aainstitutetext: Mathematical Institute, Oxford University
Andrew Wiles Building, Woodstock Road
Oxford OX2 6GG, UK bbinstitutetext: Department of Physics and
Astronomy,Uppsala University
SE-751 20 Uppsala, Sweden

# Almost contact structures on manifolds with a $G_{2}$ structure

Xenia de la Ossa b Magdalena Larfors b Matthew Magill
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We review the construction of almost contact metric (three-) structures,
abbreviated ACM(3)S, on manifolds with a $G_{2}$ structure. These are of
interest for certain supersymmetric configurations in string and M-theory. We
compute the torsion of the $SU(3)$ structure associated to an ACMS and apply
these computations to heterotic $G_{2}$ systems and supersymmetry enhancement.
We initiate the study of the space of ACM3Ss, which is an infinite dimensional
space with a local product structure and interesting topological features.
Tantalising links between ACM3Ss and associative and coassociative
submanifolds are observed.

##  1 Introduction

Low-dimensional, minimally supersymmetric vacua of string theory and M-theory
are of interest for a number of reasons. In four dimensions, such ground
states may provide effective theories for the physical world, and give
mathematically consistent UV completions of the standard models of particle
physics and cosmology. Similarly, the AdS/CFT correspondence provides further
motivation to explore supersymmetric vacua in spacetimes with constance
negative curvature in both three and four dimensions. More broadly, both four-
and three-dimensional solutions can serve as toy models where dualities,
deformations and geometric invariants may be explored through the lens of
superstring theory.

The class of manifolds with $G_{2}$ structure, which includes torsion-free
$G_{2}$ manifolds as a subclass, holds an important place in this field of
research. In M-theory they provide four-dimensional $N=1$ Minkowski vacua, and
in type II or heterotic string theory they give rise to three-dimensional
vacua with non-positive cosmological constant. These topics have been studied
for a number of years, and important results have been established. In
particular, large classes of torsion-free $G_{2}$ manifolds have been
constructed joyce1996:1 ; joyce1996:2 ; kovalev20 ; Corti:2012kd ; Corti:2013
and studied in detail by mathematicians and physicists. However, the topic of
$G_{2}$ structures and related superstring compactifications is far from
exhausted. For example, one still lacks a deformation theory of different
$G_{2}$ geometries, and $G_{2}$ instantons, beyond the infinitesimal level
established in Refs. joyce1996:1 ; joyce1996:2 ; Gutowski:2001fm ;
Grigorian:2009ge  and delaOssa:2016ivz ; delaOssa:2017pqy ; Fiset:2017auc ;
delaOssa:2018azc ; Clarke:2016qtg ; Clarke:2020erl ; there are no direct
constructions111However, see e.g. Braun:2017uku  for constructions relying on
string duality. of the singular, compact $G_{2}$ manifolds that are needed for
the existence of chiral families in M-theory vacua Atiyah:2001qf ;
Acharya:2001gy ; a method for counting associative submanifolds, important in
physics applications due to their contributions to the non-perturbative
superpotential, remains to be established Joyce:2016fij ; braun2018infinitely
; Acharya:2018nbo ; Harvey:1999as ; and proofs are lacking for conjectures
regarding duality and mirror symmetry among $G_{2}$ vacua gukov2003duality ;
Braun:2017uku ; braun2018towards ; braun2017mirror ; Eckhard:2018raj .

In this paper we discuss the existence of almost contact (three-) structures
on $G_{2}$ structure manifolds. In brief, an almost contact structure (ACS) is
related to a nowhere vanishing vector field, and an almost contact three-
structure (AC3S) is related to a triple of such vector fields (the formal
definition is given below). Both structures are guaranteed to exist on any
$G_{2}$ structure manifold thomas1969 . Our aim with this paper is, in part,
to provide a detailed review of these well-established mathematical facts,
which, to our knowledge, has been been partly lacking (see however
friedrich1997nearly  and Behrndt:2005im ). This may explain why the ACS
perspective has not been emphasized in the recent physics and mathematics
literature related to $G_{2}$ structure manifolds. In addition, we hope our
paper gives evidence that these almost contact (three-) structures provide
useful perspectives on $G_{2}$ geometry, calibrated submanifolds, string vacua
and supersymmetry.

Almost contact structures are, in some way, odd-dimensional cousins of almost
complex structures in even dimensions. Just as for almost complex structures
in even dimensions, the almost contact structures give rise to projection
operators and a decomposition of e.g. differential forms into longitudinal and
transverse components. More precisely, almost contact structures induce almost
complex structures on the transverse geometry.

As mentioned above, ACS are related to nowhere-vanishing vector fields. The
existence of nowhere vanishing vector fields, or, more generally, nowhere
vanishing differential forms, on a manifold indicates that the tangent bundle
structure group can be reduced. In string compactifications, this is
intimately tied to the amount of supersymmetry preserved by the vacuum. One
may thus suspect that the existence of an AC(3)S may lead to string solutions
with extended supersymmetry and we will demonstrate under which conditions
this holds true. More generally, almost contact structures provide additional
information that may be used in the classification of supersymmetric string
vacua. Indeed, while their related AC(3)S has not necessarily been emphasised,
$SU(2)$ and $SU(3)$ structures have been used in classifications of four-
dimensional $N=1$ M-theory vacua Behrndt:2005im , in constructing M-theory
lifts of $N=1$ type IIA solutions Andriolo:2018yrz , $N=1$ AdS type IIB vacua
Kim:2005ez ; Gran:2007ps ; Passias:2019rga , and has points in common with the
Gran–Papadopoulos classification of heterotic supersymmetric vacua Gran:2005wf
; Gran:2007kh ; Gran:2016zxk .

Nowhere-vanishing vector fields are always allowed in odd dimensions; it is
well known that the obstruction to the existence of a nowhere vanishing vector
field on a closed manifolds is a non-vanishing Euler characteristic Hopf27
.222In this paper, we will focus in particular on closed (i.e. compact,
boundaryless) manifolds. The results we state also hold for non-compact
manifolds, and manifolds with boundary. Odd-dimensional manifolds have
vanishing Euler characteristic, and hence admit an almost contact structure.
It is less obvious, but nonetheless true, that seven-dimensional manifolds
admit three linearly independent, nowhere vanishing vector fields
10.2307/45277146 ; thomas1969 ; kuo1970 . Moreover, as we will explain below,
when combined with the positive three-form that defines a $G_{2}$ structure,
these vector fields give rise to an almost contact three-structure, which is
furthermore compatible with the metric induced by a $G_{2}$ structure
friedrich1997nearly ; Arikan:2012acs ; Arikan:2011acs ; Todd:2015era .
Importantly, the AC3S vectors will not, in general, be parallel with respect
to the $G_{2}$ connection. Thus, while this shows that $G_{2}$ structure
manifolds are necessarily reduced to $SU(2)$, there is no automatic reduction
of the holonomy of the $G_{2}$ connection. A further reduction in the holonomy
would be indicative of enhanced supersymmetry and, therefore, not generically
expected.

In the rest of this paper we will provide a more detailed account of almost
contact metric three-structures on manifolds with $G_{2}$ structures. This is
in part a review of established facts, in part a derivation of new results. We
start in section 1.1 by briefly reviewing the background material and setting
our notation. In section 2 we review almost contact metric structures and the
associated $SU(3)$ structure using a somewhat novel perspective of
differential forms. We also add new observations on this topic by elaborating
on the foliation associated to the vector field of the ACMS, and the related
transverse six dimensional geometry, which indeed carries an $SU(3)$ structure
with intrinsic torsion that we may directly determine. With this result we
can, in section 3, show how $N=1$ heterotic $G_{2}$ systems may be analysed
using ACMS. We then turn, in section 4, to a review of almost contact metric
three-structures and their associated $SU(2)$ structure. Building on these
classical results, we expand, in section 4.3 upon the existence of three- and
four-dimensional foliations related to ACM3Ss and discuss the intriguing
relation between such foliations and associative and coassociative
submanifolds (which are also calibrated submanifolds if the $G_{2}$ structure
is closed and coclosed, respectively). This allows us to initiate a study of
the space of ACM3Ss and show that it has a non-trivial structure. In section 5
we give a number of examples which illustrate the concepts we have reviewed.
Finally, in section 6 we conclude and point out a number of directions for
future studies.

###  1.1 Preliminary notions

####  1.1.1 $G_{2}$ structures

A $G_{2}$ structure manifold is a seven dimensional manifold $Y$ along with a
reduction of the tangent bundle structure group to $G_{2}$ (see for example
FerGray82 ; Bryant:2005mz  and joyce2000  for more details on $G_{2}$
structures). A $G_{2}$ structure exists whenever the seven manifold is
orientable and spin.

Since the group $G_{2}$ can be identified with the group of automorphisms of
the imaginary octonions, reducing the structure group to $G_{2}$ allows us to
identify the tangent bundle as a bundle of imaginary octonions, ${\rm
Im}\,\mathbb{O}$. Making such an identification endows $Y$ with a metric and a
vector cross product, which can in turn be encoded in a stable, positive
three-form, $\varphi$.

The relation between these data is as follows. Given a metric, $g$ and cross-
product, $\times$, the three-form is defined by

$\varphi(u,v,w)=g(u\times v,w)\,.$ (1)

Locally, one can choose a trivialisation of the tangent bundle in which such a
three-form takes a standard form, which in our conventions will be

$\varphi_{0}=(e^{12}+e^{34}+e^{56})\wedge
e^{7}+e^{135}-e^{146}-e^{236}-e^{245}\,.$ (2)

On the other hand, whenever a three-form can be put in such a form, we can
extract a metric and cross product. Indeed, once we have a metric, (1) defines
a cross product. The metric can be defined, using a choice of orientation, by

$6g_{\varphi}(u,v)\,{\rm d}{\rm
vol}_{\varphi}=(u\lrcorner\varphi)\wedge(v\lrcorner\varphi)\wedge\varphi~{},$
(3)

for all vectors $u$ and $v$ in $\Gamma(TY)$. In components this means

$g_{\varphi\,ab}=\frac{\sqrt{\det
g_{\varphi}}}{3!\,4!}\,\varphi_{ac_{1}c_{2}}\,\varphi_{bc_{3}c_{4}}\,\varphi_{c_{5}c_{6}c_{7}}\,\epsilon^{c_{1}\cdots
c_{7}}=\frac{1}{4!}\,\varphi_{ac_{1}c_{2}}\,\varphi_{bc_{3}c_{4}}\,\psi^{c_{1}c_{2}c_{3}c_{4}}~{},$

where we have used the metric to define a dual four-form

$\psi=*_{\varphi}\,\varphi~{},$

and

${\rm d}x^{a_{1}\cdots a_{7}}=\sqrt{\det g_{\varphi}}\ \epsilon^{a_{1}\cdots
a_{7}}\,{\rm d}{\rm vol}_{\varphi}~{}.$

With respect to this metric, the three-form $\varphi$, and hence its Hodge
dual $\psi$, are normalised so that

$\varphi\wedge*\varphi=||\varphi||^{2}\,{\rm d}{\rm
vol}_{\varphi}~{},\qquad||\varphi||^{2}=7~{},$

that is

$\varphi\lrcorner\varphi=\psi\lrcorner\psi=7~{}.$

Choosing a local frame where $\varphi$ takes its standard form, one can verify
that the metric, $g_{\varphi}$, is the standard Euclidean metric and the four-
form will be

$\psi_{0}=e^{3456}+e^{1256}+e^{1234}-e^{2467}+e^{2357}+e^{1457}+e^{1367}$ (4)

The exterior derivative of the forms $(\varphi,\psi)$, which give the
structure equations for the $G_{2}$ structure, can be decomposed into
irreducible representations of $G_{2}$

$\displaystyle{\rm d}_{7}\varphi$
$\displaystyle=\tau_{0}\,\psi+3\,\tau_{1}\wedge\varphi+*\tau_{3}~{},$ (5)
$\displaystyle{\rm d}_{7}\psi$
$\displaystyle=4\,\tau_{1}\wedge\psi-\tau_{2}\wedge\varphi~{},$ (6)

where the torsion classes $\tau_{k}$ are $k$ forms, $\tau_{3}$ is in the $\bf
27$ irreducible representation of $G_{2}$ and $\tau_{2}$ in the $\bf 14$.

####  1.1.2 Almost contact structures

Let $Y$ be an odd dimensional Riemannian manifold with metric $g$. If the
manifold $Y$ admits the existence of an endomorphism $J$ of the tangent bundle
$TY$, a unit vector field $R$ (with respect to the metric $g$), and a one-form
$\sigma$ which satisfy

$J^{2}=-{\bf 1}+R\otimes\sigma~{},\quad\sigma(R)=1~{},$

$Y$ is said to admit an almost contact structure $(J,R,\sigma)$ sasaki1960 ;
sasaki1961 . The one-form $\sigma$ is called the contact form. The dimension
of a manifold admitting an ACS must be odd and the structure group of the
tangent space reduces to $U(n)\times{\bf 1}$, where $2n+1$ is the dimension of
$Y$. The ACS on the manifold $Y$ is said to be a contact structure if

$\sigma\wedge{\rm d}\sigma\wedge\cdots{\rm d}\sigma\neq 0~{},$

everywhere on $Y$. In this paper we are mostly interested in the existence of
almost contact structures on manifols with a $G_{2}$ structure.

A Riemannian manifold $Y$ with an ACS $(J,R,\sigma)$ has an almost contact
metric structure $(J,R,\sigma,g)$ (ACMS) if moreover

$g(Ju,Jv)=g(u,v)-\sigma(u)\,\sigma(v)~{},\qquad\forall u,v\in\Gamma(TY)~{},$
(7)

is satisfied. The fundamental two-form $\omega$ of an almost contact metric
manifold is defined by

$\omega(u,v)=g(Ju,v)~{},\qquad\forall\,u,v\in\Gamma(TY)~{},$ (8)

and satisfies

$\sigma\wedge\omega\wedge\cdots\wedge\omega\neq 0~{}.$ (9)

An almost contact 3-structure (AC3S) on a manifold $Y$ kuo1970  is defined by
three distinct almost contact structures
$(J^{\alpha},R^{\alpha},\sigma^{\alpha}),~{}\alpha=1,2,3$ on $Y$ which satisfy
the following conditions

$\begin{split}J^{\gamma}&=J^{\alpha}\,J^{\beta}-R^{\alpha}\otimes\sigma^{\beta}=-J^{\beta}J^{\alpha}+R^{\beta}\otimes\sigma^{\alpha}~{},\\\
R^{\gamma}&=J^{\alpha}(R^{\beta})=-J^{\beta}(R^{\alpha})~{},\\\
\sigma^{\gamma}&=\sigma^{\alpha}\circ J^{\beta}=-\sigma^{\beta}\circ
J^{\alpha}~{},\\\
\quad\sigma^{\alpha}(R^{\beta})&=\sigma^{\beta}(R^{\alpha})=0~{},\end{split}$
(10)

where $\\{\alpha,\beta,\gamma\\}$ are a cyclic permutation of $\\{1,2,3\\}$. A
manifold admitting an AC3S must have dimension $4n+3$ where $n$ is a non-
negative integer and the structure group of the tangent space reduces to
$Sp(n)\times{\bf 1}_{3}$. An almost contact metric 3-structure on a Riemannian
manifold $Y$ with metric $g$, is an AC3S which satifies

$g(J^{\alpha}u,J^{\alpha}v)=g(u,v)-\sigma^{\alpha}(u)\,\sigma^{\alpha}(v)~{},\forall~{}u,v\in\Gamma(TY)~{},$
(11)

for each $\alpha\in\\{1,2,3\\}$. An AC3S consisting of three contact
structures satisfying (10) is a contact 3-structure and defines a 3-Sasakian
geometry kashiwada .

##  2 $SU(3)$ structures on manifolds with a $G_{2}$ structure

A manifold $Y$ with a $G_{2}$ structure $\varphi$ has more structure than
expected. In this section we review the fact that $(Y,\varphi)$ admits an
almost contact metric structure (ACMS) and thereby reduces the structure group
to $SU(3)$ Arikan:2012acs ; Arikan:2011acs ; Todd:2015era .333In fact, we will
see in section 4 that there are least three non-zero vectors on a manifold
with a $G_{2}$ structure thomas1969 , which gives $Y$ an almost contact metric
3-structure (ACM3S) inducing an $SU(2)$ structure on $Y$ friedrich1997nearly .
The reason this happens stems from the fact that any such manifold admits a
nowhere vanishing vector field Hopf27  $R$ which can be normalized with
respect to the $G_{2}$ metric $g_{\varphi}$.

In the first two subsections we prove the following proposition.

######  Proposition 1.

friedrich1997nearly ; Todd:2015era  Let $(Y,\varphi)$ be a seven dimensional
manifold $Y$ with a $G_{2}$ structure $\varphi$ and let $g_{\varphi}$ be the
metric on $Y$ determined by $\varphi$. Then $Y$ admits an ACMS
$(J,R,\sigma,g_{\varphi})$ determined by a unit vector field $R$, where the
endomorphism $J$ is given by

$J(u)=R\times_{\varphi}u~{},\quad\forall~{}u\in\Gamma(TY)~{},$

and the one form $\sigma$ dual to $R$ with respect to the $G_{2}$ metric
$g_{\varphi}$

$\sigma(R)=1~{}.$

The ACMS determines a foliation ${\cal F}_{R}$ of $Y$ by the one dimensional
integral curves of $R$ with $G_{2}$ metric

${\rm d}s_{\varphi}^{2}=\sigma^{2}+{\rm d}s_{\perp}^{2}~{},$

where ${\rm d}s_{\perp}^{2}$ represents the metric on the transverse geometry
of ${\cal F}_{R}$ induced by the ACMS on $Y$.

Furthermore, the ACMS induces a reduction of the $G_{2}$ structure to an
$SU(3)$ structure $(\omega_{\varphi},\Omega)$ on the transverse geometry of
the foliation, where $\omega_{\varphi}$ is the fundamental two form on $Y$

$\omega_{\varphi}=i_{R}(\varphi)~{},$

and $\Omega$ is a transverse three form of type $(3,0)$ with respect to $J$.
Both forms $(\omega_{\varphi},\Omega)$ are determined uniquely by the ACS
decomposition of the $G_{2}$ structure $\varphi$ on $Y$

$\varphi=\sigma\wedge\omega_{\varphi}+\Omega_{+}~{}.$ (12)

The coassociative four form $\psi$ dual to $\varphi$ decomposes as

$\psi=*_{\varphi}\varphi=-\sigma\wedge\,\Omega_{-}+\frac{1}{2}\,\omega_{\varphi}\wedge\omega_{\varphi}~{}.$

Notice that there is no guarantee that an ACMS will be compatible with a
$G_{2}$ connection, i.e. if $\nabla$ is a given connection on $Y$ with ${\rm
Hol}(\nabla)\subseteq G_{2}$, it does not follow from a choice of ACMS that
${\rm Hol}(\nabla)\subseteq SU(3)$. These issues are discussed in section 2.3.
In appendix A we review the definition of manifolds with an $SU(3)$ structure.

###  2.1 Almost contact metric structures on a manifold with a $G_{2}$
structure

Let $Y$ be a seven dimensional compact manifold. It is known that there exists
(at least) one nowhere vanishing vector field, $R$, on $Y$ Hopf27 . This
vector field defines a one dimensional “characteristic foliation” ${\cal
F}_{R}$ of $Y$, where the one dimensional leaves are the integral curves of
$R$. Given the non-vanishing vector field $R$ on $Y$, we can choose local
coordinates on $Y$ adapted to the foliation structure

$\\{x^{a},a=1,\ldots 7\\}=\\{(r,x^{m}),m=1\ldots 6\\}~{},$ (13)

such that $r$ is the coordinate along the integral curves of $R$ (curves of
constant $x^{m}$) and such that the vector $R$ is given by

$R=\partial_{r}~{}.$ (14)

Suppose now that $Y$ has a $G_{2}$ structure $\varphi$ with metric
$g_{\varphi}$. Without loss of generality, we choose the vector $R$ to be of
unit length with respect to the $G_{2}$ metric. We define a unique one form
$\sigma$ by

$\sigma(u)=g_{\varphi}(R,u)~{},\qquad\forall\,u\in\Gamma(TY)~{}.$ (15)

Note that

$\sigma(R)=1~{},$ (16)

is the statement that $R$ has unit length in terms of $\sigma$.

The $G_{2}$ structure together with a choice of vector field, $R$, defines an
endomorphism, $J$, of $TY$ by

$J(u)=R\times_{\varphi}u~{},\qquad\forall\ u\in\Gamma(TY)~{},$ (17)

where $\times_{\varphi}$ is the cross product on $Y$ determined by the $G_{2}$
structure

$\varphi(u,v,w)=g_{\varphi}(u\times_{\varphi}v,w)~{},\qquad\forall\
u,v,w\in\Gamma(TY)~{}.$ (18)

Equivalently, we can consider $J$ to be a vector-valued one form on $Y$, in
which case it is given, in local coordinates, by

$J^{a}{}_{b}=-\varphi^{a}{}_{bc}\,R^{c}=-i_{R}(\varphi)^{a}{}_{b}~{}.$ (19)

Then, one can easily prove that

$J^{2}=-{\bf 1}+R\otimes\sigma~{},$ (20)

using the cross product identity

$u\times_{\varphi}(u\times_{\varphi}v)=-g_{\varphi}(u,u)\,v+g_{\varphi}(u,v)\,u~{},\qquad\forall\
u,v\in\Gamma(TY)~{}.$ (21)

Moreover,

$J(R)=0~{},\quad{\rm and}\qquad\sigma(J(u))=0~{},\quad\forall\
u\in\Gamma(TY)~{}.$ (22)

Therefore Todd:2015era  the $G_{2}$ structure on $Y$, together with the
existence of a (unit length) nowhere vanishing vector field $R$ on $Y$
determine an almost contact structure $(J,R,\sigma)$ on $Y$ (see section
1.1.2). The one form $\sigma$ is the contact form. The existence of the almost
contact structure (ACS) on $Y$ means that there is a reduction of the
structure group $G_{2}$ to ${\bf 1}\times U(3)$. We will see this explicitly
in this section. We remark that this ACS on $Y$ is not, in general, a contact
structure as it need not be that $\sigma$ satisfies everywhere on $Y$ the
condition

$\sigma\wedge{\rm d}_{7}\sigma\wedge{\rm d}_{7}\sigma\wedge{\rm
d}_{7}\sigma\neq 0~{}.$

Furthermore Arikan:2012acs ; Arikan:2011acs ; Todd:2015era , this ACS is
compatible with the metric, in the sense that that $g_{\varphi}$ satisfies the
necessary condition

$g_{\varphi}(Ju,Jv)=g_{\varphi}(u,v)-\sigma(u)\,\sigma(v)~{},\qquad\forall~{}u,v\in\Gamma(TY)~{}.$
(23)

and therefore $(J,R,\sigma,g_{\varphi})$ defines an ACMS.

Given the almost contact metric structure $(J,R,\sigma,g_{\varphi})$ on $Y$,
one also has the fundamental two form $\omega_{\varphi}$ on $Y$

$\omega_{\varphi}(u,v)=g_{\varphi}(Ju,v)=\varphi(R,u,v)~{},\qquad\forall\
u,v\in\Gamma(TY)~{}.$ (24)

Equivalently,

$\omega_{\varphi}=i_{R}(\varphi)~{}.$ (25)

This two form indeed satisfies

$\sigma\wedge\omega_{\varphi}\wedge\omega_{\varphi}\wedge\omega_{\varphi}\neq
0~{},$

because

${\rm d}{\rm
vol}_{\varphi}=\frac{1}{6}\,\sigma\wedge\omega_{\varphi}\wedge\omega_{\varphi}\wedge\omega_{\varphi}~{},$
(26)

which can be easily proven from (3) and (25).

Given the ACS $(J,R,\sigma)$ on $Y$, consider the bundle, ${\rm Ker}(\sigma)$,
whose fibres are the vectors which are orthogonal to $R$. This is a
codimension one subbundle of $TY$ and induces an orthogonal decomposition of
$TY$ as

$TY={\rm Span}\\{R\\}\oplus{\rm Ker}(\sigma)~{},\qquad{\rm Ker}(\sigma)={\rm
Span}\\{R\\}^{\perp}~{}.$

The endomorphism $J$ maps sections of $\Gamma(TY)$ into sections of ${\rm
Ker}(\sigma)$, as can be seen from equation (22), and thus it can be used to
construct projection operators on $TY$. Indeed, the operator

$-\,J^{2}={\bf 1}-R\otimes\sigma~{},$

is a projection operator mapping $TY$ into ${\rm Ker}(\sigma)$. The action of
the projection operators can naturally be extended to the cotangent bundle
$T^{*}Y$, which now has the orthogonal decomposition

$T^{*}Y={\rm Span}\\{\sigma\\}\oplus{\rm Span}\\{\sigma\\}^{\perp}~{},$

and, therefore, to any tensor on $Y$. We can uniquely decompose any $k$-form
$\alpha$ as

$\alpha=\sigma\wedge\alpha_{0}+\alpha_{\perp}~{},$ (27)

where $\alpha_{0}$ and $\alpha_{\perp}$ are respectively a $(k-1)$-form and a
$k$-form on $Y$ such that

$i_{R}(\alpha)=\alpha_{0}~{},\qquad i_{R}(\alpha_{0})=0~{},\qquad
i_{R}(\alpha_{\perp})=0~{}.$

Furthermore, the endomorphism $J$ induces an orthogonal decomposition of ${\rm
Ker}(\sigma)$ over $\mathbb{C}$

${\rm Ker}(\sigma)\otimes\mathbb{C}={\rm
Ker}(\sigma)_{\mathbb{C}}{}^{(1,0)}\oplus{\rm
Ker}(\sigma)_{\mathbb{C}}{}^{(0,1)}~{}.$

In fact, there are also projection operators $P$ and $Q$ on ${\rm
Ker}(\sigma)\otimes\mathbb{C}$

$\displaystyle P$ $\displaystyle=\frac{1}{2}\,({\bf
1}-i\,J-R\otimes\sigma)=-\frac{i}{2}\,J\,({\bf 1}-iJ)~{},$ $\displaystyle Q$
$\displaystyle=\frac{1}{2}\,({\bf
1}+i\,J-R\otimes\sigma)=+\frac{i}{2}\,J\,({\bf 1}+i\,J)~{},$

which map ${\rm Ker}(\sigma)$ into ${\rm Ker}(\sigma)_{\mathbb{C}}{}^{(1,0)}$
or ${\rm Ker}(\sigma)_{\mathbb{C}}{}^{(0,1)}$ respectively.

Again, the action of these operators can naturally be extended to $T^{*}Y$ and
to any tensor on $Y$. For the cotangent bundle $T^{*}Y$, ${\rm
Span}\\{\sigma\\}^{\perp}\otimes\mathbb{C}$ has the orthogonal decomposition

${\rm Span}\\{\sigma\\}^{\perp}\otimes\mathbb{C}=\left({\rm
Span}\\{\sigma\\}^{\perp}\right)^{(1,0)}\oplus\left({\rm
Span}\\{\sigma\\}^{\perp}\right)^{(0,1)}~{}.$

We can decompose any $k$-form $\alpha$ with respect to the ACS as in equation
(27) where the forms $\alpha_{0}$ and $\alpha_{\perp}$ decompose further into
$(p,q)$-type with respect to $J$.

We have arrived at the conclusion that the $G_{2}$ structure on the manifold
$Y$ is reduced to ${\bf 1}\times U(3)$. The $U(3)$ structure on $Y$ is
determined by the endomorphism $J$ which is effectively an almost complex
structure on $Y$. Moreover, we have a fundamental form $\omega_{\varphi}$
satisfying (26). We discuss below in more detail how this works and how the
$U(3)$ structure reduces further to an $SU(3)$ structure.

###  2.2 Transverse geometry and $SU(3)$ structures on $Y$

In this section we show explicitly how the structure group $G_{2}$ is reduced
to $SU(3)$. The fact that any $G_{2}$ structure manifold admits a reduction to
an $SU(3)$ structure group was already shown by friedrich1997nearly . In that
paper, the authors argued that a nowhere vanishing vector field $R$ on
$(Y,\varphi)$, along with the nowhere vanishing spinor $\eta$ implicit in the
choice of $G_{2}$ structure, induce a second spinor by Clifford
multiplication, $R\eta$. These spinors can be used to construct the $SU(3)$
structure on $Y$. In this section we instead show how the structure group is
reduced to $SU(3)$ by constructing the $SU(3)$ structure directly from the
$G_{2}$ structure three form $\varphi$ and the ACS.444Note that, although
there is always a connection $\nabla$ with $G_{2}$ holonomy on $Y$ such that
$\nabla\eta=0$, it is not necessarily the case that $\nabla(R\eta)=0$, and
therefore the holonomy group of $\nabla$ is not necessarily reduced to
$SU(3)$. A summary of the content of this section and the previous one is
given in proposition 1 at the beginning of this section 2.

We begin by reviewing the notion of the transverse geometry of the foliation
${\cal F}_{R}$. Loosely speaking, locally the transverse geometry pretends to
be the geometry of a hyperplane $X$ which is transverse to $R$. Note, however,
that there is not necessarily a six dimensional manifold $X\subset Y$ whose
tangent plane is transverse to $R$. Another way to say this is that ${\rm
Ker}(\sigma)$ is not necessarily integrable.

We say that a vector $u\in\Gamma(TY)$ is transverse to the foliation ${\cal
F}_{R}$ if $u\in\Gamma({\rm Ker}(\sigma))$. As we saw above, the vector $J(u)$
is transverse, for any $u\in TY$ (see equation (22)). For a $k$-form given as
in equation (27), we can think of $\alpha_{0}$ and $\alpha_{\perp}$ as a $k-1$
form and a $k$ form respectively on the transverse geometry of the foliation
on $Y$. We will call $\alpha_{\perp}$ the transverse component of $\alpha$,
and we call a form transverse if

$i_{R}(\alpha)=0~{}.$ (28)

Clearly, $J(\alpha)$ is transverse for any $k$-form $\alpha$ on $Y$ as
$i_{R}(J(\alpha))=0$. The endomorphism $J$ restricts to an almost complex
structure, $J_{\perp}$ on the transverse geometry of ${\cal F}_{R}$, that is,
$J_{\perp}$ is an endomorphism of ${\rm Ker}(\sigma)$ with
$J_{\perp}^{2}=-{\bf 1}~{}$. The action of $J$ on any $k$ form $\alpha$ on $Y$
can be written as

$J(\alpha)=J(\alpha_{\perp})=J_{\perp}(\alpha_{\perp})~{}.$

The transverse geometry then carries all the properties of an almost complex
structure. For instance, as mentioned before any transverse form decomposes
into $(p,q)$-type with respect to $J_{\perp}$. Furthermore, the fundamental
form $\omega_{\varphi}$ is transverse, and one can define a hermitian
structure on the transverse geometry. It is not hard to prove that
$\omega_{\varphi}$ is type $(1,1)$ with respect to $J_{\perp}$, that is

$J(\omega_{\varphi})=\omega_{\varphi}~{}.$

One can define primitive transverse forms on $Y$ and thus define the Lefshetz
decomposition, with respect to $\omega_{\varphi}$, of transverse forms.

Consider the metric $g_{\varphi}$ on $Y$. Using equation (24) and the
endomorphism $J$, one can establish a metric $g_{\perp}$ on the transverse
geometry. In fact, (24) and (20) imply that on $Y$, the $G_{2}$ metric is
given by

$g_{\varphi}(u,v)=\omega_{\varphi}(u,Jv)+\sigma(u)\,\sigma(v)~{},\qquad
u,\,v\,\in\,\Gamma(TY)~{},$ (29)

where the fact that $R$ is unit length is clearly satisfied. Also, if
$u\in\Gamma({\rm Ker}(\sigma))$, then we have by (15) that

$g_{\varphi}(u,R)=\sigma(u)=0~{}.$

Now, let $u\,,v\,\in\Gamma({\rm Ker}(\sigma))$. Then we define a metric
$g_{\perp}$ on the transverse geometry by

$g_{\varphi}(u,v)=\omega_{\varphi}(u,Jv)=g_{\perp}(u,v)~{}.$ (30)

This means we have a “bundle like” metric on $Y$ of the form

${\rm d}s_{\varphi}^{2}=\sigma^{2}+{\rm d}s_{\perp}^{2}~{},$ (31)

where ${\rm d}s_{\perp}^{2}$ represents the line element on the transverse
geometry and in the coordinate system adapted to the foliation ${\cal F}_{R}$,
the one form $\sigma$ can be written as

$\sigma={\rm d}r+\Sigma~{},\qquad\Sigma=\Sigma_{m}\,{\rm d}x^{m}~{}.$ (32)

The transverse one form $\Sigma$ behaves like a one form connection on the
transverse geometry as can be verified by performing a coordinate
transformation on $Y$. Now, recall that the $G_{2}$ structure $\varphi$
determines a metric $g_{\varphi}$ uniquely on $Y$ by equation (3). Of course,
this metric needs to be the same as that in equation (31).

To compare (31) with (3), we begin by decomposing $\varphi$ with respect to
the ACMS $(J,R,\sigma,g_{\varphi})$. Recall that the ACMS determines detemines
the fundamental two form on $Y$

$\omega_{\varphi}=i_{R}(\varphi)~{}.$

This means that $\varphi$ can be decomposed uniquely as

$\varphi=\sigma\wedge\omega_{\varphi}+\Omega_{+}~{}.$ (33)

for some well defined transverse real three form $\Omega_{+}$ on $Y$. Let
$u\,,v\,\in\Gamma(TY)$ and consider this decomposition of $\varphi$ together
with equation (3) for the metric:

$6\,g_{\varphi}(u,v)\,{\rm d}{\rm vol}_{\varphi}=i_{u}(\varphi)\wedge
i_{v}(\varphi)\wedge\varphi~{}.$

Taking $u=v=R$, and using $g_{\varphi}(R,R)=1$, we have

$6\,{\rm d}{\rm vol}_{\varphi}=i_{R}(\varphi)\wedge
i_{R}(\varphi)\wedge\varphi=\omega_{\varphi}\wedge\omega_{\varphi}\wedge(\sigma\wedge\omega_{\varphi}+\Omega_{+})~{}.$

The last term must vanish as it is a transverse seven form, while the first
term gives the volume form on $Y$ in terms of $\sigma$ and $\omega_{\varphi}$

${\rm d}{\rm
vol}_{\varphi}=\frac{1}{6}\,\sigma\wedge\omega_{\varphi}\wedge\,\omega_{\varphi}\wedge\omega_{\varphi}~{}.$
(34)

We have already seen this before, see (26). Consider now the metric components
for $u\in\Gamma({\rm Ker}(\sigma))$ and $v=R$. As in this case
$g_{\varphi}(u,R)=\sigma(u)=0$, we have

$\displaystyle 0$
$\displaystyle=\sigma\wedge\big{(}i_{u}(\Omega_{+})\wedge\omega_{\varphi}-i_{u}(\omega_{\varphi})\wedge\Omega_{+}\big{)}\wedge\omega_{\varphi}$
$\displaystyle=\sigma\wedge\left(i_{u}(\Omega_{+}\wedge\omega_{\varphi}\wedge\omega_{\varphi})+3\,i_{u}(\omega_{\varphi})\wedge\Omega_{+}\wedge\omega_{\varphi}\right)=\sigma\wedge
3\,i_{u}(\omega_{\varphi})\wedge\Omega_{+}\wedge\omega_{\varphi}~{},$

which must be true for all transverse vectors $u\in\Gamma({\rm Ker}(\sigma))$.
Hence

$\omega_{\varphi}\wedge\Omega_{+}=0~{}.$ (35)

An important consequence of this equation (35), is that $\Omega_{+}$ is a
primitive form type of $(3,0)+(0,3)$ with respect to $J$ because
$\omega_{\varphi}$ is type $(1,1)$. Finally we calculate the components
$g_{\varphi}(u,v)$ where $u$ and $v$ are both in $\Gamma({\rm Ker}(\sigma))$.
In this case, $g_{\varphi}(u,v)=g_{\perp}(u,v)$ and we have

$\begin{split}6\,g_{\perp}(u,v)\,{\rm d}{\rm
vol}_{\varphi}&=\sigma\wedge\Big{(}i_{u}(\Omega_{+})\wedge
i_{v}(\Omega_{+})\wedge\omega_{\varphi}\\\
&\qquad\qquad-\big{(}i_{u}(\omega_{\varphi})\wedge
i_{v}(\Omega_{+})+i_{v}(\omega_{\varphi})\wedge
i_{u}(\Omega_{+})\big{)}\wedge\Omega_{+}\Big{)}\\\
&=3\,\sigma\wedge\Omega_{+}\wedge\,i_{u}(\Omega_{+})\wedge\,i_{v}(\omega_{\varphi})~{},\end{split}$

where we have used the constraint (35). Using the volume form (34) and
contracting with $R$, we find

$\frac{1}{6}\,g_{\perp}(u,v)\,\omega_{\varphi}\wedge\omega_{\varphi}\wedge\omega_{\varphi}=\frac{1}{2}\,\Omega_{+}\wedge\,i_{u}(\Omega_{+})\wedge\,i_{v}(\omega_{\varphi})~{},\quad
u\,,v\,\in\Gamma({\rm Ker}(\sigma))~{}.$ (36)

As $i_{u}(\Omega_{+})$ is a transverse two form type $(2,0)+(0,2)$, for any
vector $u$, it is easy to see that

$i_{u}(\Omega_{+})=-J_{\perp}\big{(}i_{u}(\Omega_{+})\big{)}=i_{J_{\perp}u}\big{(}J_{\perp}(\Omega_{+})\big{)}~{},\quad\forall\,u\in\Gamma({\rm
Ker})(\sigma)~{}.$

Then

$\begin{split}\frac{1}{3}\,g_{\perp}(u,v)\,\omega_{\varphi}\wedge\omega_{\varphi}\wedge\omega_{\varphi}&=\Omega_{+}\wedge\,i_{J_{\perp}u}\big{(}J_{\perp}(\Omega_{+})\big{)}\wedge\,i_{v}(\omega_{\varphi})\\\
&=-i_{J_{\perp}u}\Big{(}\Omega_{+}\wedge
J_{\perp}(\Omega_{+})\wedge\,i_{v}(\omega_{\varphi})\Big{)}+i_{J_{\perp}u}(\Omega_{+})\wedge
J_{\perp}(\Omega_{+})\wedge\,i_{v}(\omega_{\varphi})\\\
&\qquad\qquad+\Omega_{+}\wedge
J_{\perp}(\Omega_{+})~{}i_{J_{\perp}u}i_{v}(\omega_{\varphi})\\\
&=g_{\perp}(u,v)\,\Omega_{+}\wedge
J_{\perp}(\Omega_{+})+J_{\perp}(\Omega_{+})\wedge
i_{J_{\perp}u}(\Omega_{+})\wedge\,i_{v}(\omega_{\varphi})~{},\end{split}$

where, by equation (30)

$i_{J_{\perp}u}i_{v}(\omega_{\varphi})=-\omega_{\varphi}(Ju,v)=g_{\perp}(u,v)~{}.$

Consider now the last term. Using the fact that $J_{\perp}^{2}=-{\bf 1}$ and
that the action of $J_{\perp}$ on a transverse six form is the identity, we
find

$\begin{split}J_{\perp}(\Omega_{+})\wedge
i_{J_{\perp}u}(\Omega_{+})\wedge\,i_{v}(\omega_{\varphi})&=-J_{\perp}\Big{(}-\Omega_{+}\wedge\,J_{\perp}\big{(}i_{J_{\perp}u}(\Omega_{+})\big{)}\wedge\,J_{\perp}\big{(}i_{v}(\omega_{\varphi})\big{)}\Big{)}\\\
&=-\Omega_{+}\wedge\,i_{J_{\perp}u}(\Omega_{+})\wedge\,i_{J_{\perp}v}(\omega_{\varphi})\end{split}$

By equations (36) and (23)

$J_{\perp}(\Omega_{+})\wedge
i_{J_{\perp}u}(\Omega_{+})\wedge\,i_{v}(\omega_{\varphi})=-\frac{1}{3}\,g_{\varphi}(Ju,Jv)\,\omega_{\varphi}\wedge\omega_{\varphi}\wedge\omega_{\varphi}=-\frac{1}{3}\,g_{\varphi}(u,v)\,\omega_{\varphi}\wedge\omega_{\varphi}\wedge\omega_{\varphi}~{}.$

Therefore

$\frac{1}{6}\,g_{\perp}(u,v)\,\omega_{\varphi}\wedge\omega_{\varphi}\wedge\omega_{\varphi}=\frac{1}{4}\,g_{\perp}(u,v)\,\Omega_{+}\wedge
J_{\perp}(\Omega_{+})~{},$

that is,

$\frac{1}{6}\,\omega_{\varphi}\wedge\omega_{\varphi}\wedge\omega_{\varphi}=\frac{1}{4}\,\Omega_{+}\wedge
J_{\perp}(\Omega_{+})~{}.$ (37)

Let $\Omega$ be a $(3,0)$ form with respect to $J_{\perp}$ such that

$\Omega_{+}=\frac{1}{2}\,(\Omega+\overline{\Omega})={\rm Re}\,\Omega~{}.$ (38)

Then we have that

$J_{\perp}(\Omega_{+})=\frac{1}{2\,i}\,(\Omega-\overline{\Omega})={\rm
Im}\,\Omega=\Omega_{-}~{}.$

Then equation (37) becomes

$\frac{1}{6}\,\omega_{\varphi}\wedge\omega_{\varphi}\wedge\omega_{\varphi}=\frac{1}{4}\,\Omega_{+}\wedge\Omega_{-}=\frac{i}{8}\,\Omega\wedge\overline{\Omega}~{}.$
(39)

One can define a Hodge-dual operator $*_{\perp}$ on the transverse geometry
using the transverse metric $g_{\perp}$. Let $\alpha$ be a $k$-form on $Y$
with decomposition

$\alpha=\sigma\wedge\alpha_{0}+\alpha_{\perp}~{},$

then, it is not too difficult to show that

$*_{\varphi}\,\alpha=(-1)^{k}\,\sigma\wedge\,*_{\perp}\,\alpha_{\perp}+*_{\perp}\,\alpha_{0}~{},$
(40)

where one needs,

$\det{g_{\varphi}}=\sigma_{0}^{2}\,\det{g_{\perp}}~{}.$

The coassociative four form $\psi$ then decomposes as

$\psi=*_{\varphi}\varphi=-\sigma\wedge\,\Omega_{-}+\rho_{\varphi}~{},$ (41)

where we have defined the transverse four form $\rho_{\varphi}$ by

$\rho_{\varphi}=*_{\perp}\,\omega_{\varphi}=\frac{1}{2}\,\omega_{\varphi}\wedge\omega_{\varphi}~{},$
(42)

and used the fact that

$*_{\perp}\Omega_{+}=\Omega_{-}~{}.$

This ends the proof of proposition 1.

It is interesting to ask at this point about the conditions for the transverse
geometry to correspond in fact to a submanifold $X\subset Y$. According to the
Frobenius Theorem, the foliation ${\cal F}_{R}$ has a global transverse
section $X$ when ${\rm Ker}(\sigma)$ is an integrable distribution, that is

$[u,v]\in\Gamma({\rm Ker}(\sigma))~{},\quad\forall~{}u\,,v\,\in\Gamma({\rm
Ker}(\sigma))~{}.$ (43)

A short computation shows that this is the case if and only if

$({\rm d}_{7}\sigma)(u,v)=0~{},\quad\forall~{}u\,,v\,\in\Gamma({\rm
Ker}(\sigma))~{},$

that is,

${\rm d}_{7}\sigma=\sigma\wedge\alpha~{},$ (44)

for some transverse one form $\alpha$, so ${\rm d}_{7}\sigma$ has no
transverse component. We will come back to this constraint later when we
discuss examples (see section 5.1).

###  2.3 Decomposing the structure equations

Let $Y$ be a manifold with a $G_{2}$ structure given by the three form
$\varphi$, and let $g_{\varphi}$ be the metric on $Y$ determined by $\varphi$.
In this subsection, we decompose the $G_{2}$ structure equations (5) and (6)
under the ACMS $(J,R,\sigma,g_{\varphi})$, to obtain the torsion classes
$W_{i}$ of the induced transverse $SU(3)$ structure in terms of those
$\tau_{i}$ of the $G_{2}$ structure. We recall that under the ACMS
$(J,R,\sigma,g_{\varphi})$ the $G_{2}$ structure is decomposed in terms of the
underlying transverse $SU(3)$ structure $(\omega_{\varphi},\Omega_{+})$ as

$\displaystyle\varphi$ $\displaystyle=\sigma\wedge\omega+\Omega_{+}~{},$ (45)
$\displaystyle\psi$ $\displaystyle=-\sigma\wedge\Omega_{-}+\rho~{},$ (46)
$\displaystyle\sigma$ $\displaystyle={\rm d}r+\Sigma~{},$ (47)

where from now on we drop the label $\varphi$ in the fundamental two form.

In order to decompose the structure equations under the ACMS, we begin with
the following lemmas regarding the decomposition of the contraction operator
$\lrcorner_{\varphi}$ and the exterior derivative. The proof of both lemmas is
straightforward.

######  Lemma 1.

Let $\alpha$ be a $p$-form and $\beta$ a $(p+q)$-form on $Y$. Let

$\alpha=\sigma\wedge\alpha_{0}+\alpha_{\perp}~{},\qquad\beta=\sigma\wedge\beta_{0}+\beta_{\perp}~{},$

be their decomposition with respect to the ACS. Then

$\alpha\lrcorner_{\varphi}\,\beta=(-1)^{p}\,\sigma\wedge\,(\alpha_{\perp}\,\lrcorner\,\beta_{0})+(\alpha_{0}\lrcorner\beta_{0}+\alpha_{\perp}\,\lrcorner\,\beta_{\perp})~{},$

where $\lrcorner_{\varphi}$ and $\lrcorner$ are the contraction operators with
respect to the $G_{2}$ metric $g_{\varphi}$ and the transverse metric
$g_{\perp}$ respectively.

######  Lemma 2.

Let $\alpha$ be a transverse p-form on $Y$, that is, $i_{R}(\alpha)=0$. The
decomposition under the ACMS of the exterior derivative ${\rm d}_{7}\alpha$ is
given by

${\rm d}_{7}\alpha=\sigma\wedge R(\alpha)+{\rm d}_{\perp}\alpha~{},$ (48)

where we have defined

${\rm d}_{\perp}\alpha={\rm d}\alpha-\Sigma\wedge R(\alpha)~{},$ (49)

and ${\rm d}$ refers to derivatives with respect to the transverse coordinates
in the coordinate system $\\{r,x^{m}\\}$ adapted to the one dimensional
foliation of $Y$ by the non-zero vector $R$ (see (13)). The operator ${\rm
d}_{\perp}$ is a derivation, in particular it satisfies the Leibnitz rule, and
its curvature ${\rm d}^{2}_{\perp}$ can be understood as the curvature of
${\rm d}_{\perp}$ given by its action on transverse forms by

${\rm d}_{\perp}^{2}=-{\rm d}_{\perp}\Sigma\wedge R~{}.$ (50)

The exterior derivative ${\rm d}_{7}$ on the one-form $\sigma$ is then

${\rm d}_{7}\sigma={\rm d}_{7}\Sigma=\sigma\wedge R(\Sigma)+{\rm
d}_{\perp}\Sigma~{},$ (51)

Finally, equations (48)-(51) imply that the exterior derivative ${\rm
d}_{7}\Lambda$ of a $p$-form $\Lambda$ on $Y$ with components

$\Lambda=\sigma\wedge\Lambda_{0}+\Lambda_{\perp}~{},$

is given by

${\rm d}_{7}\Lambda=\sigma\wedge\big{(}-{\rm
d}\Lambda_{0}+R(\Lambda_{\perp}+\Sigma\wedge\Lambda_{0})\big{)}+{\rm
d}_{\perp}\Lambda_{\perp}+{\rm d}_{\perp}\Sigma\wedge\Lambda_{0}~{}.$ (52)

Using Lemma 2, the decomposition of ${\rm d}_{7}\varphi$ and ${\rm d}_{7}\psi$
is given by

$\displaystyle{\rm d}_{7}\varphi$ $\displaystyle=\sigma\wedge\big{(}-{\rm
d}\omega+R(\Omega_{+}+\Sigma\wedge\omega)\big{)}+{\rm
d}_{\perp}\Omega_{+}+{\rm d}_{\perp}\Sigma\wedge\omega~{},$ (53)
$\displaystyle{\rm d}_{7}\psi$ $\displaystyle=\sigma\wedge\big{(}{\rm
d}\Omega_{-}+R(\rho-\Sigma\wedge\Omega_{-})\big{)}+{\rm d}_{\perp}\rho-{\rm
d}_{\perp}\wedge\Omega_{-}~{}.$ (54)

Equating these with the decomposition under the ACMS of the right hand side of
the $G_{2}$ structure equations (5) and (6), we find the following relations
for the transverse $SU(3)$ structure

$\displaystyle{\rm d}\Omega_{+}$
$\displaystyle=\tau_{0}\,\rho+3\,\tau_{1\,\perp}\wedge\Omega_{+}-\big{(}J(\tau_{3\,0})+{\rm
d}\Sigma\big{)}\wedge\omega+\Sigma\wedge R(\Omega_{+}+\Sigma\wedge\omega)~{},$
$\displaystyle{\rm d}\Omega_{-}$
$\displaystyle=4(\tau_{1\,0}\,\rho+\tau_{1\,\perp}\wedge\Omega_{-})-R(\rho-\Sigma\wedge\Omega_{-})-\tau_{2\,0}\wedge\Omega_{+}-\tau_{2\,\perp}\wedge\omega~{},$
$\displaystyle{\rm d}\omega$
$\displaystyle=\tau_{0}\,\Omega_{-}-3\,\tau_{1\,0}\,\Omega_{+}+3\,\tau_{1\,\perp}\wedge\omega+*\tau_{3\,\perp}+R(\Omega_{+}+\Sigma\wedge\omega)~{},$
$\displaystyle{\rm d}\rho$ $\displaystyle=4\,\tau_{1\,\perp}\wedge\rho+{\rm
d}\Sigma\wedge\Omega_{-}+\Sigma\wedge
R(\rho-\Sigma\wedge\Omega_{-})-\tau_{2\,\perp}\wedge\Omega_{+}~{},$

where we have used Lemma 1 and decomposed the $G_{2}$ torsion classes with
respect to the ACS

$\tau_{1}=\sigma\,\tau_{1\,0}+\tau_{1\,\perp}~{},\qquad\qquad\tau_{2}=\sigma\wedge\tau_{2\,0}+\tau_{2\,\perp}\qquad\qquad\tau_{3}=\sigma\wedge\tau_{3\,0}+\tau_{3\,\perp}~{}.$
(55)

The condition that $\tau_{3}\in\Omega^{3}_{\bf 27}(Y)$, implies that
$\tau_{3\,\perp}$ is type $(2,1)+(1,2)$, $\tau_{3\,0}$ is primitive and

$\omega\lrcorner\tau_{3\,\perp}=\tau_{3\,0}\lrcorner\,\Omega_{+}~{}.$ (56)

Similarly, the condition that $\tau_{2}\in\Omega^{2}_{\bf 14}(Y)$ implies that
$\tau_{2\,\perp}$ is primitive and

$\tau_{2\,\perp}\lrcorner\,\Omega_{-}=\tau_{2\,0}~{}.$ (57)

We can now compare these relations with the $SU(3)$ structure equations (171)
and (172), and deduce formulas for the torsion classes of the ACMS-induced
$SU(3)$ structure on $Y$, $W_{i}$, in terms of the $G_{2}$ torsion classes and
the flow of the $SU(3)$ structure along $R$. After a somewhat lengthy
computation555These computations are very similar to those in reference
delaOssa:2014lma ., we find

$\begin{split}{\rm
Re}\,W_{0}&=\frac{2}{3}\,\tau_{0}+\frac{1}{6}\,\Omega_{-}\lrcorner
R\big{(}\Omega_{+}+\Sigma\wedge\omega\big{)}~{},\\\\[5.0pt] {\rm
Im}\,W_{0}&=2\,\tau_{1\,0}-\frac{1}{6}\,\Omega_{+}\lrcorner
R\big{(}\Omega_{+}+\Sigma\wedge\omega\big{)}~{},\\\\[5.0pt]
2\,W_{1}&=4\,\tau_{1\,\perp}-J(\tau_{2\,0})+{\rm
d}_{\perp}\Sigma\lrcorner\,\Omega_{-}+\omega\lrcorner(\Sigma\wedge
R(\omega))~{},\\\\[5.0pt] 2\,{\rm
Re}\,\theta&=4\,\tau_{1\,\perp}+\frac{1}{2}\,J(\tau_{2\,0})+\frac{1}{2}\,\omega\lrcorner\,R(\Omega_{+})+\Omega_{-}\lrcorner
R(\Sigma\wedge\Omega_{-})\\\\[5.0pt] {\rm
Re}\,W_{2}&=-\tau_{3\,0}^{(1,1)}-{\cal P}\Big{(}{\rm
d}_{\perp}\Sigma-\omega\lrcorner\big{(}\Sigma\wedge
R(\Omega_{+})\big{)}\Big{)}^{(1,1)}~{},\\\\[5.0pt] {\rm
Im}\,W_{2}&=-\tau_{2\,\perp}^{(1,1)}-{\cal
P}\Big{(}R(\omega)-\omega\lrcorner\big{(}\Sigma\wedge
R(\Omega_{-})\big{)}\Big{)}^{(1,1)}~{},\\\\[5.0pt] W_{3}&={\cal
P}\Big{(}J(\tau_{3\,\perp})+R\big{(}\Omega_{+}+\Sigma\wedge\omega\big{)}^{(2,1)+(1,2)}\Big{)}~{}.\end{split}$
(58)

In these equations, $\cal P$ denotes that the primitive part of the form is
taken. Two additional identities appear in this computation:

$\displaystyle 2\,\omega\lrcorner{\rm d}_{\perp}\Sigma$
$\displaystyle=-\tau_{0}-\Omega_{-}\lrcorner\,R(\Omega_{+})~{},$ (59)
$\displaystyle\big{(}{\rm d}_{\perp}\Sigma\big{)}^{(2,0)+(0,2)}$
$\displaystyle=\tau_{3\,0}{}^{(2,0)+(0,2)}+\left(\tau_{1\,\perp}+\frac{1}{2}\,J(\tau_{2\,0})+R(\Sigma)+\frac{1}{2}\,\omega\lrcorner
R(\Omega_{+})\right)\lrcorner\Omega_{-}~{}.$ (60)

Let $\nabla$ be a connection compatible with the $G_{2}$ structure with
intrinsic torsion classes $\\{\tau_{i},i=0,1,2,3\\}$. Equations (58) imply
that one can construct on $Y$ a connection $\hat{\nabla}$ with $SU(3)$
holonomy with intrinsic torsion classes $\\{{\rm
Re}(\theta),W_{0},W_{1},W_{2},W_{3}\\}$. We remark however that this does not
necessarily imply that $\nabla\omega=0$ and $\nabla\Omega=0$. In fact, one can
prove, recalling $\omega=i_{R}(\varphi)$, that $\nabla\omega=0$ if and only if
$\nabla\sigma=0$. In section, 3 we discusss an application of this condition
in the context of supersymmetry enhancement in heterotic string theories.

##  3 Heterotic $G_{2}$ systems under the ACMS

In this section we explore the effect of ACMS on a class of minimally
supersymmetric compactifications of the heterotic string Gunaydin:1995ku ;
Gauntlett:2001ur ; 2001math……2142F ; Gauntlett:2002sc ; Firedrich:2003 ;
Gauntlett:2003cy ; Ivanov:2003nd ; Lukas:2010mf ; Gray:2012md  that have
received attention lately due to, on the one hand, their connection to $G_{2}$
instanton bundles, and on the other hand, their interesting deformation theory
delaOssa:2016ivz ; delaOssa:2017pqy ; Fiset:2017auc ; delaOssa:2018azc ;
Clarke:2016qtg ; Clarke:2020erl . In particular, we decompose the description
of this class of solutions with respect to an ACMS. We will see that this
extra structure allows to make contact with string compactifications with
enhanced supersymmetry. We use this to write down the necessary and sufficient
constraints on the ACMS for supersymmetry enhancement.

###  3.1 Heterotic $G_{2}$ systems

Let $Y$ be a seven dimensional manifold with a $G_{2}$ structure $\varphi$ and
let $V$ be a vector bundle on $Y$ with connection $A$. We are interested in
the decomposition of ten dimensional heterotic superstring backgrounds on
$(Y,V)$ that preserve minimal supersymmetry under the ACMS, i.e. heterotic
$G_{2}$ systems. A heterotic $G_{2}$ system is defined to be the quadruple

$[(Y,\varphi),(V,A),(TY,\Theta),H]~{},$ (61)

where

  * •

$\varphi$ is an integrable $G_{2}$ structure on the seven dimensional manifold
$Y$, that is $\tau_{2}=0$. In this case the structure equations can be written
as

$\displaystyle{\rm d}_{7}\varphi$
$\displaystyle=\tau_{0}\,\psi+3\,\tau_{1}\wedge\varphi+*\tau_{3}=i_{T}(\varphi)~{},$
(62) $\displaystyle{\rm d}_{7}\psi$
$\displaystyle=4\,\tau_{1}\wedge\psi=i_{T}(\psi)~{},$ (63)

where $T$ is the totally antisymmetric torsion given by

$T(\varphi)=\frac{1}{6}\,\tau_{0}\,\varphi-\tau_{1}\lrcorner\,\psi-\tau_{3}~{}.$
(64)

Note that a $G_{2}$ structure admits a totally antisymmetric torsion if and
only if ${\rm d}\psi\in\Omega^{5}_{{\bf 7}}$, and $T(\varphi)$ is in fact the
torsion of the unique metric connection with a totally antisymmetric torsion.

  * •

$V$ is a gauge bundle with connection $A$ that is an instanton, i.e. its
curvature $\cal F$ satisfies

${\cal F}\wedge\psi=0~{}.$ (65)

  * •

$\Theta$ is a connection on the tangent bundle $TY$ of $Y$ which is also an
instanton

${\cal R}(\Theta)\wedge\psi=0~{},$ (66)

where ${\cal R}(\Theta)$ is the curvature of $\Theta$.

  * •

$H$ is a three form defined by

$H={\rm d}_{7}B+\frac{\alpha^{\prime}}{4}\,({\cal CS}(A)-{\cal
CS}(\Theta))~{},$ (67)

where ${\cal CS}(A)$ is the Chern-Simons form of the connection $A$

${\cal CS}(A)={\rm tr}\left(A\wedge{\rm d}_{7}A+\frac{2}{3}\,A^{3}\right)~{},$
(68)

with a similar definition for ${\cal CS}(\Theta)$, and $B$ is the $B$-field.
The fields $H$, $A$, $B$ and $\Theta$ are constrained such that

$H=T(\varphi)~{},$ (69)

where $T(\varphi)$ is given in equation (64) and it is the totally
antisymmetric torsion of a (unique) connection $\nabla$ compatible with the
$G_{2}$ structure.

###  3.2 Decomposition of the $G_{2}$ structure equations with respect to the
ACMS

In section 2.3 we presented the decomposition of the structure equations of a
general $G_{2}$ structure under the ACMS. We do not find it necessary to write
these relations here as for an integrable $G_{2}$, all we need to do is to set
$\tau_{2}=0$.

###  3.3 Instanton conditions

We want to discuss how the instanton conditions are decomposed under the ACMS
on $Y$.

Let

$A=\sigma\,a_{0}+a~{},$ (70)

be the composition under the ACMS of the gauge connection $A$. Then, the
curvature $\cal F$ of the connection $A$ decomposes accordingly as

${\cal F}=\sigma\wedge{\cal F}_{0}+{\cal F}_{\perp}~{},$ (71)

where

$\displaystyle{\cal F}_{0}$ $\displaystyle=-{\rm
d}_{a}a_{0}+R(a+\Sigma\,a_{0})~{},$ (72) $\displaystyle{\cal F}_{\perp}$
$\displaystyle={\cal F}(a)+{\rm d}\Sigma\,\,a_{0}-\Sigma\wedge
R(a+\Sigma\,a_{0})~{}.$ (73)

It is not too difficult to show that the instanton condition ${\cal
F}\wedge\psi=0$ for the curvature $\cal F$ is equivalent to the constraints

$\displaystyle\omega\lrcorner{\cal F}_{\perp}$ $\displaystyle=0~{},$ (74)
$\displaystyle{\cal F}_{\perp}\lrcorner\Omega_{-}$ $\displaystyle={\cal
F}_{0}~{}.$ (75)

It is instructive to note that in terms of

$\hat{a}=a+\Sigma\,a_{0}~{},$ (76)

the components of the curvature are

${\cal F}_{0}=-{\rm d}_{\hat{a}}\,a_{0}+R(\hat{a})~{},\qquad\qquad{\cal
F}_{\perp}={\cal F}(\hat{a})-\Sigma\wedge{\cal F}_{0}~{},$ (77)

and the instanton conditions become

$\omega\lrcorner{\cal F}(\hat{a})=-{\cal F}_{0}\lrcorner
J(\Sigma)~{},\qquad\qquad{\cal F}(\hat{a})\lrcorner\,\Omega_{-}={\cal
F}_{0}+{\cal F}_{0}\lrcorner(\Sigma\lrcorner\,\Omega_{-})~{}.$ (78)

These conditions do not correspond to $\hat{a}$ instantons on the transverse
geometry unless, for example, ${\cal F}_{0}=0$.

The component ${\cal F}_{0}$ of the field strength $\cal F$ can be interpreted
as a trivial deformation of the gauge connection $A$ under the one parameter
group of diffeomorphisms generated by the vector $R$. To see this, recall
delaOssa:2017pqy 666In delaOssa:2017pqy  it was proven that trivial
deformations of the heterotic $G_{2}$ system are exact in the cohomology of a
nilpotent operator. We refer the reader to this reference for details. that a
trivial deformation of $A$ due to a gauge transformation with parameter
$\epsilon$ together with a diffeomorphism generated by a vector $V$ is given
by

$(\delta A)_{triv}={\rm d}_{A}\epsilon+i_{V}({\cal F})~{}.$

The second term gives the component ${\cal F}_{0}$ when $V=R$. In other words,
${\cal F}_{0}$ gives the variation of $\hat{a}$ along the integral curves of
$R$ up to a gauge transformation of $\hat{a}$ generated by $a_{0}$. Note that
if $R$ is a Killing vector then ${\cal F}_{0}=0$ and, in this case, $\hat{a}$
is an $SU(3)$ instanton. We will return to this case in section 3.6.

Similar considerations apply to the instanton connection $\Theta$ on the
tangent bundle of $Y$. In this case we decompose the connection $\Theta$ on
the tangent bundle of $Y$ under the ACMS as

$\Theta=\sigma\wedge\theta_{0}+\theta_{\perp}~{}.$

The curvature of this connection is thus decomposed as

${\cal R}(\Theta)=\sigma\wedge{\cal R}(\Theta)_{0}+{\cal
R}(\Theta)_{\perp}~{},$

where, in terms of $\hat{\theta}=\theta+\Sigma\,\theta_{0}\,$, the instanton
conditions become

$\omega\lrcorner{\cal R}(\hat{\theta})=-{\cal R}_{0}\lrcorner
J(\Sigma)~{},\qquad\qquad{\cal R}(\hat{\theta})\lrcorner\,\Omega_{-}={\cal
R}_{0}+{\cal R}_{0}\lrcorner(\Sigma\lrcorner\,\Omega_{-})~{}.$ (79)

Just as before, the component ${\cal R}_{0}$ of the curvature ${\cal
R}(\Theta)$ is interpreted as a trivial deformation of the instanton
connection $\Theta$ on $TY$ under the one parameter group of diffeomorphisms
generated by the vector $R$. When $R$ is a Killing vector, this component
vanishes and $\hat{\theta}$ is an instanton.

###  3.4 Anomaly cancellation condition

Let

$H=\sigma\wedge H_{0}+H_{\perp}~{},$ (80)

be the decomposition of the flux $H$ with respect to the ACS. Then, recalling
equations (67) and (68), the terms in the flux decomposition are given by

$\displaystyle H_{0}$ $\displaystyle=-{\rm
d}b_{0}+R(\hat{b})+\frac{\alpha^{\prime}}{4}\,\left({\rm tr}\big{(}a_{0}\,{\rm
d}\hat{a}-\hat{a}\wedge{\cal F}_{0}\big{)}-{\rm tr}\big{(}\theta_{0}\,{\rm
d}\hat{\theta}-\hat{\theta}\wedge{\cal R}_{0}\big{)}\right)~{},$ (81)
$\displaystyle H_{\perp}$ $\displaystyle={\rm
d}\hat{b}+\frac{\alpha^{\prime}}{4}\,\Big{(}{\cal CS}(\hat{a})-{\cal
CS}(\hat{\theta})\Big{)}-\Sigma\wedge H_{0}~{},$ (82)

where

$\hat{b}=B_{\perp}+\Sigma\wedge B_{0}~{},\qquad b_{0}=B_{0}~{}.$

We can interpret $H_{0}$ as a trivial deformation of the $B$ field due to the
diffeomorphism generated by $R$. In fact, up to an ambiguity of an exact two
form, a trivial deformation of the $B$-field due to a diffeomorphism generated
by the a vector $V$ and a gauge transformation with parameter $\epsilon$, is
given by delaOssa:2017pqy

$(\delta B)_{triv}=i_{V}\,(H)+\frac{\alpha^{\prime}}{2}\,{{\rm
tr}}(\epsilon{\cal F})~{}.$ (83)

When $V=R$, the first term is indeed $H_{0}$ as claimed. If $R$ is an
isometry, then $H_{0}$ vanishes up to an exact two form. We will see the
implications of this result in section 3.6 as well as in an example in section
5.1.

The anomaly cancellation condition is the requirement that the flux $H$ equals
the torsion $T(\varphi)$ of the connection with $G_{2}$ holonomy determined
uniquely by $\varphi$ (see equations (64) and (69)). Under the ACMS we have
then

$H_{0}=T_{0}({\varphi})~{},\qquad H_{\perp}=T_{\perp}(\varphi)~{}.$ (84)

where

$T=\sigma\wedge T_{0}+T_{\perp}~{},$ (85)

and

$\displaystyle T_{0}$
$\displaystyle=\frac{1}{6}\,\tau_{0}\,\omega-\tau_{1\,\perp}\lrcorner\
\Omega_{-}-\tau_{3\,0}~{},$ (86) $\displaystyle T_{\perp}$
$\displaystyle=\frac{1}{6}\,\tau_{0}\,\Omega_{+}+\tau_{1\,0}\,\Omega_{-}+J(\tau_{1\,\perp})\wedge\omega-\tau_{3\,\perp}~{}.$
(87)

One can write expressions for $T_{0}$ and $T_{\perp}$ in terms of the torsion
classes of $SU(3)$ structure induced by the ACMS using equations (58) with
$\tau_{2}=0$.

###  3.5 $N=1$ superpotential in terms of the ACMS

A compactification of the heterotic string on a heterotic $G_{2}$ system leads
to minimally supersymmetric ($N=1$) effective field theory on either $AdS_{3}$
or three dimensional Minkowski space time. It has been shown de_Ia_Ossa_2020
that, up to an overall constant, the superpotential $W$ of the $N=1$ effective
theory is given by

$W=\int_{Y}e^{-2\phi}\ \left((H+h\,\varphi)\wedge\psi-\frac{1}{2}\,{\rm
d}_{7}\varphi\wedge\varphi\right)~{},$ (88)

where $h$ is the three dimensional flux which is related to the curvature of
the three dimensional space-time and $\phi$ is the dilaton field. It was also
shown in de_Ia_Ossa_2020  that this superpotential is a functional of the
fields whose critical points give the conditions for preservation of $N=1$
supersymmetry in three dimensions, or equivalently, the conditions defining
the $G_{2}$ structure. Furthermore, it was shown that supersymmetry requires,
apart from the constraints described above, that $h$ be proportional to the
torsion class $\tau_{0}$, and that $\tau_{1}$ is ${\rm d}$-exact, specifically

$h=\frac{1}{3}\,\tau_{0}~{},\qquad\tau_{1}=\frac{1}{2}\,{\rm d}\phi~{}.$

As we have seen in section 2, the ACMS $(J,R,\sigma,g_{\varphi})$ on a
manifold $Y$ with a $G_{2}$ structure $\varphi$, implies that the $G_{2}$
structure can be decomposed in terms of an underlying transverse $SU(3)$
structure $(\omega_{\varphi},\Omega_{+})$. This is summarised in Proposition
1. In a similar way, we may use equations (34), (39), (53), and (80), to
decompose the various terms in the superpotential (88). A short computation
gives

$\begin{split}W&=\int_{Y}e^{-2\phi}\,\sigma\wedge{\rm
Im}\left(\left[\,\hat{H}+i\,{\rm d}_{\perp}\omega\right.\right.\\\\[5.0pt]
&\qquad\left.\left.+\frac{1}{8}\,\left(\omega\lrcorner(H_{0}-{\rm
d}_{\perp}\Sigma)+7\,h-\left(\Sigma\wedge
H_{0}+\frac{i}{2}\,R(\Omega_{+})\right)\lrcorner\bar{\Omega}\,\right)\,\bar{\Omega}\right]\wedge\Omega\right)~{},\end{split}$
(89)

where

$\hat{H}={\rm d}\hat{b}+\frac{\alpha^{\prime}}{4}\,\big{(}{\cal
CS}(\hat{a})-{\cal CS}(\hat{\theta})\big{)}~{}.$ (90)

This decomposition provides links between the heterotic $G_{2}$ system and the
six dimensional Strominger–Hull system Strominger:1986uh ; Hull:1986kz
.777Such links were discussed from a different perspective in delaOssa:2017gjq
. In particular, this is evident from the first two terms in the square
bracket in (89), where we recognize the $SU(3)$ superpotential of the latter
system.888The remaining terms of $W$ are related to the non-transverse
objects. This is consistent with the fact that the systems need not be related
by a circle reduction, and that the heterotic $G_{2}$ system need only
preserve half of the supersymmetry of the Strominger–Hull system. Indeed, in
contrast to Strominger–Hull system, we cannot identify a holomorphic
superpotential. As a further comment we note that, as is true for any four
dimensional $N=1$ theory, the Strominger–Hull superpotential captures the
F-term constraints, but not the D-terms. In contrast, for three dimensional
$N=1$ theories, there is no decomposition into F- and D-terms, and indeed the
$G_{2}$ superpotential captures also the D-term constraints of the
Strominger–Hull system.

###  3.6 Three dimensional theory with $N=2$

Consider now the following question: What are the constraints on the heterotic
$G_{2}$ system to have supersymmetry enhanced to $N=2$? This question was
already addressed in Gran:2005wf ; Gran:2007kh ; Gran:2016zxk . In this
section, we approach the question using an almost contact metric structure and
show that supersymmetry enhancement is equivalent to the existence of a
particular kind of ACMS on $Y$.

We know that any manifold $Y$ with a $G_{2}$ structure admits a covariantly
constant nowhere vanishing spinor $\eta$. Given the existence of a nowhere
vanishing vector $R$, the manifold $Y$ admits another nowhere vanishing spinor
$R\eta$ which is not, however, necessarily covariantly constant. In this
section we deduce the geometric conditions under which the spinor $R\eta$ is
covariantly constant too. As we will see in this section, this has interesting
applications, as for example, to the construction of three dimensional
theories with $N=2$ supersymmetry by constraining further the heterotic
$G_{2}$ system such that $R\eta$ is indeed covariantly constant. As we will
see, with two covariantly constant spinors at hand, the structure group of $Y$
reduces further to a certain type of $SU(3)$ structure.

The requirement that $\nabla(R\eta)=0$ is equivalent to the condition that $R$
is itself covariantly constant. Equivalently (as the connection $\nabla$ is
metric), we require that $\sigma$ is covariantly constant

$\nabla_{a}\sigma_{b}=0~{}.$ (91)

As we remarked earlier, we now have $\nabla\omega=0$ and $\nabla\Omega=0$ so
the holonomy of the $G_{2}$ compatible connection $\nabla$ is reduced to
$SU(3)$. Symmetrising this equation with respect to the indices $a,b$, we
obtain that $R$ must be a Killing vector. When $R$ is a Killing vector,
$\varphi$ becomes independent of the coordinate $r$ and hence, the transverse
forms $\omega$, $\Omega$, and $\Sigma$ are also independent of $r$. The
antisymmetric part of equation (91) gives a further constraint

${\rm d}_{7}\sigma=i_{T}(\sigma)=T_{0}~{}.$ (92)

The authors of Gran:2005wf ; Gran:2007kh  obtain precisely condition (91) as
part of the requirements to obtain supersymmetry preserving backgrounds. In
fact, a ten dimensional backgound on $AdS_{3}\times Y$, where $Y$ is a
manifold with a $G_{2}$ structure, demands the existence of an extra
covarianly constant one form to have an extra supersymmetry and hence reducing
the the holonomy of the $G_{2}$ connection to $SU(3)$.

The relations between the $G_{2}$ torsion classes and the induced $SU(3)$
torsion classes given in equations (93) greatly simplify because we now have
to set $\tau_{2}=0$ and, $R$ being a Killing vector, means that $R(\Sigma)=0$,
$R(\omega)=0$ and $R(\Omega)=0$. We then find

$\begin{split}{\rm
Re}\,W_{0}&=\frac{2}{3}\,\tau_{0}=-\frac{4}{3}\,\omega\lrcorner\,{\rm
d}\Sigma~{},\qquad\qquad{\rm Im}\,W_{0}=2\,\tau_{1\,0}~{},\\\\[5.0pt]
2\,W_{1}&=4\,\tau_{1\,\perp}+{\rm
d}\Sigma\lrcorner\,\Omega_{-}~{},\qquad\qquad 2\,{\rm
Re}\,\theta=4\,\tau_{1\,\perp}~{},\\\\[5.0pt] {\rm
Re}\,W_{2}&=-\tau_{3\,0}^{(1,1)}-{\cal P}\big{(}{\rm
d}\Sigma\big{)}^{(1,1)}~{},\qquad\qquad{\rm
Im}\,W_{2}=-\tau_{2\,\perp}^{(1,1)}~{},\\\\[5.0pt] W_{3}&={\cal
P}\big{(}J(\tau_{3\,\perp})\big{)}~{},\\\\[5.0pt] \big{(}{\rm
d}\Sigma\big{)}^{(2,0)+(0,2)}&=\tau_{3\,0}{}^{(2,0)+(0,2)}+\tau_{1\,\perp}\lrcorner\Omega_{-}~{}.\end{split}$
(93)

As discussed above, a Killing vector is necessary but not sufficient for an
enhancement of the supersymmetry, we need furthermore to impose the condition
(92). Using (93) in (86), the condition (92) becomes

${\rm d}\Sigma=T_{0}=-\frac{1}{3}\,(\omega\lrcorner{\rm d}\Sigma)\,\omega+{\rm
Re}W_{2}+{\cal P}({\rm d}\Sigma)^{(1,1)}-({\rm d}\Sigma)^{(2,0)+(0,2)}~{}.$
(94)

Therefore ${\rm d}\Sigma$ must be a primitive $(1,1)$ form and ${\rm
Re}W_{2}=0$. Moreover, the fact that $\omega\lrcorner{\rm d}\Sigma=0$ means
that ${\rm Re}W_{0}=0$, that is $\tau_{0}=0$. For the heterotic string we need
to add the constraint that $2\,\tau_{1}={\rm d}_{7}\phi$. This implies that

$\tau_{1\,0}=0~{},\qquad 2\,\tau_{1\,\perp}={\rm d}\phi~{}.$ (95)

Therefore

${\rm Im}\,W_{0}=0~{}.$ (96)

In summary, the torsion classes of the $SU(3)$ on the transverse geometry
satisfy

$\displaystyle W_{0}$ $\displaystyle=0~{},\qquad W_{2}=0~{},$ $\displaystyle
W_{1}$ $\displaystyle={\rm Re}\,\theta={\rm
d}\phi=-\tau_{3\,0}\lrcorner\,\Omega_{-}~{},\qquad$ $\displaystyle W_{3}$
$\displaystyle={\cal P}\big{(}J(\tau_{3\,\perp})\big{)}~{},$

and we also have

$\tau_{0}=0~{},\qquad{\rm d}\Sigma=-\tau_{3\,0}^{(1,1)}~{},$

where $\tau_{3\,0}$ is primitive.

As an example, we note that, if the manifold $Y$ has $G_{2}$ holonomy, then
the transverse bundle ${\rm Ker}(\sigma)$ is necessarily integrable, so $Y$ is
a codimenion one foliation with leaves which are Calabi–Yau three folds.

More generally, the vanishing of $W_{0}$ and $W_{2}$ means that the almost
complex structure $J$ on the transverse geometry is integrable, and the fact
that $W_{1}$ (and hence ${\rm Re}\,\theta$) are exact imply that the
transverse geometry is conformally balanced. Therefore the transverse geometry
has the $SU(3)$ structure of the Strominger-Hull system. Moreover, the fact
that ${\rm d}\Sigma$ is a primitive $(1,1)$ form means that the manifold $Y$
has the structure of a $U(1)$ principal bundle with a holomorphic connection
$\Sigma$ over the transverse geometry. Finally, we note that the vanishing of
$\tau_{0}$ implies that the three dimensional spacetime is Minkowski space.

Consider now the instanton conditions. Following the discussion at the end of
section 3.3, we recall that ${\cal F}_{0}$ represents the change of the
connection $A$ with respect to the vector $R$. As $R$ is, in our case, a
Killing vector, we have ${\cal F}_{0}=0$ and it follows that $\hat{a}$ is a
holomorphic instanton on the transverse geometry. Similarly, ${\cal R}_{0}=0$
and $\hat{\theta}$ is also an instanton.

For the anomaly cancellation condition, we earlier saw that if $R$ is a
Killing vector, then $H_{0}$ must be an exact two form, a conclusion that was
arrived at by studying the symmetries of the heterotic system. Interestingly,
the condition that $H_{0}$ be an exact form is precisely the content of
equation (94), which fixes this exact form to be

$H_{0}=T_{0}={\rm d}\Sigma~{},$

where we recall that the second equality comes from the fact that $R$ is not
just a Killing vector, but is also covariantly constant. For completeness we
note that the transverse part of the flux becomes

$H_{\perp}=-\Sigma\wedge{\rm d}\Sigma+\hat{H}~{},$

where $\hat{H}$ is defined in equation (90). Note, furthermore, that the
Bianchi identity of the anomaly cancellation condition becomes

${\rm d}H_{\perp}=-{\rm d}\Sigma\wedge{\rm
d}\Sigma+\frac{\alpha^{\prime}}{4}\Big{(}{\rm tr}({\cal F}(\hat{a})\wedge{\cal
F}(\hat{a}))-{\rm tr}({\cal R}(\hat{\theta})\wedge{\cal
R}(\hat{\theta}))\Big{)}~{}.$ (97)

We remark that it has not been necessary to require the integrability of ${\rm
Ker}(\sigma)$. That is, it is not necessary to require that ${\rm d}\Sigma=0$
to have $N=2$ in three dimensions, as one might have expected.

##  4 $SU(2)$ structures on manifolds with a $G_{2}$ structure

The $SU(3)$ structure discussed in the previous section is not the only
“bonus” restriction on the topology of $G_{2}$ structure manifolds. Indeed,
there are _two_ non-vanishing vector fields on a manifold $Y$ with a $G_{2}$
structure 10.2307/1970439 ; thomas1969 , which may be combined with the
$G_{2}$ compatible spinor to form two additional nowhere-vanishing spinors.
These three spinors reduce the structure group of the manifold to $SU(2)$
friedrich1997nearly . Moreover, the existence of two well-defined vectors
implies that there is an almost contact metric 3-structure (ACM3S) on $Y$
(that is a reduction of the structure group to ${\bf 1}_{3}\times Sp(1)$)
kuo1970  (see also Todd:2015era ). In this section, we will expand on this
topic, and show how the $Sp(1)\cong SU(2)$ structure is embedded in the
$G_{2}$ structure. As in the preceding section, we will describe the ACM3S in
terms of differential forms. We will also clarify when the ACM3S leads to a
involutive decomposition of the tangent bundle $TY$, and expand on the
relation to associative and coassociative submanifolds. Finally, we will study
the space of ACM3S.

###  4.1 Almost contact 3-structure

It is a classical result by E. Thomas, that any compact, orientable
7-dimensional manifold $Y$ admits two globally defined, everywhere linearly
independent vector fields $R^{1},R^{2}\in\Gamma(TY)$ 10.2307/1970439 ;
thomas1969 . Suppose now that $Y$ has $G_{2}$ structure $\varphi$ with metric
$g_{\varphi}$. We may then assume, without loss of generality, that the
2-frame $(R^{1},R^{2})$ consists of vectors that are orthonormal. Indeed,
given two non-orthogonal, but linearly independent, vectors $(R^{1},R^{2})$,
we can form two orthogonal ones using the $G_{2}$ cross product:
$(R^{1},R^{1}\times_{\varphi}R^{2})$. We can also normalise the vectors by
dividing by their norms as calculated with the $G_{2}$ metric. Thus any
$G_{2}$ manifolds allow orthonormal 2-frames of vectors.

Each vector $R^{\alpha}$ will be associated to an ACMS, in the sense discussed
in section 2.1. We thus have Todd:2015era  (see also Arikan:2012acs ;
Arikan:2011acs ) two dual one-forms $\sigma^{\alpha}$ so that
$(J^{\alpha},R^{\alpha},\sigma^{\alpha},g_{\varphi})$, for $\alpha=1,2$, are
two ACMS on $Y$. If the structures furthermore satisfy the relations

$\begin{split}&\sigma^{1}(R^{2})=\sigma^{2}(R^{1})=0\\\
&J^{1}(R^{2})=-J^{2}(R^{1})\\\ &\sigma^{1}\circ J^{2}=-\sigma^{2}\circ
J^{1}\\\
&J^{1}J^{2}-R^{1}\otimes\sigma^{2}=-J^{2}J^{1}+R^{2}\otimes\sigma^{1}\;,\end{split}$
(98)

Kuo kuo1970  has shown that $Y$ admits a third ACMS given by

$J^{3}=J^{1}J^{2}-R^{1}\otimes\sigma^{2}\;,\;R^{3}=J^{1}(R^{2})\;,\;\sigma^{3}=\sigma^{1}\circ
J^{2}\;.$ (99)

Together, these three ACMS then satisfy (10), and hence define an almost
contact 3-structure (AC3S) kuo1970 . In fact, one can show that, for two ACS
associated to the same metric, the last constraint in (98) implies the other
three constraints kuo1970 .

From the definition of an AC3S, we have the following useful identities

$\sigma^{\alpha}(R^{\beta})=\delta^{\alpha\beta}\;\;,\;\;J^{\alpha}(R^{\beta})=\epsilon^{\alpha\beta\gamma}R^{\gamma}\;\;,\;\;\sigma^{\alpha}\circ
J^{\beta}=-\sigma^{\beta}\circ J^{\alpha}\;.$ (100)

In addition,

$R^{3\,a}=[J^{1}(R^{2})]^{a}=\varphi^{a}{}_{bc}R^{1\,b}R^{2\,c}\;\mbox{ i.e.
}\;R^{3}=R^{1}\times_{\varphi}R^{2}$ (101)

or, equivalently,

$\sigma^{\gamma}=\frac{1}{2}\epsilon^{\alpha\beta\gamma}i_{R^{\beta}}i_{R^{\alpha}}\varphi\;,$
(102)

which will be used below when we discuss the $SU(2)$ decomposition of the
$G_{2}$ structure.

Finally, it was recently proven by Todd Todd:2015era  that on any $G_{2}$
structure manifold, two ACMS’s
$(J^{\alpha},R^{\alpha},\sigma^{\alpha},g_{\varphi})$ will automatically
satisfy the last constraint of (98). Thus, we have

######  Theorem 1 (Todd, Todd:2015era ).

Let $(Y,\varphi)$ be a compact and boundary-less 7-manifold with $G_{2}$
structure $\varphi$. Then $Y$ admits an almost contact metric 3-structure
which is compatible with the G2-metric.

Note that the three ACMS have different contact forms $\sigma^{\alpha}$, but
the fact that they are associated to the same $G_{2}$ metric,

$g_{\varphi}(J^{\alpha}u,J^{\alpha}v)=g(u,v)-\sigma^{\alpha}(u)\sigma^{\alpha}(v)\;\;,\;\;{\alpha}=1,2,3\;\;{\rm(no\;sum)}\;,$
(103)

implies the resulting almost contact 3-structure is indeed an almost contact
metric 3-structure (ACM3S).

We observe, in particular, that the ACM3S involves fixing a distinguished
orthonormal three-frame, $(R^{1},R^{2},R^{3})$, which induces a decomposition
of the tangent bundle

$TY\cong\mathcal{T}\oplus\mathcal{T}^{\perp},$ (104)

where $\mathcal{T}$ is trivial rank three bundle with a distinguished
trivialisation induced by the three-frame $(R^{1},R^{2},R^{3})$. The second
factor, $\mathcal{T}^{\perp}$, is then the orthogonal complement. With this
data, we are able to identify $\mathcal{T}$ as a trivial bundle of imaginary
quaternions, with a product induced by the $G_{2}$ cross product. This follows
immediately from (101).

The choice of ACM3S therefore reduces our structure group
$G_{2}\rightarrow{\bf 1}_{3}\times H,$ for some $H\subset G_{2}$. In fact,
results of Kuo, kuo1970 , show that $H=Sp(1)\cong SU(2)$. One way to see this,
from the $G_{2}$ perspective, is to observe that the subgroup of $G_{2}$ that
preserves three orthonormal vectors is, indeed, $SU(2)$, from which the result
follows. Note, in particular, that the rank four bundle, $\mathcal{T}^{\perp}$
has reduced structure group $SO(4)\rightarrow SU(2)$. A reduction of structure
group leads to distinguished differential forms living in irreducible
representations of the reduced group and we will explicitly exhibit the forms
induced by the reduction of $\mathcal{T}^{\perp}$’s structure group. These
structure forms will be familiar to readers that have studied four dimensional
$SU(2)$ structure manifolds, but it is important to bear in mind that
$\mathcal{T}^{\perp}$ need not be tangent to any underlying four
manifold.999The conditions that $\mathcal{T}^{\perp}$ must satisfy in order
for such manifolds to exist are reviewed in section 4.2. This decomposition is
purely at the level of the bundle.

The splitting of the tangent bundle, (104) induces an analogous decomposition
of the cotangent bundle

$T^{*}Y=\mathcal{T}^{*}\oplus\mathcal{T}^{*\,\perp}\equiv{\rm
Span}\\{\sigma^{1},\sigma^{2},\sigma^{3}\\}\oplus{\rm
Span}\\{\sigma^{1},\sigma^{2},\sigma^{3}\\}^{\perp}\;,$ (105)

and, consequently, a decomposition of the differential forms. In particular,
we will refer to a $k$-form, $\lambda$, as being $\alpha$-transverse if it
satisfies

$i_{R^{\alpha}}(\lambda)=0\;.$ (106)

More generally, an arbitrary form $k$-form, $\lambda$, can be decomposed as

$\lambda=\sum_{\alpha}\sigma^{\alpha}\wedge\lambda_{0}^{\alpha}+\lambda_{\perp}~{},$
(107)

where

$i_{R^{\alpha}}(\lambda)=\lambda_{0}^{\alpha}~{},\qquad
i_{R^{\alpha}}(\lambda_{0}^{\alpha})=0~{},\quad\mbox{ and,
}\;i_{R^{\alpha}}(\lambda_{\perp})=0\,,\;\forall\alpha~{}.$

Recall that each ACS has an associated fundamental two-form, (25),

$\omega^{\alpha}=i_{R^{\alpha}}(\varphi)\,.$ (108)

Evidently, $\omega^{1}$ is a $1$-transverse two-form, but it is neither $2$\-
nor $3$-transverse; as a consequence it is not in
$\Gamma(\Lambda^{2}\mathcal{T}^{*\perp})$, but is instead a linear combination

$\omega^{\alpha}=\sum_{\beta\neq\alpha}\sigma^{\beta}\wedge\omega_{0\,\beta}^{\alpha}+\omega^{\alpha}_{\perp}\;.$
(109)

where we recall that $\omega^{\alpha}_{\perp}$ is, in fact, transverse with
respect to all three ACMS. Moreover, (102) implies that e.g.

$\sigma^{3}=i_{R^{2}}\omega^{1}=-i_{R^{1}}\omega^{2}$ (110)

and we thus have that $\omega_{0\,1}^{2}=-\sigma^{3}$, or, in general,

$\omega^{\alpha}=\frac{1}{2}\epsilon^{\alpha\beta\gamma}\sigma^{\beta}\wedge\sigma^{\gamma}+\omega^{\alpha}_{\perp}\;.$
(111)

Next, we decompose the $G_{2}$ structure form, $\varphi$ with respect to the
ACM3S. Using that the trivial bundle, $\mathcal{T}\subset TY$, can be
interpreted, fibrewise, as the imaginary quaternions sitting inside the
imaginary octonions, we can quickly deduce that $\varphi_{\perp}=0$. This is
because the octonionic product of any two elements in
$\operatorname{Im}(\mathbb{H})^{\perp}\subset\operatorname{Im}(\mathbb{O})$
will necessary land back in $\operatorname{Im}(\mathbb{H})$, while
$\varphi_{\perp}$ projects this product back into the complement
$\operatorname{Im}(\mathbb{H})^{\perp}$.

More concretely, we can consider a local frame, $e^{i}$, such that
$e^{4+\alpha}:=R^{\alpha}$, and $e^{1},e^{2},e^{3},e^{4}$ are chosen such that
$\varphi$ takes the standard form

$\varphi_{0}=(e^{12}+e^{34}+e^{56})\wedge
e^{7}+e^{135}-e^{146}-e^{236}-e^{245}\,.$ (112)

In such a local frame, $\varphi_{\perp}$ will be those terms of $\varphi_{0}$
in which none of $e^{5},e^{6},e^{7}$ appear and one observes that there are no
such terms.

Furthermore, comparing the terms that appear in (112) with the expansion
(111), we can explicitly expand $\varphi$:

$\varphi=\frac{1}{3!}\epsilon_{\alpha\beta\gamma}\sigma^{\alpha}\wedge\sigma^{\beta}\wedge\sigma^{\gamma}+\sum_{\alpha}\sigma^{\alpha}\wedge\omega^{\alpha}_{\perp}\;,$
(113)

The expansion of $\psi$ can be straightforwardly computed, using either a
local frame and its standard form, or applying the Hodge star. Either way, one
obtains

$\psi={\rm d}{\rm
vol}_{\perp}+\frac{1}{2}\epsilon_{\alpha\beta\gamma}\sigma^{\alpha}\wedge\sigma^{\beta}\wedge\omega^{\gamma}_{\perp}\;.$
(114)

Note that we use ${\rm d}{\rm vol}_{\perp}$ to refer to the canonical section
of $\Lambda^{4}(\mathcal{T}^{\perp,*})$, although there may not be a four
dimensional manifold, even locally, for which ${\rm d}{\rm vol}_{\perp}$ is a
volume form. Whenever $\mathcal{T}^{\perp}$ is integrable (see the next
subsection), then ${\rm d}{\rm vol}_{\perp}$ is indeed the volume form for the
leaves of the corresponding foliation.

In the preceding, we have come across a triple of real two-forms
$\omega^{\alpha}_{\perp}$, $\alpha=1,2,3$. These characterise the reduced
$SU(2)$ structure of the rank four bundle, $\mathcal{T}^{\perp}$. This will be
familiar to readers comfortable with $SU(2)$ structures in dimension four and
we briefly review this setting in Appendix B. To be sure that these really are
the correct differential forms, we must check that

$\omega^{\alpha}_{\perp}\wedge\omega^{\beta}_{\perp}=2\delta^{\alpha\beta}{\rm
d}{\rm vol}_{\perp}\;.$ (115)

That this holds follows directly from the decompositions (114) (113). Indeed,
on the one hand, we have

$0\neq 7{\rm d}{\rm
vol}_{\varphi}=\varphi\wedge\psi=\sigma^{1}\wedge\sigma^{2}\wedge\sigma^{3}\wedge({\rm
d}{\rm vol}_{\perp}+3\omega^{\alpha}_{\perp}\wedge\omega^{\alpha}_{\perp})\;,$
(116)

showing that $\omega^{\alpha}_{\perp}\wedge\omega^{\alpha}_{\perp}=2{\rm
d}{\rm vol}_{\perp}$. On the other hand, we have

$0=\varphi\wedge\varphi\sim\sigma^{\alpha}\wedge\sigma^{\beta}\wedge\omega^{\alpha}_{\perp}\wedge\omega^{\beta}_{\perp}$
(117)

so $\omega^{\alpha}_{\perp}\wedge\omega^{\beta}_{\perp}$ must vanish when
$\alpha\neq\beta$.

We now return to the role that the unit vector fields, $R^{\alpha}$ played in
the above computations. In particular, it is important that fixing the
splitting (104) does not fix the ACM3S. The extra data required is a global
identification of $\mathcal{T}$ with the imaginary quaternions,
$\operatorname{Im}(\mathbb{H})$, along with the multiplication induced by
$\varphi$. This is equivalent to a choice of three orthonormal vector fields,
satisfying (101). In particular, after a choice of splitting, (104), there
remains an interesting space of compatible almost contact metric 3-structures,
which we study in Subsection 4.3. To the authors’ knowledge, this has not been
explored in the literature and we provide some initial results below.

In contrast, a decomposition
$TY\cong\mathcal{T}_{1}\oplus\mathcal{T}_{1}^{\perp}$ into a trivial one-
dimensional bundle, $\mathcal{T}_{1}$, and its orthogonal complement,
$\mathcal{T}^{\perp}_{1}$, has a much less interesting space of almost contact
structures. Indeed, recall from Section 2 that the data needed for an ACMS on
a $G_{2}$ structure manifold is precisely that of a unit vector
field.101010Just as an ACM3S allows us to identify $\mathcal{T}$ with the
imaginary quaternions, an ACMS allows us to identify $\mathcal{T}_{1}$ with
the imaginary complex numbers. The rank one bundle, $\mathcal{T}_{1}$ admits
precisely two such vector fields, say $R$ and $-R$. Choosing either, say $R$
for concreteness, induces an ACMS such that $\mathcal{T}_{1}={\rm Span}(R)$
and $\mathcal{T}_{1}={\rm Ker}\sigma$.

To further contrast with the ACMSs of Section 2, the trivial subbundle,
$\mathcal{T}$, is no longer guaranteed to be tangent to a foliation, i.e. it
need not be an integrable distribution. As a consequence, we can not generally
expect to find adapted coordinates in the same vein as (13). In other words,
we can not expect there to be even a local product structure of the geometry.
We turn to this question now.

###  4.2 Integrability

In this subsection we will investigate the conditions for the distributions
$\mathcal{T}$ and $\mathcal{T}^{\perp}$ to be tangent to a foliation of the
seven manifold, $Y$. We will begin with the trivial bundle, $\mathcal{T}$. It
is a standard result in the study of foliations that this is true if and only
if $\mathcal{T}$ is involutive, for which it suffices that the Lie bracket of
the distinguished vector fields, $R^{\alpha}$, closes. That is, there are real
functions $f^{\alpha\beta}_{\;\;\;\gamma}$,
$\alpha,\beta,\gamma\in\\{1,2,3\\}$ such that

$[R^{\alpha},R^{\beta}]_{x}=f^{\alpha\beta}_{\;\;\;\gamma}(x)R^{\gamma}_{x}\quad\;\forall\,x\in
Y\,.$ (118)

Observe that the analagous condition on the rank one bundle induced by an
ACMS, ${\rm Span}(R)$, is trivially satisfied since $[R,R]=0$. This is a
reflection of the well-known fact that vector fields always admit integrable
curves or, to put it another way, that ordinary differential equations always
have locally unique solutions. The possible failure of integrability of a
higher rank foliation is related to the comparative subtlety of partial
differential equations.

Note also that (118) is independent of the distinguished vector fields,
$R^{\alpha}$, which can be checked by straightforward computation. In
particular, integrability is a property of the subbundle, not of its
trivialisation.

Condition (118) can be equivalently formulated in terms of the kernel
$\mathcal{T}^{*,\perp}$. Recalling that a one-form, $\xi$ is in
$\Gamma(\mathcal{T}^{*,\perp})$ if and only if $i_{R^{\alpha}}\xi=0$ for each
$\alpha=1,2,3$, then, $\mathcal{T}$ is involutive if and only if ${\rm
d}(\mathcal{T}^{*,\perp})\subset\mathcal{T}^{*,\perp}\otimes\Omega^{1}(Y)$,
i.e. that $i_{R^{\alpha}}i_{R^{\beta}}{\rm d}\xi=0$ for all
$\xi\in\Gamma(\mathcal{T}^{*,\perp})$.

We can now review the conditions under which $\mathcal{T}^{\perp}$ is
integrable. Since the presentation of $\mathcal{T}^{\perp}$ is not as
convenient as that of $\mathcal{T}$, it is the second of the above
characterisations that is most convenient, viz. $\mathcal{T}^{\perp}$ is
integrable if and only ${\rm
d}(\mathcal{T}^{*})\subset\mathcal{T}^{*}\otimes\Omega^{1}(Y)$. Using that
$\mathcal{T}^{*}={\rm Span}\\{\sigma^{1},\sigma^{2},\sigma^{3}\\}$, we
conclude that $\mathcal{T}^{\perp}$ is involutive if and only if ${\rm
d}\sigma^{\alpha}=\sigma^{\beta}\wedge\mu_{\beta}^{\alpha}$, for one-forms
$\mu_{\beta}^{\alpha}$. This is the ACM3S analogue of the condition (44) for
ACMS’s.

We can now ask about the relation between integrability of $\mathcal{T}$,
respectively $\mathcal{T}^{\perp}$, and the $G_{2}$ structure on $Y$. Let us
begin by assuming that $\mathcal{T}$ is integrable, so that $Y$ has a three
dimensional foliation with leaves, say $\mathcal{L}_{x}$, such that
$T_{y}\mathcal{L}_{x}=\mathcal{T}_{x}$. Here, $x$ is any point on the leaf, so
there is a huge degeneracy in this labelling and, unless it will lead to
confusion, we will omit recording this point. We observe that, by
construction, $\mathcal{L}$ has a volume form which is, at each point
$y\in\mathcal{L}_{x}$, the wedge of the dual one-forms ${\rm d}{\rm
vol}_{\mathcal{L},y}=(\sigma^{1}\wedge\sigma^{2}\wedge\sigma^{3})_{y}$.
Further, this wedge product is nothing but the $G_{2}$ structure three-form,
$\varphi$, restricted to $T_{y}\mathcal{L}$, as can be easily seen by choosing
a local frame extending $\sigma^{\alpha}$, cf. (113). This is precisely the
condition that $\mathcal{L}$ be an associative manifold, harvey1982calibrated
; mclean1998deformations .

We have just shown that $\mathcal{T}$ is integrable if and only if $Y$ is
foliated by associative submanifolds. In the case that our $G_{2}$ structure
is closed, ${\rm d}\varphi=0$, then $\varphi$ is a calibration and the
associative submanifolds are the corresponding calibrated cycles, which
minimize volume within their homology class. In string theoretic
compactification scenarios, calibrated cycles contribute to non-perturbative
effects in the effective field theory and it is a long-standing open problem
to properly account for these contributions, see Harvey:1999as ;
braun2018infinitely ; Acharya:2018nbo ; Braun:2017uku  for instance, or
Eckhard:2018raj  for a brane world-sheet perspective. For more general $G_{2}$
structures, in particular for non-closed $G_{2}$ structures, the positive
three-form $\varphi$ is longer a calibration, the volume-minimizing cycles and
corresponding non-perturbative effects are, in general, less understood
Joyce:2016fij . The surprising link between three-structures and associatives
may offer a tool to be applied to both of these problems and we comment on
this possibility in the conclusions, Section 6.

At any rate, the condition that $\mathcal{T}$ be integrable is clearly a
strong one. As a final comment on this property, we remark that the leaf,
$\mathcal{L}$, is evidently parallelisable and, once we fix the ACM3S, is
equipped with a canonical trivialisation. In three dimensions, every oriented
manifold is parallelisable, so this does not add extra conditions on the
topology of the leaf.

The case for $\mathcal{T}^{\perp}$ is similar. Were $\mathcal{T}^{\perp}$ to
be integrable, $Y$ would have a four-dimensional foliation, whose leaves will
be denoted $\mathcal{L}^{\perp}$,
$T_{x}\mathcal{L}^{\perp}=\mathcal{T}^{\perp}_{x}$. Choosing a local
orthonormal frame extending $(R^{1},R^{2},R^{3})$, say with
$e^{1},e^{2},e^{3},e^{4}$ then we have that in this frame the volume of
$\mathcal{L}^{\perp}$ will be given by ${\rm d}{\rm
vol}_{x}(\mathcal{L}^{\perp})=e^{1}\wedge e^{2}\wedge e^{3}\wedge e^{4}$,
which is precisely the restriction of the coassociative four form.
Submanifolds of a $G_{2}$ structure manifold with volume form given by the
restriction of the $G_{2}$ structure four form, $\psi$, are called
coassociative submanifolds. When $\psi$ is closed, it is a calibration of the
seven manifold and coassociative submanifolds are the corresponding calibrated
manifolds. Therefore, much of what we said on the interest in associative
three-cycles can be said, mutatis mutandis, for coassociatives.

It is interesting to note that the setting where $\mathcal{T}^{\perp}$ is
integrable is reminiscent of the study of $G_{2}$ manifolds fibred by
coassociative cycles which has been promoted by Donaldson 2016arXiv160308391D
, Baraglia 2010JGP….60.1903B  and Kovalev 2005math…..11150K , among others.
K3-fibrations are also relevant in M-theory/heterotic duality and have been
utilised in defining a conjectural generalisation of mirror symmetry to the
seven dimensional case gukov2003duality ; braun2018towards ; braun2017mirror .
It would be interesting to explore how these topics interact with the ACM3Ss.

###  4.3 Space of ACM3S

We now return to the study of the space of ACM3 structures compatible with a
given $G_{2}$ structure on a seven manifold, $(Y,\varphi)$. We will denote
this space by $\mathscr{C}$, or $\mathscr{C}(Y,\varphi)$ if context makes it
necessary to emphasise the $G_{2}$ structure manifold. This is an interesting
space in its own right, but it is also necessary to understand from a physics
perspective, since it is far from clear that different choices of ACM3S would
yield equivalent effective field theory descriptions of physics. In other
words, there may be certain choices of ACM3S that provide better descriptions
of the low-energy physics. At the moment, the precise meaning of these choices
for physics are unclear and we will simply give a mathematical description of
this space. It seems likely that physics considerations will refine this study
and we will comment on these possibilities in the conclusion, see section 6.
Mathematically, this space, $\mathscr{C}$, can be seen as a very coarse
invariant of the $G_{2}$ structure, essentially depending only on the
topological class of the associated $G_{2}$ bundle.

One indication of this interdependence is the way parallelisability is
reflected by ACM3Ss. This connection comes about as follows: consider any two
splittings $\mathcal{T}\oplus\mathcal{T}^{\perp}\cong
TY\cong\mathcal{T}^{\prime}\oplus\mathcal{T}^{\prime,\perp}$. Observe that the
rank of the sum $\mathcal{T}+\mathcal{T}^{\prime}$ must have at least one
fibre (and therefore, also all fibres in an open neighbourhood) with dimension
greater than three in order that the splittings be different. On the other
hand, if _all_ the fibres have rank greater than three, then the manifold is
in fact parallelizable. Indeed, since we assume
$\mathcal{T}+\mathcal{T}^{\prime}$ is everywhere at least rank 4, then we can
conclude that there is at least four orthonormal vector fields, say
$R^{\mu},\,\mu=1,2,3,4$. Regarding these as unit, imaginary octonions with the
$G_{2}$ cross product, we then have the basic fact that any four orthogonal,
imaginary octonions generate the space of all imaginary octonions. Turning
this statement around: whenever the underlying $G_{2}$ structure manifold is
_not_ parallelizable, any two splittings will overlap in at least one point.

We will now give a concrete expression for the space of all ACM3Ss induced by
the $G_{2}$ structure. We have seen that these are in one-to-one
correspondence with orthonormal three-frames satisfying (102), i.e. that
$R^{3}=R^{1}\times_{\varphi}R^{2}$. Since this implies that the third
orthonormal vector is completely fixed by the first two, we conclude that
$\mathscr{C}$ is simply the space of all orthonormal, ordered pairs of vector
fields. Fibrewise, the space of orthonormal pairs of vectors inside
$T_{x}Y\cong\mathbb{R}^{7}$ is a so-called Stiefel manifold which we can
denote $V_{2}(T_{x}Y)$. This space has a description as a homogeneous space
$G_{2}/SU(2)$ harvey1982calibrated , expressing the fact that $G_{2}$ acts
transitively on orthonomal pairs of vectors, with stabiliser $SU(2)$.
Globally, there is a fibre bundle associated to the tangent bundle with
typical fibre $V_{2}(\mathbb{R}^{7})$. We will denote this bundle by
$\mathcal{V}_{2}(TY)$. A section of $\mathcal{V}_{2}(TM)$ is the same as an
orthonormal two-frame or, equivalent in our situation, an ACM3S, and therefore
the space of ACM3S is the space of sections of $\mathcal{V}_{2}(TY)$,
$\mathscr{C}=\Gamma(Y,\mathcal{V}_{2}(TY))$.111111Thomas’ proof that any
$G_{2}$ structure manifold admits a 2-vector field, 10.2307/1970439 ;
thomas1969  used obstruction theory to show this space of sections is non-
empty. This space may have non-trivial topology, including non-trivial
homotopy groups, which we will investigate in some examples, see Section 5.

This space of sections, $\mathscr{C}$, has a locally trivial fibre bundle
structure, where the base corresponds to the space of splittings of the form
(104) and the fibres are the trivialisations of this bundle. The projection
map is simply taking the span fibrewise. We will now make this more concrete.

The space of decompositions, $TY\cong\mathcal{T}\oplus\mathcal{T}^{\perp}$,
with $\mathcal{T}$ a trivial bundle with associative fibres is, fibrewise,
equivalent to choosing a three-plane in the seven-dimensional tangent space
with a further constraint that enables it to be regarded as a copy of the
imaginary quaternions. The space of such choices at a given point is the so-
called associative Grassmanian, $G(\varphi_{x})$, and is, like
$V_{2}(T_{x}Y)$, a homogeneous space with $G_{2}$ total space,
$G(\varphi_{x})=G_{2}/SO(4)$, harvey1982calibrated . Once again, we can
consider a fibre bundle associated to $TY$, now with typical fibre
$G(\varphi_{0})$. We will call this fibre bundle
$\mathcal{G}(\varphi)\rightarrow Y$ and let
$\tilde{\mathscr{S}}:=\Gamma(Y,\mathcal{G}(\varphi))$ be the space of
sections. By construction, a section of this bundle corresponds to a rank
three subbundle of $TY$ with associative fibres, and thus a splitting
$TY\cong\mathcal{T}\oplus\mathcal{T}^{\perp}$, but it is not obvious that the
bundle $\mathcal{T}$ must be trivial. Thus, $\tilde{\mathscr{S}}$ is not
precisely the space we need, it is too big. The subspace of trivial bundles,
however, consists of a union of path-connected components of
$\tilde{\mathscr{S}}$. Indeed, the bundle associated to a section in the same
path component as that of a trivial bundle is, by definition, homotopic to the
trivial bundle and consequentially isomorphic. Therefore, the relevant
subspace of all sections will be a disjoint union of certain disconnected
components of the space of sections, which we will denote by
$\mathscr{S}\subset\tilde{\mathscr{S}}$.

We can now look at the fibres of the projection
$\mathscr{C}\rightarrow\mathscr{S}$, which consists of the the space of
orthonormal, associative trivialisations of the corresponding trivial rank
three bundle. Since we assume this is not an empty space, we can fix an
initial ACM3S by making an arbitrary choice of orthonormal vector fields,
$(R^{1},R^{2},R^{3})$ satisfying (101), i.e.
$R^{3}=R^{1}\times_{\varphi}R^{2}$.

Any other trivialisation of $\mathcal{T}$ is given by an orthonormal framing,
$(S^{1},S^{2},S^{3})$, and it follows that there is a unique $\Theta\in{\rm
Maps}(Y,O(3))$ such that each $S^{\alpha}=\Theta^{\alpha}_{\beta}R^{\beta}$.
Imposing that $(S^{1},S^{2},S^{3})$ satisfies condition (101), implies that
$\Theta$ in fact takes values in $SO(3)$.

We see, then, that the space of orthonormal trivialisations of $\mathcal{T}$,
compatible with (101), has a free, transitive action of the topological group
${\rm Maps}(Y,SO(3))$; in other words it is a torsor for this group. After
making an arbitrary choice of basepoint in $Y$, this group factorises into a
product, with one factor the space of constant $SO(3)$-valued maps and the
other the basepoint-preserving maps. The first factor can be seen as an
overall rotation of the vector fields and it seems best to regard this as a
redundancy, which we will quotient out. More precisely, we make a choice of
basepoint, $x_{0}\in Y$, which gives us a canonical homomorphism ${\rm
Maps}(Y,SO(3))\cong{\rm Maps}_{*}(Y,SO(3))\times SO(3)$ where ${\rm Maps}_{*}$
denotes those maps which send the basepoint to the identity, $x_{0}\mapsto
1\in SO(3)$. Explicitly, this map is given by
$\Theta\mapsto((\Theta\cdot\Theta(x_{0})^{-1}),\Theta(x_{0}))$ and can be
interpreted as using a global rotation to ensure that any given ACM3S agrees
with that induced by $(R^{1},R^{2},R^{3})$ at $x_{0}$.

We will denote this space of trivialisations by $\mathscr{T}$, which is
manifestly independent of the underlying splitting, $s\in\mathscr{S}$. We will
now show that the projection $p:\mathscr{C}\rightarrow\mathscr{S}$ is indeed
locally trivial. This argument is reminiscent of the standard construction of
the universal bundle over the classifying space $BU(n)$, see for instance
hatcher2016vector ; Magill1103965 .

The topology that we take for both spaces is that induced by compact-open
topology on the space of all maps. Focusing on the total space, $\mathscr{C}$,
for concreteness, the compact-open topology has a subbase given by the sets
$V(K,U)=\\{f:Y\rightarrow\mathcal{V}_{2}(TY)\,|\,f(K)\subset U\\}$ for
$K\subset Y$ compact and $U\subset\mathcal{V}_{2}(TY)$ open. An open set in
the space of sections will be the intersection of an open set on the space of
all maps, with the set of intersections. The same definitions apply, mutatis
mutandis, for the base space, $\mathscr{S}$. In particular, for an arbitrary
section $s\in\mathscr{S}$, we can take an open neighbourhood to be one of the
subbasis generators, $V(K,U)$. We need $U$ to be some open neighbourhood such
that, each $s^{\prime}\in U$ is, fibrewise, a three-plane that forms a graph
over $s_{x}$. In other words, the plane corresponding to $s^{\prime}(x)$ is
just a small rotation of the initial three-plane, $s(x)$, for all $x\in Y$. We
can take $K\subset Y$ to be any nonempty, compact subset, possibly $Y$ itself.
The claim is that $p^{-1}(V(K,U))\cong V(K,U)\times{\rm Maps}(Y,SO(3))$.
Indeed, if we fix a trivialisation over $s$, then by projection and Gram-
schmidt, we are able to continuously assign an orthonormal trivialisation to
each section in this neighbourhood and from there extract the isomorphism. We
therefore have a fibre bundle,
$\mathscr{T}\rightarrow\mathscr{C}\rightarrow\mathscr{S}$, or more explicitly

${\rm
Maps}(Y,SO(3))\rightarrow\Gamma(Y,\mathcal{V}_{2}(TY))\rightarrow\Gamma(Y,\mathcal{G}(\varphi))\,.$
(119)

We do not know when, if ever, this space is in fact a trivial product, but we
will show an example, 5.4, in which it is non-trivially fibred. More
generally, it is natural to expect that ${\rm Maps}(Y,SO(3))$ has infinitely
many components, because both $Y$ and $SO(3)$ have freely generated summands
in third cohomology, while $G_{2}/SU(2)$ has only torsion, so it is plausible
that the space of sections of $\mathcal{V}_{2}(TY)$ also has only finitely
many components. This leads us to expect that the fibre bundle is generically
non-trivial, although this question is not settled here.

##  5 Examples

In this section we will present example geometries that serve to illustrate
concepts related with almost contact structures. The selected examples admit
$G_{2}$ structures with different properties, and some have been used in
physics for the construction of supersymmetric solutions of string or
M-theory. We also explore the connection between ACM3S and associative
submanifolds in a class of non-compact examples with $G_{2}$ holonomy.

###  5.1 Heterotic $G_{2}$ systems on nilmanifolds

Parallelizable nilmanifolds provide explicit examples of $G_{2}$ structure
manifolds, that can be analysed in great detail using their left-invariant
one-forms Fernandez:2008wla ; delBarco:2020ddt . Here we will recapitulate, in
some detail, a particular example from Ref. Fernandez:2008wla : the
nilmanifold $N(3,1)$, which solves the heterotic $G_{2}$ system presented in
3.1 and which appears particularily apt to illustrate that almost contact
structures give added insight to the physical properties of string
compactifications.

Let $H(3,1)$ denote the seven dimensional generalised Heisenberg group.,
consisting of nilpotent real matrices of the form (cf. page 12 of
Fernandez:2008wla )

$H(3,1)=\left\\{\left(\begin{array}[]{lllll}1&x_{1}&x_{2}&x_{3}&z\\\
0&1&0&0&y_{1}\\\ 0&0&1&0&y_{2}\\\ 0&0&0&1&y_{3}\\\ 0&0&0&0&1\\\
\end{array}\right)|x_{i},y_{i},z\in\mathbb{R},1<i<3\right\\}\,.$ (120)

From this, we may construct a compact nilmanifold
$N(3,1)=\Gamma(3,1)\backslash H(3,1)$, where $\Gamma(3,1)\in H(3,1)$ consists
of integer matrices of the above form. As all nilmanifolds, $N(3,1)$ is
parallelizable, and its geometric features can be derived using a basis of
left-invariant one-forms $e^{a}$, $a=1,..,7$ on $H(3,1)$ which satisfy the
structure equations

${\rm d}e^{i}=0\;,1<i<6\;\;,\;\;{\rm d}e^{7}=ae^{12}+be^{34}+ce^{56}\;,$ (121)

where we use the abbreviation $e^{ij}=e^{i}\wedge e^{j}$ and $a,b,c$ are real
non-zero constants. Also, in this section, we will not distinguish between
${\rm d}_{7}$ and ${\rm d}$, since they coincide on $N(3,1)$. The structure
equations clearly show that the $N(3,1)$ can be viewed as a twisted torus

$S^{1}\hookrightarrow N(3,1)\rightarrow T^{6}\;,$

and that $e^{7}$ acts as a connection that encodes the twisting of $S^{1}$
over the six-torus $T^{6}$.

In Ref. Fernandez:2008wla , Lemma 5.5 shows that $N(3,1)$ admits a three-
parameter family of $G_{2}$ structures, defined by

$\varphi=(e^{12}+e^{34}+e^{56})\wedge e^{7}+e^{135}-e^{146}-e^{236}-e^{245}$
(122)

with associated diagonal metric $g_{\varphi}=e^{1}\otimes
e^{1}+...+e^{7}\otimes e^{7}$. An explicit calculation shows that the Hodge
dual fourform is

$\psi=*_{\varphi}\varphi=e^{3456}+e^{1256}+e^{1234}-e^{2467}+e^{2357}+e^{1457}+e^{1367}\;.$

Moreover,

${\rm d}\psi=0\;,\;{\rm d}\varphi=(a+b)e^{1234}+(a+c)e^{1256}+(b+c)e^{3456}$

and thus the torsion is contained in the classes $\tau_{0}$ and $\tau_{3}$.
Explicitly, we have

$\tau_{0}=\frac{2}{7}(a+b+c)\;,\text{and}\;\tau_{3}=*{\rm
d}\varphi-\tau_{0}\varphi\;.$ (123)

Note that the case where $c=-(a+b)$ leads to a vanishing $\tau_{0}$ and hence
a Minkowski solution to the heterotic $G_{2}$ system of section 3.1.
Furthermore, in this case, $T$ simplifies to

$T=-\tau_{3}=-(a+b)e^{567}+be^{347}+ae^{127}\;.$ (124)

We will restrict to this case in the following.

####  5.1.1 Solving the heterotic $G_{2}$ system

In order to solve the heterotic $G_{2}$ system of section 3.1, the nilmanifold
$Y$ must admit $G_{2}$ instanton connections on $TY$ and $V$. Ref.
Fernandez:2008wla  shows that this can be accomplished on $N(3,1)$, provided
that $c=-(a+b)$. This is seen as follows: recall that all nilmanifolds have
Levi-Civita connection 1-forms completely specified by their structure
constants $f^{i}{}_{jk}$ :

$(\kappa^{LC})^{i}_{j}=\frac{1}{2}(f^{i}{}_{jk}-f^{k}{}_{ij}+f^{j}{}_{ki})e^{k}$
(125)

which can be read off from (121) using ${\rm d}e^{i}=f^{i}{}_{jk}e^{jk}$.

Moreover, $\nabla^{+}=\nabla^{LC}+\frac{1}{2}T$ is the unique $G_{2}$
compatible connection, and has associated connection one-forms
Fernandez:2008wla

$(\kappa^{+})^{1}_{2}=-ae^{7}\;\;,\;\;(\kappa^{+})^{3}_{4}=-be^{7}\;\;,\;\;(\kappa^{+})^{5}_{6}=(a+b)e^{7}\;\;.\;\;$
(126)

As always, the curvature two-forms are given by $(\Omega)^{i}_{j}={\rm
d}(\kappa)^{i}_{j}+(\kappa)^{i}_{k}\wedge(\kappa)^{k}_{j}$, so

$(\Omega^{+})^{1}_{2}=-a{\rm d}e^{7}\;,\;(\Omega^{+})^{3}_{4}=-b{\rm
d}e^{7}\;,\;(\Omega^{+})^{5}_{6}=-(a+b){\rm d}e^{7}\;.$ (127)

Now, as discussed in section 3.1, supersymmetry requires that $\nabla^{+}$ is
a $G_{2}$ instanton, and so should satisfy (65). This follows from

${\rm d}e^{7}\wedge\psi=(a+b-a-b)e^{123456}=0\;.$

Thus, $a+b+c=0$ implies that $\nabla^{+}$ is a $G_{2}$ instanton. In fact, the
same holds for any other connection with one-form components proportional to
$e^{7}$.

In the same manner, we may then construct a vector bundle connection $A$
satisfying (65). We assign connection one-forms $(\kappa^{A})^{i}_{j}\sim
e^{7}$, which will satisfy (65). As shown in Ref. Fernandez:2008wla , the
nilmanifold thus admits a 3-parameter family of $G_{2}$ instanton connections
$A^{\lambda,\mu,\tau}$. Finally, with this choice of connections, the Bianchi
identity associated to the anomaly cancellation condition (67) admits a
solution as long as $(\lambda,\mu,\tau)\neq(0,0,0)$ and
$\lambda^{2}+\mu^{2}+\tau^{2}<a^{2}+b^{2}+c^{2}=2(a^{2}+b^{2}+ab)$
Fernandez:2008wla .

####  5.1.2 Left-invariant contact structure and N=2 supersymmetry

The left-invariant one-forms $e^{i}$ have dual vectors $E_{i}$ defined by

$e^{j}(E_{i})=\delta_{i}^{j}\;.$

Having seven globally defined vectors is the hallmark of a parallelizable
seven-dimensional manifold. Picking any of these vectors defines an ACMS on
N(3,1), and any three-frame defines an ACM3S.

In this subsection, we will determine the properties of the ACMS defined by

$R=E_{7}\mbox{ and }\sigma=e^{7}\;.$

By definition, we have $R(\sigma)=1$. Moreover, by (121) we see that ${\rm
d}\sigma$ is purely transverse, so $\rm{Ker}(\sigma)$ is non-integrable and,
therefore, not tangent to a 6-dimensional foliation (cf. section 2.2). Indeed,
we have that

$\sigma\wedge{\rm d}\sigma\wedge{\rm d}\sigma\wedge{\rm
d}\sigma=-3!a\,b\,(a+b)\,e^{1234567}\neq 0\;.$ (128)

Thus, the almost contact structure induced by $\sigma$ is in fact a contact
structure. The fundamental two-form is

$\omega=i_{R}(\varphi)=ae^{12}+be^{34}-(a+b)e^{56}={\rm d}\sigma\;.$

This contact structure furthermore leads to enhanced supersymmetry. Recall,
from section 3.6, that when $R$ is covariantly constant, there exist two
covariantly constant spinors on $Y$. We can phrase the covariant constancy of
$R$ as a question on the connection one-forms. Indeed, the connection one-form
is defined by

$(\kappa^{+})_{ji}(E_{k}):=g(\nabla^{+}_{E_{k}}E_{j},E_{i})$ (129)

and, consequentially, $\nabla^{+}E_{7}=0$ if and only $(\kappa^{+})_{7i}\equiv
0$. That this is true follows immediately from the discussion leading to
(126).

We may go on to check the compatibility of the $G_{2}$ instanton connections
and $H$-flux, constructed in the previous subsection, with the possible
existence of $N=2$ supersymmetry enhancement induced by the covariantly
constant vector field, $E_{7}$. The connection one-forms clearly lack a
transverse piece, whereas the associated curvature is completely transverse,
i.e.

$(\kappa^{A})^{i}_{j}\sim\sigma\;,\;(\Omega^{A})^{i}_{j}\sim
ae^{12}+be^{34}-(a+b)e^{56}\;.$

We thus read off that $F_{0}=0$, as we have argued in section 3.6 should be
the case for $N=2$ enhanced solutions. Likewise, the anomaly cancellation
condition requires, for $N=2$ SUSY, that $H$ is completely transverse up to an
exact contribution. Indeed, the torsion (124) for this nilmanifold example
lacks a transverse piece, and has

$T_{0}={\rm d}\sigma={\rm d}\Sigma\;.$

We thus conclude that the $N(3,1)$ solution to the $N=1$ heterotic $G_{2}$
system, that was constructed in Ref. Fernandez:2008wla , in fact preserves
$N=2$ supersymmetry. We will discuss this further in the next subsection.

####  5.1.3 Left-invariant almost contact 3-structures

In the preceding section we showed that $N(3,1)$ admits a left-invariant CS
which is associated to two covariantly constant spinors, leading to $N=2$
supersymmetry. A natural question to pose is whether the ACS associated to the
remaining left-invariant one-forms, $e^{i}$, for $i=1,..,6$, give a further
enhancement of supersymmetry. To answer this question we are led to construct,
and explore, the space of left-invariant ACM3S on this nilmanifold. As
discussed in section 4.3, since we are dealing with a parallelisable manifold,
we expect several distinguishable ACM3S, corresponding to the $7\choose 2$
different ways of choosing 2-frames among the seven left-invariant vectors
$E_{i}$.

As a first remark, we have ${\rm d}e^{i}=0$ for $i\neq 7$. Consequently, the
ACS associated to $e^{i}$ for $i\neq 7$ are clearly not contact structures.
Thus $N(3,1)$ does not admit a left-invariant contact 3-structure (see the
discussion at the end of 1.1.2).

Let us then start by identifying $R^{1}=E_{7}$. Then, we may choose
$R^{2}=E_{1}$ say, after which $R^{3}$ will be determined by (101) to be
$R^{3}=E_{2}$. Choosing instead $R^{2}=E_{3}(E_{5})$ gives
$R^{3}=E_{4}(E_{6})$, and if we start with picking $R^{2}=E_{2n}$ we find the
same trivialisation up to an overall sign. The upshot is that, once
$R^{1}=E_{7}$ is chosen, there are three inequivalent trivialisations
$(R^{1},R^{2},R^{3})=(E_{7},E_{1},E_{2})$,$(E_{7},E_{3},E_{4})$,$(E_{7},E_{5},E_{6})$.
As discussed in section 4.3, these partially overlapping ACM3S are a
consequence of the parallelizability of the nilmanifold. We will see below
that these different trivializations all have the same qualitative properties.

For concreteness, let us study the trivialisation
$(R^{1},R^{2},R^{3})=(E_{7},E_{1},E_{2})$. We may now determine whether the
associated splitting $TY=\mathcal{T}\oplus\mathcal{T}^{\perp}$ lead to
involutive $\mathcal{T},\mathcal{T}^{\perp}$. As discussed in Section 4.2,
$\mathcal{T}$ is involutive if for any one-form $\xi\in\mathcal{T}^{*,\perp}$
we have

$i_{R^{\alpha}}i_{R^{\beta}}{\rm d}\xi=0\;.$

This follows directly from the fact that
$\mathcal{T}^{*,\perp}=\text{Span}\\{e^{3},e^{4},e^{5},e^{6}\\}$, and

${\rm d}e^{i}=0\;\;\forall\;\;i\neq 7.$

In contrast, it is evident from the discussion around equation (128), that
$\mathcal{T}^{\perp}$ fails to be involutive since

$i_{R^{1}}i_{R^{2}}{\rm d}\sigma^{1}=a\neq 0\;.$

This conclusion clearly holds also for the ACM3S given by
$(R^{1},R^{2},R^{3})=(E_{7},E_{3},E_{4})$ or $(E_{7},E_{5},E_{6})$. We thus
conclude that these ACM3S are associated three-dimensional foliations of $Y$,
but no four-dimensional foliation. The leaves of the three-dimansional
foliation are associative submanifolds (however, they are not volume-
minimizing since we lack a calibrating three-form).

Let us now explore an ACM3S that does not include $E_{7}$ in the three-frame
$(R^{1},R^{2},R^{3})$. This leads to rather different properties. For example,
let us take $(R^{1},R^{2},R^{3})=(E_{1},E_{3},E_{5})$. The associated
splitting $TY=\mathcal{T}\oplus\mathcal{T}^{\perp}$ then lead to involutive
$\mathcal{T}$ and $\mathcal{T}^{\perp}$. Clearly, since
$\mathcal{T}^{*,\perp}=\text{Span}\\{e^{2},e^{4},e^{6},e^{7}\\}$, for any
$\xi\in\Gamma(\mathcal{T}^{*,\perp})$ we have either ${\rm d}\xi=0$ or

$i_{R^{\alpha}}i_{R^{\beta}}{\rm d}\xi\sim i_{R^{\alpha}}i_{R^{\beta}}{\rm
d}e^{7}=0$

showing that $\mathcal{T}$ is involutive. Moreover, $\mathcal{T}^{\perp}$ is
involutive, since $(\sigma^{1},\sigma^{2},\sigma^{3})=(e^{1},e^{3},e^{5})$ are
all closed. Thus, with this choice of ACM3S, we see that Y admits both a
three- and a four-dimensional foliation. The leaves of these foliations are,
respectively, associative and coassociative submanifolds. Furthermore, since
${\rm d}\psi=0$, the leaves of the latter foliation correspond to calibrated
submanifolds.

Supersymmetry enhancement cares less about these different ACM3Ss on $N(3,1)$.
Among the left-invariant basis vectors, only $E^{7}$ is covariantly closed.
This follows directly from the definition of the connection one-form, (129),
in conjunction with (126). Thus, none of the left-invariant ACM3S under study
imply an enhanced supersymmetry to $N=4$. Just like in section 3.6, where we
saw that a reduction of structure group need not be compatible with a given
connection, the parallelisability of the nilmanifold does not imply
supersymmetry is maximally enhanced. Barring extra covariant vector fields on
this nilmanifold, the holonomy group stays $SU(3)$.

Finally, before closing this discussion, let us remark that we have not
exhausted the possible almost contact structures. Our discussion is limited to
left-invariant ACM3Ss, and even here one can imagine constructing ACM3S using
position dependent linear combination of the left-invariant forms which may
lead to different conclusions.

###  5.2 Barely $G_{2}$ examples

In this section we demonstrate the features of an $SU(2)$ structure in a
special class of $G_{2}$ structure manifolds known as barely $G_{2}$
manifolds, joyce1996:2 ; Grigorian:2009nx ; Harvey:1999as . These manifolds
are constructed from Calabi-Yau threefolds endowed with a real structure. For
our purposes, we will focus on the subclass where the real structure is freely
acting and where the initial Calabi-Yau manifold has vanishing Euler
characteristic. Although this is expected to be an extremely small class
(there are only two complete intersection Calabi-Yau manifolds with these
properties, Grigorian:2009nx , with Betti numbers (15,15) and (19,19)), the
restriction is purely for convenience. We will begin reviewing the
construction and key features of barely $G_{2}$ manifolds, before moving on to
$SU(2)$ structures.

The starting point for a barely $G_{2}$ manifold is a Calabi-Yau threefold,
$Z$, endowed with a real structure, that is, an antiholomorphic involution,
$\zeta$. To avoid having to resolve singularities, we will assume that $\zeta$
is freely acting. This real structure is used to endow $Z\times S^{1}$ with a
$\mathbb{Z}_{2}$ action, $\hat{\zeta}=\zeta\times(-1)$, where $(-1)$ acts on
$S^{1}\subset\mathbb{C}$ via $e^{i\theta}\mapsto(e^{-i\theta})$, i.e.
reflection about the real axis. The barely $G_{2}$ manifold, $Y$ is the
quotient by this action,

$Y=(Z\times S^{1})/(\zeta\times(-1))\,.$

For us, it will be convenient to give an alternative definition for $Y$:

$Y=(Z\times[0,1])/\sim$ (130)

where we identify $(z,0)\sim(\zeta(z),1)$ and
$\partial_{t}|_{(z,0)}\sim-\partial_{t}|_{\zeta(z),1}$.

Observe that this space has only a single constant spinor, but nevertheless
the holonomy is a proper subgroup of $G_{2}$, being
$SU(3)\rtimes\mathbb{Z}_{2}$. The induced $G_{2}$ structure forms are induced
by the Calabi-Yau Kahler form, $\omega$, and holomorphic three-form $\Omega$:

$\displaystyle\varphi$ $\displaystyle=\omega\wedge{\rm
d}t+\operatorname{Re}\Omega$ (131) $\displaystyle\psi$
$\displaystyle=\rho-{\rm d}t\wedge\operatorname{Im}\Omega\,.$ (132)

Observe that these product forms indeed survive the quotient, since:

$\displaystyle\zeta^{*}\Omega$ $\displaystyle=\bar{\Omega}$ (133)
$\displaystyle\zeta^{*}\omega$ $\displaystyle=-\omega$ (134)
$\displaystyle(-1)^{*}{\rm d}t$ $\displaystyle=-{\rm d}t\,.$ (135)

Now, by assumption, $Z$ has vanishing Euler characteristic and consequently
the underlying smooth manifold admits a vector field, $v$. We will assume that
we are given such a vector field, say $v$, and, without loss of generality,
that it is of unit norm and invariant under the real structure,
$\zeta^{*}v=v$. This will guarantee that $v$ induces a vector field on the
barely $G_{2}$ manifold, $Y$, say $R^{1}$.

On the other hand, if $I$ denotes the complex structure on $Z$, then $w:=Iv$
is another unit vector field, but is not invariant under the real structure,
instead acted on by a sign, $\zeta^{*}w=-w$. Similarly, the unit tangent
vector field on the interval, $\partial_{t}$, is not invariant so is not well-
defined on equivalence classes.

The three vector fields, $v,w,\partial_{t}$, obviously induce vector fields on
the product $Z\times[0,1]$, but as observed above, neither $w$ nor
$\partial_{t}$ will survive the quotient individually. It is, however,
possible to form invariant combinations using both vector fields, in
particular by rotating through the two-frame they span. More precisely,
consider

$R^{2}=\cos(\pi t)w-\sin(\pi t)\partial_{t}\,,$ (136)

which is well-defined on the quotient since

$R^{2}_{(z,0)}=w_{z}=R^{2}_{(\zeta(z),1)}\,.$

Note, further, $R^{1}$ is everywhere orthogonal to $R^{2}$ so, in particular,
it is everywhere linearly independent.

The third vector field in the three-frame necessary to define an $SU(2)$
structure is obtained with the $G_{2}$ induced cross product:

$R^{3}:=R^{1}\times_{\varphi}R^{2}$ (137)

though it is in fact easiest to compute first the dual one-form. Indeed, by
definition $\sigma^{3}:=g(R^{3},-)$ can be computed to be

$\sigma^{3}=i_{R^{2}}i_{R^{1}}\varphi=\cos(\pi t){\rm d}t-\sin(\pi
t)g(w,-)\,.$ (138)

This is now easy to dualise back to a vector field and we obtain

$R^{3}=-\sin(\pi t)w+\cos(\pi t)\partial_{t}\,.$ (139)

We have now fixed three linearly independent vector fields that fix the
$SU(2)$ structure and can now directly compute the induced structure forms.
Before doing so, we remark that these vector fields can not be covariantly
constant, since we have assumed that the initial Calabi-Yau manifold had full
$SU(3)$ holonomy. As a consequence, the $G_{2}$ connection does not descend to
a connection of the reduced $SU(2)$ structure.

Each vector induces an $SU(3)$ structure, for which we can construct a two-
form $\omega_{i}:=i_{R^{i}}\varphi$ and three-form
$\Omega_{-}^{i}=i_{R^{i}}\psi$, a dual one-form $\sigma_{i}=g(R^{i},-)$ and
endomorphism $J^{i}$ such that $g(J^{i}-,-)=\omega_{i}$. We will focus on
computing $\omega_{i}$ and $\Omega_{-}^{i}$ since we do not need a metric for
this data. The computations are straightforward and we obtain:

$\displaystyle\omega_{1}$ $\displaystyle=-w^{\\#}\wedge{\rm
d}t+i_{v}\operatorname{Re}\Omega$ (140) $\displaystyle\omega_{2}$
$\displaystyle=-\sin(\pi t)\omega+\cos(\pi t)(v^{\\#}\wedge{\rm
d}t+i_{v}\operatorname{Im}\Omega)$ (141) $\displaystyle\omega_{3}$
$\displaystyle=\cos(\pi t)\omega-\sin(\pi t)(v^{\\#}\wedge{\rm
d}t+\operatorname{Im}\Omega)$ (142) $\displaystyle\Omega_{-}^{1}$
$\displaystyle=-w^{\\#}\wedge\omega+{\rm d}t\wedge
i_{v}\operatorname{Im}\Omega$ (143) $\displaystyle\Omega_{-}^{2}$
$\displaystyle=\sin(\pi t)\operatorname{Im}\Omega+\cos(\pi
t)(v^{\\#}\wedge\omega+{\rm d}t\wedge i_{v}\operatorname{Re}\Omega)$ (144)
$\displaystyle\Omega_{-}^{3}$ $\displaystyle=-\cos(\pi
t)\operatorname{Im}\Omega-\sin(\pi t)(v^{\\#}\wedge\omega+{\rm d}t\wedge
i_{v}\operatorname{Re}\Omega)$ (145)

where we have introduced $v^{\\#}=g(v,-)$ and $w^{\\#}=g(w,-)$ and recall that
$\omega,\Omega$ are the Calabi-Yau structure forms. The particular vector
field that we give here can be viewed as rotating through the local frame
$(v,w,\partial_{t})$, a fact that is also apparent in the structure forms. On
a generic barely $G_{2}$-manifold we would not expect to find vector fields
analogous to $v$ and $w$, but even so we are guaranteed to find a ACM3S. One
would expect that any such three-framing would intertwine the Calabi-Yau and
interval directions much more intricately than exhibited here.

We can now consider the space of trivialisations that are compatible with the
splitting induced by $(R^{1},R^{2},R^{3})$. We will show that there are at
least two connected components, which we distinguish by map induced on the
first homology groups. As is well-known, the first homology group of $SO(3)$
is $H_{1}(SO(3),\mathbb{Z})\cong\mathbb{Z}_{2}$, with generator induced by the
generator of $SO(2)\cong S^{1}$ under the inclusion $SO(2)\hookrightarrow
SO(3)$, i.e.

$\displaystyle\rho:I\rightarrow SO(3)\,,$
$\displaystyle\rho(t)=\left(\begin{array}[]{ccc}1&0&0\\\ 0&\cos(2\pi
t)&-\sin(2\pi t)\\\ 0&\sin(2\pi t)&\cos 2\pi t\end{array}\right).$ (149)

Using standard techniques in algebraic topology, it can be shown that
$H_{1}(Y,\mathbb{Z})\cong\mathbb{Z}_{2}$ also, with generator induced by the
open path on $Z\times S^{1}$

$\chi(t)=\left\\{\begin{array}[]{cc}(\gamma_{0}(2t),1)&\quad t\in[0,1/2]\\\
(\zeta(x_{0}),e^{i\pi(2t-1)})&\quad t\in[1/2,1]\end{array}\right.$ (150)

with $\gamma_{0}$ an arbitrary path in $Z$ between a fixed $x_{0}$ and
$\zeta(x_{0})$. Note that since $Z$ is simply connected, the homotopy class of
$\chi$ is independent of $\gamma_{0}$.

Consider, then, the map $\Theta:Y\rightarrow SO(3)$, induced by the map,
$\tilde{\Theta}$ on $Z\times I$

$\tilde{\Theta}(x,t)=\left(\begin{array}[]{ccc}1&0&0\\\ 0&\cos(2\pi
t)&-\sin(2\pi t)\\\ 0&\sin(2\pi t)&\cos 2\pi t\end{array}\right).$ (151)

Then, it is straightforward to check that

$\Theta_{*}\chi(t)=\left\\{\begin{array}[]{cc}1&\quad t\in[0,1/2]\\\
\Theta(\sigma(x_{0}),2t-1)&\quad t\in[1/2,1]\end{array}\right.\sim\rho(t).$

Consequentially, $\Theta_{*}:H_{1}(Y;\mathbb{Z})\rightarrow
H_{1}(SO(3),\mathbb{Z})$ is an isomorphism and $\Theta$ can not be homotopic
to the constant map $1:Y\rightarrow SO(3)$. The trivialisation corresponding
to $\Theta$ is given by the vector fields $(S^{1},S^{2},S^{3})$, which, by
direct computation, are:

$\displaystyle S^{1}$ $\displaystyle=R^{1}=v$ (152) $\displaystyle S^{2}$
$\displaystyle=\cos(2\pi t)R^{2}-\sin(2\pi t)R^{3}=\cos(\pi t)w-\sin(3\pi
t)\partial_{t}$ (153) $\displaystyle S^{3}$ $\displaystyle=\sin(2\pi
t)R^{2}+\cos(2\pi t)R^{3}=-\sin(\pi t)w+\cos(3\pi t)\partial_{t}$ (154)

The fact that $\Theta$ does not induce the zero map on first homology, or
equivalently, that it does not induce the zero map on the fundamental group,
implies that $\Theta$ can not be lifted to the universal cover of $SO(3)$,
which is $SU(2)$. This indicates that the spin structure that is canonically
associated to the trivialised bundle, $\mathcal{T}$, differs between the $R$
and $S$ trivialisations.

Although $\Theta$ induces countably many trivialisations via
$\Theta^{n},\,n\,\in\,\mathbb{Z}$, the first homology groups are only able to
distinguish between the cases that $n$ is even or odd. The group $SO(3)$ has
only torsion homotopy groups in dimensions zero through seven, with the
exception of $\pi_{3}$, $\pi_{3}(SO(3))\cong\mathbb{Z}$. Therefore, the third
homotopy group is a natural arena in which to attempt to distinguish the
components of these maps, though we do not attempt that here.

Further, we have only shown that these two trivialisations are non-homotopic
inside the fixed trivial 3-bundle and it does not necessarily follow that they
are non-homotopic in the space of all ACM3S’s. This would only follow if the
locally trivial product structure that we identified in Subsection 4.3 is in
fact trivial. Working with the space of all ACM3S’s is more subtle because
they correspond to sections of a possibly non-trivial bundle, with twisting
dictated by the initial $G_{2}$ structure and we have nothing conclusive to
say about this.

###  5.3 A compact $G_{2}$ holonomy example

Let us now turn to manifolds with $G_{2}$ holonomy, and explore almost contact
structures on such spaces. In this section, we review an example of compact
seven-manifolds with $G_{2}$ holonomy that is due to Joyce. As we will see,
this construction allows us to specify a canonical AC(3)S that is suitable to
illustrate the topological nature of the space of ACM3S. In the next section,
we will explore this topic in more detail in a non-compact setting.

The construction of this type of $G_{2}$ holonomy manifolds is modelled on the
Kummer construction of $SU(2)$ holonomy metrics on the orbifold
$T^{4}/\mathbb{Z}^{2}$. The construction was originally proposed in
joyce1996:1 ; joyce1996:2 , has been presented in great detail in the books
joyce2000 , and a clear summary of the construction can be found in Ref.
joyce:proc . For the sake of completeness, we will recapitulate the
construction algorithm here.

To begin, we start with the 7-dimensional torus $T^{7}$, with coordinates
$x^{a},a=1,..7$ satisfying $x^{a}\sim x^{a}+1$. This space may be equipped
with the standard $G_{2}$ structure $\varphi_{0}$ as in (2), by the global
identication $e^{a}={\rm d}x^{a}$. The metric $g_{\varphi_{0}}$ is naturally
flat. By quotienting $T^{7}$ by a finite automorphism group $\Gamma$ that
respects $\varphi_{0}$, a new $G_{2}$ space is obtained. In general,
$T^{7}/\Gamma$ will be an orbifold, with singular set $S$ specified by the fix
points of $\Gamma$. We will see an example of this momentarily.

In order to obtain a smooth $G_{2}$ manifold, there is then need to resolve
the singularities. In Joyce’s construction, this is accomplished by noticing
that for certain groups $\Gamma$ the singular set $S$ decomposes into
connected components that may be resolved using standard procedures in complex
algebraic geometry (see below). This resolution gives a non-singular
7-manifold $Y$, which can be shown to admit a 1-parameter family of $G_{2}$
structures $\varphi_{t}$ with torsion
$|\nabla\varphi_{t}|={\mathcal{O}}(t^{4})$. This 1-parameter family of $G_{2}$
structures is obtained by gluing together, using a partition of unity, the
flat $G_{2}$ structure $(\varphi_{0},g_{0})$ in the non-singular “bulk” of $Y$
with local $G_{2}$ structures $(\varphi_{i},g_{i})$ valid near the various
resolved orbifold singularities. The non-zero torsion is localized in the
regions with non-trivial derivatives for the partition of unity, i.e. where
the resolved singular spaces adjoin with the bulk.

It is then possible to prove that for all sufficiently small parameters $t$,
one may deform $(\varphi_{t},g_{t})$ to $(\tilde{\varphi},\tilde{g})$ with
vanishing torsion. Finally, given the specific choices made in the
construction, one may show that the holonomy of $\tilde{g}$ is indeed $G_{2}$,
and not a subgroup thereof. Thus $Y$ is a compact $G_{2}$ holonomy manifold.

Our purpose in this section is to explore AC(3)S that are compatible with
Joyce’s construction. Let us therefore describe in some more detail the
singular set $S$, following joyce:proc  (see also Kronheimer:1989zs ;
kronheimer1989 ; ROAN1996489 ). By careful selection of $\Gamma$, one may
ascertain that $S$ decomposes into connected components $S_{i}$ that are
locally isomorphic to either $T^{3}\times\mathbb{C}^{2}/G$, for $G\subset
SU(2)$ finite, or $S^{1}\times\mathbb{C}^{3}/G$, for $G\subset SU(3)$ finite
and freely acting on $\mathbb{C}^{3}\backslash 0$. We may then use that

  1. 1.

$\mathbb{C}^{2}/G$, for $G\subset SU(2)$ finite, may be resolved to a smooth
Asymptotically Locally Euclidean (ALE) space $U_{2}$, with Kähler metric of
$SU(2)$ holonomy

  2. 2.

$\mathbb{C}^{3}/G$, for $G\subset SU(3)$ as above, may be resolved to a smooth
ALE space $U_{3}$, with Kähler metric of $SU(3)$ holonomy

Then, we may resolve $Y$ by locally excising the connected components
$S^{\alpha}$ of the singular set $S$, and then glue in smooth product spaces
$\hat{S}^{\alpha}$, which have the form $T^{3}\times U_{2}$ or $S^{1}\times
U_{3}$, depending on the nature of the singularity.

The product spaces $\hat{S}^{\alpha}$ admit $G_{2}$ structures, which we will
discuss in detail for the example below. An obvious effect of this
construction is that these local $G_{2}$ structures are reduced to $SU(2)$ and
$SU(3)$ structures, respectively, and the holonomy of the local metric will be
either $SU(2)$ or $SU(3)$. This provides a first link to the AC(3)S that we
have discussed in this paper. Let’s explore that in more detail in an example.

The following example is taken from joyce1996:1 : as above, we construct an
orbifold by quotienting $T^{7}$ by a finite group, which in this example is
the $\mathbb{Z}_{2}^{3}$ generated by121212There is a change in convention in
the definition of the local form of $\varphi$ with respect to joyce1996:1 ;
this requires changing a sign in the $\gamma$ action.

$\displaystyle\alpha((x_{1},...,x_{7}))$
$\displaystyle=(-x_{1},-x_{2},-x_{3},-x_{4},x_{5},x_{6},x_{7})$ (155)
$\displaystyle\beta((x_{1},...,x_{7}))$
$\displaystyle=(-x_{1},\frac{1}{2}-x_{2},x_{3},x_{4},-x_{5},-x_{6},x_{7})$
(156) $\displaystyle\gamma((x_{1},...,x_{7}))$
$\displaystyle=(\frac{1}{2}-x_{1},x_{2},\frac{1}{2}-x_{3},x_{4},x_{5},-x_{6},-x_{7})\;.$
(157)

One can readily show that this group preserves the $G_{2}$ threeform
$\varphi_{0}$ given in (2). The singular set $S$ of $T^{7}/\mathbb{Z}^{3}_{2}$
is determined by the fix points of the generators $\alpha,\beta$ and $\gamma$,
and a little thought reveals that $S$ decomposes into 12 disjoint components
of the form $T^{3}\times\mathbb{C}^{2}/\\{\pm 1\\}$ joyce1996:1 .

Now, we’re in the first situation described in the above list, and for each
simply connected component of singular set, we may blow up
$\mathbb{C}^{2}/\\{\pm 1\\}$ to $U^{A}$, where $U^{A}$ is a smooth ALE space
that agrees with $\mathbb{C}^{2}/\\{\pm 1\\}$ on its boundary. We may
construct this geometry explicitly. Denoting by $(z^{1},z^{2})$ the
coordinates on $\mathbb{C}^{2}$, $U^{A}$ admits a hyper-Kähler triple of two-
forms $\\{\omega^{\alpha}(t)\\}$ satisfying

$\omega^{1}(t)=\frac{i}{2}\partial\bar{\partial}f_{t}\;,\;\omega^{2}(t)+i\omega^{3}(t)={\rm
d}z^{1}\wedge{\rm d}z^{2}$ (158)

where we have introduced a suitable Kähler potential that reduces to
$f_{t}\to|z_{1}|^{2}+|z_{2}|^{2}$ at the boundary of $U^{A}$, and hence
guarantees that the two-forms $\omega^{\alpha}(t)$ become flat
$\hat{\omega}^{\alpha}$ at this boundary.

We then find that the smooth manifold $Y$, that result from desingularising
$T^{7}/\mathbb{Z}^{2}_{3}$ in the manner just described, admit a nowhere
vanishing three-form $\varphi_{t}$. Using the above defined two-forms, and
taking one-forms $\sigma^{\alpha}$ as sections of $\Lambda^{*}T^{3}$, this can
be written as

${\varphi}_{t}={\varphi}_{0}=\frac{1}{3!}\epsilon_{\alpha\beta\gamma}\sigma^{\alpha}\wedge\sigma^{\beta}\wedge\sigma^{\gamma}+\sum_{\alpha}\sigma^{\alpha}\wedge\hat{\omega}^{\alpha}\;,$

in the non-singular bulk of $T^{7}/\mathbb{Z}^{2}_{3}$, and

${\varphi}_{t}=\frac{1}{3!}\epsilon_{\alpha\beta\gamma}\sigma^{\alpha}\wedge\sigma^{\beta}\wedge\sigma^{\gamma}+\sum_{\alpha}\sigma^{\alpha}\wedge\omega^{\alpha}(t)\;,$

in an open neighbourhood that contains $S\times U^{A}$. This is a smooth
three-form, since $\\{\omega^{\alpha}(t)\\}\to\\{\hat{\omega}^{\alpha}\\}$ at
the boundary of $U^{A}$, and it is closed for all $t$. However, a subtlety of
the construction is that it is only for small $t$ that $\varphi_{t}$ is
guaranteed to be a positive three-form, and hence defines a $G_{2}$ structure,
in the interpolating region joyce1996:1 . Finally, while $\varphi_{t}$ is
closed for all $t$, it fails to be coclosed in this region. As discussed
above, a closed and coclosed $G_{2}$ structure may be constructed, in the
$t\to 0$ limit, as $\varphi=\varphi_{t}+{\rm d}\eta_{t}$, where the two-form
$\eta$ satisfies a certain elliptic differential equation that we will not
discuss in detail.

Thus, in the parametric limit of small $t$, we have constructed a $G_{2}$
structure $\varphi_{t}$ which is of the form (113), and provide an example of
an ACM3S. In particular, there is a set of globally defined two-forms,
$\\{\omega^{\alpha}(t)\\}$, which, when combined with the $G_{2}$ three-form,
uniquely defines three one-forms $\\{\sigma^{\alpha}\\}$, which in turn
defines an orthonormal three-frame $\\{R^{1},R^{2},R^{3}\\}$.

So far, we have seen that there are obvious similarities between Joyce’s
torsionful $G_{2}$ structure $\varphi_{t}$ and the ACM3S decomposition that we
know should exist for any $G_{2}$ structure. A more interesting question to
ask is whether Joyce’s construction automatically provides information about
the AC(3)S of the torsion-free $G_{2}$ structure $\varphi_{t}$. This is true,
but only to some extent. Indeed, any nowhere vanishing vector field $R$ will
provide an ACS when combined either with $\varphi_{t}$ or $\varphi$. However
the fundamental two-forms given by (24) depend on the three-form, and hence
the transverse geometry will differ between these ACS. Clearly, comparing to
(58), the $SU(3)$ torsion classes will also depend on whether the torsionful
or torsionfree $G_{2}$ structure is selected.

For AC3S the situation is similar, but here one also has to take into account
that while linear independence of vector fields does not depend on the $G_{2}$
structure, orthonormality of vectors do. Thus, an orthonormal three-frame
$\\{R^{1},R^{2},R^{3}\\}$ associated to $\varphi_{t}$ will not be orthonormal
with respect to $\varphi$. However, the space of AC3S, $\mathscr{C}$, is
topological, and hence the same for $\varphi_{t}$ and $\varphi$. Thus, we may
hope to determine this space using only the explicit, torsionful $G_{2}$
structure $\varphi_{t}$. This is interesting, because we can then hope to even
say something new about how to count associatives on a $G_{2}$ manifold. We
hope to return to this intriguing problem in a future publication.

###  5.4 A class of non-compact $G_{2}$ holonomy examples

In this section we will consider the local neighbourhood of an associative
three-cycle in the context of ACM3Ss. Associative cycles are known to be
relevant to M-theory compactifications and we earlier found a surprising
relation with ACM3Ss. In particular, the associated trivial bundle,
$\mathcal{T}$, can only be tangent to a submanifold if it is an associative
submanifold.131313Note that $\mathcal{T}$ may be tangent to a submanifold
without everywhere being a foliation, that is, the involutivity condition