\widetilde{X}=(Y\times S^{1})/{\mathbb{Z}}_{2}$.
Setting $\widetilde{Y}=Y\times S^{1}$, we have the cohomology groups

$\displaystyle H^{0}(\widetilde{Y},{\mathbb{R}})$
$\displaystyle=\left({\mathbb{R}}\right)_{\text{e}}~{},$ $\displaystyle
H^{1}(\widetilde{Y},{\mathbb{R}})$
$\displaystyle=\left({\mathbb{R}}\right)_{\text{o}}~{},$ $\displaystyle
H^{2}(\widetilde{Y},{\mathbb{R}})$
$\displaystyle=\left({\mathbb{R}}^{10}\right)_{\text{e}}\oplus\left({\mathbb{R}}^{12}\right)_{\text{o}}~{},$
$\displaystyle H^{5}(\widetilde{Y},{\mathbb{R}})$
$\displaystyle=\left({\mathbb{R}}\right)_{\text{o}}~{},$ $\displaystyle
H^{4}(\widetilde{Y},{\mathbb{R}})$
$\displaystyle=\left({\mathbb{R}}\right)_{\text{o}}~{},$ $\displaystyle
H^{3}(\widetilde{Y},{\mathbb{R}})$
$\displaystyle=\left({\mathbb{R}}^{10}\right)_{\text{o}}\oplus\left({\mathbb{R}}^{12}\right)_{\text{e}}~{}.$
(4.6)

The action of $R_{5}\Pi e^{i\pi{{{\boldsymbol{F}}}}_{\text{\tiny{L}}}}$ is

field $\displaystyle g$ $\displaystyle\phi$ $\displaystyle B$ $\displaystyle
C_{1}$ $\displaystyle C_{3}$ $\displaystyle
R_{5}e^{i\pi{{{\boldsymbol{F}}}}_{\text{\tiny{L}}}}\Pi~{}$ $\displaystyle+$
$\displaystyle+$ $\displaystyle-$ $\displaystyle-$ $\displaystyle+$

To obtain the massless bosonic spectrum on $\widetilde{X}$ we again take the
invariants obtained by reducing these fields, supplemented by $31$
deformations of the metric. We see that $B_{2}$ and $C_{3}$ lead to $11$
vectors: one of these resides in the gravity multiplet, while the others are
in $10$ $5$-dimensional vector multiplets. The dilaton yields $1$ scalar,
while the reduction of $B_{2}$, $C_{3}$ and $C_{1}$ leads to, respectively,
$12$, $13$, and $1$ scalars. All in all, then, we find a metric $11$ vectors,
and $58$ scalars, just as in the IIB description.

###  4.2 Pinors and spacetime supersymmetry

The worldsheet construction makes it clear that the IIA string yields a
supersymmetric compactification on $\widetilde{X}$, but just as in the IIB
discussion, we would like to have a spacetime interpretation of this, given
that $\widetilde{X}$ is not orientable. The point is that nevertheless
$\widetilde{X}$ carries a $\text{Pin}^{-}$ structure, and this allows the
existence of the requisite supersymmetric fermions.

As a first step we note that $\widetilde{X}$ is a circle bundle over the
Enriques surface $X$: the projection $p_{1}:\widetilde{X}\to X$ simply forgets
the circle direction. The bundle has a section, where
$i_{1}:X\to\widetilde{X}$ is given by $i_{1}(x)=(x,0)$, and of course
$i_{1}p_{1}=\text{id}_{\widetilde{X}}$, $p_{1}i_{1}=\text{id}_{X}$. It follows
that

$\displaystyle T_{\widetilde{X}}=p_{1}^{\ast}(T_{X})\oplus
p_{1}^{\ast}(L)~{},$ (4.8)

where $L=\det T_{X}$. We can use $p_{1}$ to pull back Stiefel-Whitney classes
to conclude

$\displaystyle w_{1}(T_{\widetilde{X}})$
$\displaystyle=p_{1}^{\ast}w_{1}(L)~{},$ $\displaystyle
w_{2}(T_{\widetilde{X}})$
$\displaystyle=p_{1}^{\ast}\left(w_{2}(T_{X})+w_{1}(T_{X})\cup
w_{1}(L)\right)=p_{1}^{\ast}w_{2}(T_{X})~{}.$ (4.9)

The last equality follows because $X$ is oriented, so that $w_{1}(T_{X})=0$.
Because $i_{1}^{\ast}p_{1}^{\ast}=\text{id}_{\widetilde{X}}$, we see that
$w_{1}(L)\neq 0$ and $w_{2}(T_{\widetilde{X}})\neq 0$. Next, note that
$\widetilde{X}$ itself is the base of a circle bundle $Z=(Y\times S^{1}\times
S^{1})/{\mathbb{Z}}_{2}$ — the Enriques Calabi-Yau $3$-fold we encountered
above. Using the projection $p_{2}:Z\to\widetilde{X}$ and section
$i_{2}:\widetilde{X}\to Z$ defined in an analogous fashion, we then see

$\displaystyle 0$
$\displaystyle=w_{1}(T_{Z})=p_{2}^{\ast}\left(w_{1}(T_{\widetilde{X}})+p_{1}^{\ast}w_{1}(L)\right)~{},$
$\displaystyle 0$
$\displaystyle=w_{2}(T_{Z})=p_{2}^{\ast}\left(w_{2}(T_{\widetilde{X}})+w_{1}(T_{\widetilde{X}})\cup
p_{1}^{\ast}w_{1}(L)\right)~{}.$ (4.10)

Using $i_{2}^{\ast}p_{2}^{\ast}=\text{id}_{\widetilde{X}}$ we conclude

$\displaystyle w_{2}(T_{\widetilde{X}})$ $\displaystyle\neq 0~{},$
$\displaystyle w_{2}(T_{\widetilde{X}})+w_{1}(T_{\widetilde{X}})\cup
w_{1}(T_{\widetilde{X}})$ $\displaystyle=0~{}.$ (4.11)

But now we recall [40] that $w_{2}$ is the obstruction to the existence of a
$\text{Pin}^{+}$ structure, while $w_{2}+w_{1}\cup w_{1}$ is the obstruction
to the existence of a $\text{Pin}^{-}$ structure. Thus, we see that
$\widetilde{X}$ admits a $\text{Pin}^{-}$ structure but does not admit a
$\text{Pin}^{+}$ structure. In fact, with our simple quotient construction it
is not hard to check explicitly that a reflection in the circle direction
squares to $-{\mathbbm{1}}$ on the pinor over $\widetilde{X}$, which is the
defining feature of a $\text{Pin}^{-}$ structure.

The $\text{Pin}^{-}$ structure on $\widetilde{X}$ can be thought of as a lift
to the pinor bundle of the holonomy group of
$\widetilde{X}$—$\operatorname{SU}(2)\rtimes{\mathbb{Z}}_{2}$, which is
contained in $\operatorname{O{}}(5)$ but not in $\operatorname{SO}(5)$: this
bundle has non-zero covariantly constant sections, and this is responsible for
the preservation of spacetime supersymmetry.

###  4.3 M-theory lift

What is the M-theory lift of IIA compactified on $\widetilde{X}$? Since
M-theory compactification only makes sense on a manifold with a
$\text{Pin}^{+}$ structure, by the preceding discussion the lift cannot simply
be $\widetilde{X}\times S^{1}$. On the other hand, the duality with F-theory
provides a clear answer: the lift is simply M-theory compactified on the
Enriques Calabi-Yau $Z$. The IIA orientifold action can then be understood
from the M-theory perspective as follows. The appearance of
$e^{i\pi{{{\boldsymbol{F}}}}_{\text{\tiny{L}}}}$ in the IIA reduction is
explained by the observation [41] that the reflection symmetry of the M-theory
circle corresponds to $e^{i\pi{{{\boldsymbol{F}}}}_{\text{\tiny{L}}}}$ in the
IIA string. On the other hand, the GSO projection in the IIA worldsheet
description requires the reflection in the $10$-dimensional spacetime to be
accompanied by worldsheet parity.

This construction can be compared with M-theory compactified on the Klein
bottle [42]. In that case we start with M-theory on a non-oriented manifold
with a $\text{Pin}^{+}$ structure—as explained in, for example [43, 8, 44],
this is required for a consistent M-theory background—and its IIA reduction is
the orbifold of $S^{1}$ that combines
$e^{i\pi{{{\boldsymbol{F}}}}_{\text{\tiny{L}}}}$ and a half-circle shift.
Here, on the other hand, the M-theory background is spin, and its IIA
reduction is a compactification on a non-orientable manifold with a
$\text{Pin}^{-}$ structure and an orientable but non-spin IIB-dual.

##  5 Outlook

In this short note we pointed out a simple class of supersymmetric IIB string
compactifications that share a number of interesting properties: first, they
have a perturbative closed string orientifold description; second, they are
described as flat F-theory backgrounds, where the axio-dilaton profile remains
constant over a Kähler base; finally, they have a supergravity description as
compactifications of IIB supergravity on non-spin manifolds. The latter
feature depends on the non-trivial transformations of IIB fermion fields under
the IIB duality group. With the addition of a circle the backgrounds also have
a IIA interpretation as compactification on unoriented manifolds.

While these backgrounds are rather trivial from the point of view of generic
F/M-theory compactifications, they deserve a closer look precisely because,
while preserving various amounts of supersymmetry, they have a perturbative
string-theoretic description and a geometric interpretation. They thus offer a
perfect framework for exploring aspects of F/M-theory away from the large
radius limit, as well as string duality along the lines of [45]. For instance,
it may be possible to find dual descriptions of some of our $5$-dimensional
examples as heterotic compactification on compact flat manifolds that has been
recently studied in [46]. Perhaps they may help to explore new spacetime
perspectives on string orbifold constructions such as that pursued in [47].

We envision several ways to explore and expand on these observations. Perhaps
the most immediate would be to explore the appropriate generalization of
spacetime T-duality that would account for the non-trivial transformation from
a compactification on a manifold with non-vanishing $w_{2}$ Stiefel-Whitney
class to a T-dual compactification on a non-orientable manifold. While the
appearance of non-spin manifolds has been discussed in the context of
topological T-duality, for example in [48], the resolution of the attending
supersymmetry puzzle involved a non-trivial flux and a corresponding
$\text{Spin}_{\text{c}}$ structure. It would be very useful to rephrase our
T-dual pairs in a purely spacetime language that generalizes the results of
[48].

It will also be good to have a better understanding of the F- and M-theory
lifts of these string backgrounds. In each case we see that the lift gives a
“geometrization” of the action of a spacetime gauge symmetry, such as $\Pi
e^{i\pi{{{\boldsymbol{F}}}}_{\text{\tiny{L}}}}$ in the string
compactification, resulting in a locally constant but globally non-trivial
fibration of the M-theory circle or the F-theory torus. What are the general
rules?

In addition it will be interesting to look for string dual descriptions. Are
there manifestations of the “non-spin” nature of these compactifications in
the dual description? It would also be interesting to explore the
constructions more systematically. For example, one could use the
classification of compact flat Kähler manifolds [49] to identify suitable non-
spin manifolds and study in detail the resulting orientifolds.

More generally, the global structures studied here can also arise in less
trivial examples that also involve fluxes and might lack an explicit
worldsheet description. To handle such situations it would be useful to
develop a theory akin to that of G-structures that would also incorporate the
global effects. Finally, it should also be possible to find other routes to
non-spin compactifications. For instance, by compactifying the heterotic
string on $T^{2}$ one also obtains a duality group action, and further
compactification could make use of this. Studying such compactifications,
their geometries, non-geometric limits, and dual descriptions should provide
valuable markers in the string landscape.

##  Appendix A $G$-structures and the Enriques Calabi-Yau manifold

In this short appendix we make some remarks on the topology of the Enriques
Calabi-Yau $Z=(Y\times T^{2})/{\mathbb{Z}}_{2}$. A detailed study of the full
homology was carried out in [50]. We will not need all of those details and
just note two basic facts: the fundamental group is given by
$\pi_{1}(Z)=({\mathbb{Z}})^{2}\rtimes{\mathbb{Z}}_{2}$, while its holonomy
group is
$\Gamma=\operatorname{SU}(2)\rtimes{\mathbb{Z}}_{2}\subset\operatorname{SU}(3)$.

To explain the first result we can work on the covering space
$\widetilde{Z}=Y\times{\mathbb{R}}^{2}$. To obtain $Z$ we quotient by a group
with generators $h[\vec{m}]$,
$\vec{m}\in{\mathbb{Z}}^{2}\subset{\mathbb{R}}^{2}$, and $g$, which act on a
point $z=(p,\vec{x})\in\widetilde{Z}$ as

$\displaystyle h[\vec{m}](p,\vec{x})$ $\displaystyle=(p,\vec{x}+\vec{m})~{},$
$\displaystyle g(p,\vec{x})$ $\displaystyle=(\phi(p),-\vec{x})~{},$ (A.1)

where $\phi:Y\to Y$ is the Enriques involution. From this form of the
generators we easily see that $h[\vec{m}]$ generate a normal subgroup
${\mathbb{Z}}^{2}$, while $g$ generates a subgroup ${\mathbb{Z}}_{2}$, and the
group product is the advertised semi-direct product:

$\displaystyle h[\vec{m}]gh[\vec{n}]g^{k}=h[\vec{m}-\vec{n}]g^{k+1}~{}.$ (A.2)

Moving on to the holonomy, the first factor is associated to $\Gamma_{0}$, the
holonomy group generated by parallel transport around null-homotopic loops,
while the second is associated to the non-trivial homotopy class corresponding
to the generator $g\in\pi_{1}(Z)$.111111As reviewed in [51],
$\Gamma_{0}\subset\Gamma$ is a normal subgroup, and in general there is a
surjective homomorphism $\pi_{1}({\cal Y})\to\Gamma/\Gamma_{0}$. In our
example the holonomy for any loop associated to the ${\mathbb{Z}}^{2}$ factor
is contained in $\operatorname{SU}(2)$, and so is trivial in
$\Gamma/\Gamma_{0}$.  We can understand the appearance of the semi-direct
product by considering the holonomy action on $Z$ in a little more detail.
First, fixing a base-point and a holomorphic frame for the cotangent bundle
$\\{\boldsymbol{p}_{1},\boldsymbol{p}_{2},\boldsymbol{p}_{3}\\}$, where the
first two basis vectors are as in (2.47) while $\boldsymbol{p}_{3}$ is the
holomorphic frame associated to the torus fiber, the holonomy generated by the
null-homotopic loops is $\Gamma_{0}=\operatorname{SU}(2)$, and its elements
are of the form

$\displaystyle A[M]$ $\displaystyle=\begin{pmatrix}M&0\\\
0&1\end{pmatrix}~{},$ (A.3)

where $M$ is an $\operatorname{SU}(2)$ matrix in the fundamental
representation. On the other hand, a representative for the holonomy along the
loop corresponding to $g$ is (up to conjugation by an $A[M]$) given by

$\displaystyle B$ $\displaystyle=\begin{pmatrix}-\sigma_{3}&0\\\
0&-1\end{pmatrix}~{}.$ (A.4)

Again, the explicit product of elements of the form $A[M]B$ shows the semi-
direct product structure. Note that we obtain a very similar discussion if we
drop the $T^{2}$ fiber all together: we will then find that the Enriques
surface has fundamental group ${\mathbb{Z}}_{2}$ and holonomy
$\operatorname{SU}(2)\rtimes{\mathbb{Z}}_{2}\subset\operatorname{U{}}(2)\subset\operatorname{SO}(4)$.

It is easy to extend this analysis to the $5$-fold example at the end of
section 3.2, where now the holonomy group is generated by

$\displaystyle A_{1}[M]$ $\displaystyle=\begin{pmatrix}M&0&0\\\
0&{\mathbbm{1}}_{2}&0\\\ 0&0&1\end{pmatrix}~{},$ $\displaystyle A_{2}[M]$
$\displaystyle=\begin{pmatrix}{\mathbbm{1}}_{2}&0&0\\\ 0&M&0\\\
0&0&1\end{pmatrix}~{},$ $\displaystyle B_{1}$
$\displaystyle=\begin{pmatrix}-\sigma_{3}&0&0\\\ 0&{\mathbbm{1}}_{2}&0\\\
0&0&-1\end{pmatrix}~{},$ $\displaystyle B_{2}$
$\displaystyle=\begin{pmatrix}{\mathbbm{1}}_{2}&0&0\\\ 0&-\sigma_{3}&0\\\
0&0&-1\end{pmatrix}~{}.$ (A.5)

Since $A_{1}[M]$ and $B_{1}$ each commute with $A_{2}[M]$ and $B_{2}$, the
holonomy is $(\operatorname{SU}(2)\rtimes{\mathbb{Z}}_{2})^{2}$.

Finally, we make one remark concerning the relationship between the
preservation of multiple $G$-structures and supersymmetry. The quotient
preserves two $\operatorname{SU}(3)$ structures on $Z$, one with $3$-form
$\Omega=\omega\wedge dz$, and another with $3$-form
$\Omega^{\prime}=\omega\wedge d\overline{z}$. This is a feature often seen in
supergravity solutions with enhanced supersymmetry: different spinors
stabilize different $\operatorname{SU}(3)$ structures, and their intersection
can lead to a smaller structure group. In this case the two
$\operatorname{SU}(3)$ structures indeed intersect in a smaller group
$\operatorname{SU}(2)\rtimes{\mathbb{Z}}_{2}\subset\operatorname{SU}(3)$, but
it is just large enough to avoid enhanced supersymmetry.

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