Generalised Chern–Simons Theory and ${\rm G}_{2}$-instantons over Associative
Fibrations

Generalised Chern–Simons Theory
and $\boldsymbol{{\rm G}_{2}}$-Instantons over Associative Fibrations

Henrique N. SÁ EARP

H.N. Sá Earp

Imecc - Institute of Mathematics, Statistics and Scientific Computing,
Unicamp, Brazil<EMAIL_ADDRESS>http://www.ime.unicamp.br/~hqsaearp/

Received January 29, 2014, in final form August 07, 2014; Published online
August 11, 2014

Adjusting conventional Chern–Simons theory to ${\rm G}_{2}$-manifolds, one
describes ${\rm G}_{2}$-instantons on bundles over a certain class of
$7$-dimensional flat tori which fiber non-trivially over $T^{4}$, by a
pullback argument. Moreover, if $c_{2}\neq 0$, any (generic) deformation of
the ${\rm G}_{2}$-structure away from such a fibred structure causes all
instantons to vanish. A brief investigation in the general context of
(conformally compatible) associative fibrations $f:Y^{7}\to X^{4}$ relates
${\rm G}_{2}$-instantons on pullback bundles $f^{*}E\to Y$ and self-dual
connections on the bundle $E\to X$ over the base, a fact which may be of
independent interest.

Chern–Simons; Yang–Mills; ${\rm G}_{2}$-manifolds; associative fibrations

53C07; 53C38; 58J28

##  1 Introduction

This article fits in the context of gauge theory in higher dimensions,
following the seminal works of S. Donaldson & R. Thomas, G. Tian and others
[4, 16]. The common thread to such generalisations is the presence of a closed
differential form on the base manifold $Y$, inducing an analogous notion of
anti-self-dual connections, or _instantons_ , on bundles over $Y$. In the case
at hand, ${\rm G}_{2}$-manifolds are $7$-dimensional Riemannian manifolds with
holonomy in the Lie group ${\rm G}_{2}$, which implies the existence of
precisely such a structure. This allows one to make sense of ${\rm
G}_{2}$-_instantons_ as the energy-minimising gauge classes of connections,
solutions to the corresponding Yang–Mills equation.

Heuristically, ${\rm G}_{2}$-instantons are somewhat analogous to flat
connections in dimension 3. Given a bundle over a compact $3$-manifold, with
space of connections $\mathcal{A}$ and gauge group $\mathcal{G}$, the
_Chern–Simons functional_ is a multi-valued real function on the quotient
$\mathcal{B=A}/\mathcal{G}$, with integer periods, whose critical points are
precisely the flat connections [3, § 2.5]. Similar theories can be formulated
in higher dimensions in the presence of a suitable closed differential form
[4, 15]; e.g. on a ${\rm G}_{2}$-manifold $(Y,\varphi)$, the coassociative
$4$-form $\ast\varphi$ allows for the definition of a functional of
Chern–Simons type111In fact only the condition $d\ast\varphi=0$ is required,
so the discussion extends to cases in which the ${\rm G}_{2}$-structure
$\varphi$ is not necessarily torsion-free.. Its ‘gradient’, the Chern–Simons
1-form, vanishes precisely at the ${\rm G}_{2}$-instantons, hence it detects
the solutions to the Yang–Mills equation. These gauge-theoretic preliminaries
are covered in Section 2.

On the other hand, one may understand ${\rm G}_{2}$-manifolds as a particular
case of the rich theory of calibrated geometries [6], for which the ${\rm
G}_{2}$-structure $\varphi$ is a calibration 3-form. Then a 3-dimensional
submanifold $P$ is said to be _associative_ if it is calibrated by $\varphi$,
i.e., if $\varphi|_{P}=d\operatorname{Vol}|_{P}$. The deformation theory of
associative submanifolds is known to be obstructed [9], so their occurrence in
families, e.g. fibering over a 4-manifold, is nongeneric and somewhat exotic.
Nonetheless, we may consider theoretically, at first, the existence of
instantons over associative fibrations $f:Y^{7}\to X^{4}$. Given a bundle
$E\to X$, a connection $\mathbf{A}$ on its pullback $\mathbf{E}$ is locally of
the form

$\displaystyle\mathbf{A}\overset{\rm
loc}{=}A_{t}(x)+\sum\limits_{i=1}^{3}\sigma_{i}(x,t)dt^{i},$

where $\\{A_{t}\\}$ is a family of connections on $E$ parametrised by the
associative fibers $P_{x}:=f^{-1}(x)$ and
$\sigma_{i}\in\Omega^{0}(Y,f^{*}\mathfrak{g}_{E})$. In Section 3.1 I prove the
following relation between ${\rm G}_{2}$-instantons and families of self-dual
connections over the base:

######  Theorem 1.1.

Let $f:Y\to X$ define an associative fibration and $\mathbf{E}\to Y$ be the
pullback from an indecomposable vector bundle $E\to X$.

  1. $(i)$

If a connection $\mathbf{A}$ on $\mathbf{E}$ is a ${\rm G}_{2}$-instanton,
then $\\{A_{t}\\}$ is a family of self-dual connections on $E$, satisfying

$\displaystyle\frac{\partial A_{t}}{\partial t^{i}}=d_{A_{t}}\sigma_{i},\qquad
i=1,2,3.$

  2. $(ii)$

If, moreover, the family $A_{t}\equiv A_{t_{0}}$ is constant, then
$\mathbf{A}=f^{*}A_{t_{0}}$ is a pullback.

NB: We denote henceforth by $\mathcal{M}^{4}_{+}$ the moduli space of SD
connections on the base and by $\mathcal{M}^{7}_{\varphi}$ the moduli space of
${\rm G}_{2}$-instantons relative to ${\rm G}_{2}$-structure $\varphi$.

Finally, over the remaining of Section 3, these ideas are applied to a
concrete example of certain $T^{3}$-fibrations over $T^{4}$, topologically
equivalent to the 7-torus, which I will call _${\rm G}_{2}$ -torus fibrations_
[11]. Deforming the metric (i.e. the lattice) on $T^{4}$ induces a change on
the fibration map and hence on the ${\rm G}_{2}$-structure, and one can use
Chern–Simons formalism to see how this affects the moduli of ${\rm
G}_{2}$-instantons:

######  Theorem 1.2.

Let $f:\mathbb{T}\rightarrow T^{4}$ be a ${\rm G}_{2}$-torus fibration,
$\mathbf{E}\to\mathbb{T}$ be the pullback of an indecomposable vector bundle
$E\to T^{4}$ and $\varphi$ denote the ${\rm G}_{2}$-structure of $\mathbb{T}$;
then

  1. $(i)$

every SD connection on $E$ lifts to a ${\rm G}_{2}$-instanton on $\mathbf{E}$,
i.e.,

$\displaystyle f^{*}\mathcal{M}^{4}_{+}\subset\mathcal{M}^{7}_{\varphi};$

  2. $(ii)$

if, moreover, $c_{2}(E)\neq 0$, then any perturbation $\varphi+\phi$ away from
the class of fibred structures causes the moduli space of ${\rm
G}_{2}$-instantons to vanish, i.e.,

$\displaystyle\mathcal{M}^{7}_{\varphi+\phi}=\varnothing.$

The construction of ${\rm G}_{2}$-instantons is a recent and active research
area. Indeed Theorem 1.2 yields nontrivial, albeit nongeneric, examples of
${\rm G}_{2}$-instanton moduli, whenever a complex vector bundle $E\to T^{4}$
admits SD connections. The interested reader will find other examples in works
of Walpuski, Clarke and the author [2, 11, 12, 13, 17]. In the high-energy
physics community, solutions to a very similar problem in the context of ${\rm
G}_{2}$-structures with torsion have been found eg. for cylinders over nearly-
Kähler homogeneous spaces [5] and more generally for cones over nontrivial
manifolds admitting real Killing spinors [7].

Finally, a paper just published by Wang [18] makes significant progress
towards a Donaldson theory over higher-dimensional foliations, which seems to
encompass our ${\rm G}_{2}$-torus fibration as a special, codimension 4 tight
foliation, whose leaf space is the smooth $4$-manifold $X$. It is inspiring to
speculate whether an invariant of the corresponding foliated moduli space can
be explicitly computed for some suitable bundle $\mathbf{E}\to\mathbb{T}$, or
indeed if that space coincides with our definition of $\mathcal{M}^{7}$.

##  2 Gauge theory over $\boldsymbol{{\rm G}_{2}}$-manifolds

I will concisely recall the essentials of gauge theory on ${\rm
G}_{2}$-manifolds, while referring the interested reader to a more detailed
exposition in [12].

Let $Y$ be an oriented smooth $7$-manifold; a ${\rm G}_{2}$-_structure_ is a
smooth $3$-form $\varphi\in\Omega^{3}(Y)$ such that, at every point $p\in Y$,
one has $\varphi_{p}=f_{p}^{\ast}(\varphi_{0})$ for some frame
$f_{p}:T_{p}Y\rightarrow\mathbb{R}^{7}$ and (adopting the conventions of [14])

$\displaystyle\varphi_{0}=e^{567}+\omega_{1}\wedge e^{5}+\omega_{2}\wedge
e^{6}+\omega_{3}\wedge e^{7}$ (1)

with

$\displaystyle\omega_{1}=e^{12}-e^{34},\qquad\omega_{2}=e^{13}-e^{42}\qquad\text{and}\qquad\omega_{3}=e^{14}-e^{23}.$

Moreover, $\varphi$ determines a Riemannian metric $g(\varphi)$ induced by the
pointwise inner-product

$\displaystyle\langle u,v\rangle e^{1\dots
7}:=\frac{1}{6}(u\lrcorner\varphi_{0})\wedge(v\lrcorner\varphi_{0})\wedge\varphi_{0},$
(2)

under which $\ast_{\varphi}\varphi$ is given pointwise by

$\displaystyle\ast\varphi_{0}=e^{1234}-\omega_{1}\wedge
e^{67}-\omega_{2}\wedge e^{75}-\omega_{3}\wedge e^{56}.$ (3)

Such a pair $(Y,\varphi)$ is a  _${\rm G}_{2}$ -manifold_ if $d\varphi=0$ and
$d\ast_{\varphi}\varphi=0$.

###  2.1 The $\boldsymbol{{\rm G}_{2}}$-instanton equation

The ${\rm G}_{2}$-structure allows for a $7$-dimensional analogue of
conventional Yang–Mills theory, yielding a notion of (anti-)self-duality for
$2$-forms. Under the usual identification between $2$-forms and matrices, we
have $\mathfrak{g}_{2}\subset\mathfrak{so}(7)\simeq\Lambda^{2}$, so we denote
$\Lambda_{14}^{2}:=\mathfrak{g}_{2}$ and $\Lambda_{7}^{2}$ its orthogonal
complement in $\Lambda^{2}$:

$\displaystyle\Lambda^{2}=\Lambda_{7}^{2}\oplus\Lambda_{14}^{2}.$ (4)

It is easy to check that $\Lambda_{7}^{2}=\langle
e_{1}\lrcorner\varphi_{0},\dots,e_{7}\lrcorner\varphi_{0}\rangle$, hence the
orthogonal projection onto $\Lambda_{7}^{2}$ in (4) is given by

$\displaystyle L_{\ast\varphi_{0}}:\ \ $
$\displaystyle\Lambda^{2}\rightarrow\Lambda^{6},$
$\displaystyle\eta\mapsto\eta\wedge\ast\varphi_{0}$

in the sense that [1, p. 541]

$\displaystyle L_{\ast\varphi_{0}}|_{\Lambda_{7}^{2}}:\ \
\Lambda_{7}^{2}\;\tilde{\rightarrow}\;\Lambda^{6}\qquad\text{and}\qquad
L_{\ast\varphi_{0}}|_{\Lambda_{14}^{2}}=0.$

Furthermore, since (4) splits $\Lambda^{2}$ into irreducible representations
of ${\rm G}_{2}$, a little inspection on generators reveals that
$(\Lambda^{2})_{{}^{7}_{14}}$ is respectively the ${}_{+1}^{-2}$-eigenspace of
the ${\rm G}_{2}$-equivariant linear map

$\displaystyle T_{\varphi_{0}}:\ \ $
$\displaystyle\Lambda^{2}\rightarrow\Lambda^{2},$ $\displaystyle\eta\mapsto
T_{\varphi_{0}}\eta:=\ast(\eta\wedge\varphi_{0}).$

Consider now a vector bundle $E\rightarrow Y$ over a compact ${\rm
G}_{2}$-manifold $(Y,\varphi)$; the curvature $F:=F_{A}$ of some connection
$A$ decomposes according to the splitting (4):

$\displaystyle F_{A}=F_{7}\oplus F_{14},\qquad
F_{i}\in\Omega_{i}^{2}(\operatorname{End}E),\qquad i=7,14.$

The $L^{2}$-norm of $F_{A}$ is the _Yang–Mills functional_ :

$\displaystyle{\rm YM}(A):=\|F_{A}\|^{2}=\|F_{7}\|^{2}+\|F_{14}\|^{2}.$ (5)

It is well-known that the values of ${\rm YM}(A)$ can be related to a certain
characteristic class of the bundle $E$, given (up to choice of orientation) by

$\displaystyle\kappa(E):=-\int_{Y}\operatorname{tr}\big{(}F_{A}^{2}\big{)}\wedge\varphi.$

Using the property $d\varphi=0$, a standard argument of Chern–Weil theory [10]
shows that the de Rham class $[\operatorname{tr}(F_{A}^{2})\wedge\varphi]$ is
independent of $A$, thus the integral is indeed a topological invariant. The
eigenspace decomposition of $T_{\varphi}$ implies (up to a sign)

$\displaystyle\kappa(E)=-2\|F_{7}\|^{2}+\|F_{14}\|^{2},$

and combining with (5) we get

$\displaystyle{\rm
YM}(A)=-\frac{1}{2}\kappa(E)+\frac{3}{2}\|F_{14}\|^{2}=\kappa(E)+3\|F_{7}\|^{2}.$

Hence ${\rm YM}(A)$ attains its absolute minimum at a connection whose
curvature lies either in $\Lambda_{7}^{2}$ or in $\Lambda_{14}^{2}$. Moreover,
since ${\rm YM}\geq 0$, the sign of $\kappa(E)$ obstructs the existence of one
type or the other, so we fix $\kappa(E)\geq 0$ and define _${\rm G}_{2}$
-instantons_ as connections with $F\in\Lambda_{14}^{2}$, i.e., such that ${\rm
YM}(A)=\kappa(E)$. These are precisely the solutions of the ${\rm
G}_{2}$-_instanton equation_ :

$\displaystyle F_{A}\wedge\ast\varphi=0$ (6a) or, equivalently, $\displaystyle
F_{A}-\ast(F_{A}\wedge\varphi)=0.$ (6b)

If instead $\kappa(E)\leq 0$, we may still reverse orientation and consider
$F\in\Lambda_{14}^{2}$, but then the above eigenvalues and energy bounds must
be adjusted accordingly, which amounts to a change of the $(-)$ sign in (6b).

###  2.2 Definition of the Chern–Simons functional $\boldsymbol{\vartheta}$

Gauge theory in higher dimensions can be formulated in terms of the geometric
structure of manifolds with exceptional holonomy [4]. In particular,
instantons can be characterised as critical points of a Chern–Simons
functional, hence zeroes of its gradient 1-form [3]. The explicit case of
${\rm G}_{2}$-manifolds, which we now describe, was first examined in the
author’s thesis [11].

Let $E\to Y$ be a vector bundle; the space $\mathcal{A}$ is an affine space
modelled on $\Omega^{1}(\mathfrak{g}_{E})$ so, fixing a reference connection
$A_{0}\in\mathcal{A}$,

$\displaystyle\mathcal{A}=A_{0}+\Omega^{1}(\mathfrak{g}_{E})$

and, accordingly, vectors at $A\in\mathcal{A}$ are 1-forms $a,b,\ldots\in
T_{A}\mathcal{A}\simeq\Omega^{1}(\mathfrak{g}_{E})$ and vector fields are maps
$\alpha,\beta,\ldots:\mathcal{A}\to\Omega^{1}(\mathfrak{g}_{E})$. In this
notation we define the _Chern–Simons functional_ by

$\displaystyle\vartheta(A):=\tfrac{1}{2}\int_{Y}\operatorname{tr}\left(d_{A_{0}}a\wedge
a+\frac{2}{3}a\wedge a\wedge a\right)\wedge\ast\varphi,$

fixing $\vartheta(A_{0})=0$. This function is obtained by integration of the
_Chern–Simons $1$-form_

$\displaystyle\rho(\beta)_{A}=\rho_{A}(\beta_{A}):=\int_{Y}\operatorname{tr}(F_{A}\wedge\beta_{A})\wedge\ast\varphi.$
(7)

We find $\vartheta$ explicitly by integrating $\rho$ over paths
$A(t)=A_{0}+ta$, from $A_{0}$ to any $A=A_{0}+a$:

$\displaystyle\vartheta(A)-\vartheta(A_{0})=\int_{0}^{1}\rho_{A(t)}(\dot{A}(t))dt=\int_{0}^{1}\left(\int_{Y}\operatorname{tr}\big{(}\big{(}F_{A_{0}}+td_{A_{0}}a+t^{2}a\wedge
a\big{)}\wedge a\big{)}\wedge\ast\varphi\right)dt$
$\displaystyle\phantom{\vartheta(A)-\vartheta(A_{0})}=\frac{1}{2}\int_{Y}\operatorname{tr}\left(d_{A_{0}}a\wedge
a+\frac{2}{3}a\wedge a\wedge a\right)\wedge\ast\varphi+K,$

where $K=K(A_{0},a)$ is a constant and vanishes if $A_{0}$ is an instanton.

The co-closedness condition $d\ast\varphi=0$ implies that the $1$-form (7) is
closed, so the procedure doesn’t depend on the path $A(t)$. Indeed, given
tangent vectors $a,b\in\Omega^{1}(\mathfrak{g}_{E})$ at $A$, the leading term
in the expansion of $\rho$,

$\displaystyle\rho_{A+a}(b)-\rho_{A}(b)=\int_{Y}\operatorname{tr}(d_{A}a\wedge
b)\wedge\ast\varphi+O\big{(}|b|^{2}\big{)},$

is symmetric by Stokes’ theorem:

$\displaystyle\int_{Y}\operatorname{tr}(d_{A}a\wedge b-a\wedge
d_{A}b)\wedge\ast\varphi=\int_{Y}d(\operatorname{tr}(b\wedge
a)\wedge\ast\varphi)=0.$

We conclude that

$\displaystyle\rho_{A+a}(b)-\rho_{A}(b)=\rho_{A+b}(a)-\rho_{A}(a)+O\big{(}|b|^{2}\big{)}$

and, comparing reciprocal Lie derivatives on parallel vector fields
$\alpha\equiv a$, $\beta\equiv b$ near a point $A$, we have:

$\displaystyle
d\rho(\alpha,\beta)_{A}=(\mathcal{L}_{b}\rho)_{A}(a)-(\mathcal{L}_{a}\rho)_{A}(b)=\lim\limits_{h\rightarrow
0}\frac{1}{h}\big{\\{}\rho_{A+hb}(a)-\rho_{A}(a))-(\rho_{A+ha}(b)-\rho_{A}(b))\big{\\}}$
$\displaystyle\phantom{d\rho(\alpha,\beta)_{A}}=\lim\limits_{h\rightarrow
0}\frac{1}{h^{2}}\underset{O\big{(}|h|^{3}\big{)}}{\underbrace{\big{\\{}(\rho_{A+hb}(ha)-\rho_{A}(ha))-(\rho_{A+ha}(hb)-\rho_{A}(hb))\big{\\}}}}=0.$

Since $\mathcal{A}$ is contractible, by the Poincaré lemma $\rho$ is the
derivative of some function $\vartheta$. Again by Stokes, $\rho$ vanishes
along $\mathcal{G}$-orbits $\operatorname{im}(d_{A})\simeq
T_{A}\\{\mathcal{G}.A\\}$. Thus $\rho$ descends to the quotient $\mathcal{B}$
and so does $\vartheta$, locally.

###  2.3 Periodicity of $\boldsymbol{\vartheta}$

Consider the gauge action of $g\in\mathcal{G}$ and some path
$\\{A(t)\\}_{t\in[0,1]}\subset\mathcal{A}$ connecting an instanton $A$ to
$g.A$. The natural projection $p_{1}:Y\times[0,1]\rightarrow Y$ induces a
bundle

$\displaystyle\begin{array}[]{@{}ccc}\mathbf{E}_{g}&\overset{\widetilde{p_{1}}}{\longrightarrow}&E\\\
\downarrow&&\downarrow\\\
Y\times[0,1]&\overset{p_{1}}{\longrightarrow}&Y\end{array}$

and, using $g$ to identify the fibres
$(\mathbf{E}_{g})_{0}\overset{g}{\simeq}(\mathbf{E}_{g})_{1}$, one may think
of $\mathbf{E}_{g}$ as a bundle over $Y\times S^{1}$. Moreover, in some local
trivialisation, the path $A(t)=A_{i}(t)dx^{i}$ gives a connection
$\mathbf{A}=\mathbf{A}_{0}dt+\mathbf{A}_{i}dx^{i}$ on $\mathbf{E}_{g}$:

$\displaystyle(\mathbf{A}_{0})_{(t,p)}=0,\qquad(\mathbf{A}_{i})_{(t,p)}=A_{i}(t)_{p}.$

The corresponding curvature $2$-form is
$F_{\mathbf{A}}=(F_{\mathbf{A}})_{i0}dx^{i}\wedge
dt+(F_{\mathbf{A}})_{jk}dx^{j}\wedge dx^{k}$, where

$\displaystyle(F_{\mathbf{A}})_{i0}=\dot{A}_{i}(t),\qquad(F_{\mathbf{A}})_{jk}=(F_{A})_{jk}.$

The periods of $\vartheta$ are then of the form

$\displaystyle\vartheta(g.A)-\vartheta(A)=\int_{0}^{1}\rho_{A(t)}(\dot{A}(t))dt=\int_{Y\times[0,1]}\operatorname{tr}(F_{A(t)}\wedge\dot{A}_{i}(t)dx^{i})\wedge
dt\wedge\ast\varphi$
$\displaystyle\phantom{\vartheta(g.A)-\vartheta(A)}{}=\int_{Y\times
S^{1}}\operatorname{tr}F_{\mathbf{A}}\wedge
F_{\mathbf{A}}\wedge\ast\varphi=\frac{1}{8\pi^{2}}\big{\langle}c_{2}(\mathbf{E}_{g})\smallsmile[\ast\varphi],Y\times
S^{1}\big{\rangle}.$

The Künneth formula for $Y\times S^{1}$ gives

$\displaystyle H^{4}\big{(}Y\times
S^{1},\mathbb{R}\big{)}=H^{4}(Y,\mathbb{R})\oplus
H^{3}(Y,\mathbb{R})\otimes\underset{\mathbb{Z}}{\underbrace{H^{1}\big{(}S^{1},\mathbb{R}\big{)}}}$

and obviously $H^{4}(Y)\smallsmile[\ast\varphi]=0$ so, denoting by
$c_{2}^{\prime}(\mathbf{E}_{g})$ the component lying in $H^{3}(Y)$ and by
$S_{g}:=[\frac{1}{8\pi^{2}}c_{2}^{\prime}(\mathbf{E}_{g})]^{PD}$ its
normalised Poincaré dual, we are left with

$\displaystyle\vartheta(g.A)-\vartheta(A)=\langle[\ast\varphi],S_{g}\rangle.$

Consequently, the periods of $\vartheta$ lie in the set

$\displaystyle\left\\{\int_{S_{g}}\ast\varphi\,\bigg{|}\,S_{g}\in
H_{4}(Y,\mathbb{R})\right\\}.$

That may seem odd at first, because $\ast\varphi$ is not, in general, an
integral class and so the set of periods is _dense_. However, as long as our
interest remains in the study of the moduli space
$\mathcal{M}=\operatorname{Crit}(\rho)$ of ${\rm G}_{2}$-instantons, there is
not much to worry, for the gradient $\rho=d\vartheta$ is unambiguously defined
on $\mathcal{B}$.

##  3 Instantons over $\boldsymbol{{\rm G}_{2}}$-torus fibrations

Instances of ${\rm G}_{2}$-manifolds fibred by associative submanifolds in the
literature are relatively scarce, not least because their deformation theory
is zero-index elliptic [9] and therefore any new examples will be somewhat
exotic. A few trivial cases include the products $T^{7}=T^{4}\times T^{3}$ and
$K3\times T^{3}$ and also $CY^{3}\times S^{1}$ given a family of curves in the
Calabi–Yau [8, § 10.8]. The example I will propose is unique in the sense that
the total space is not a Riemannian product.

###  3.1 Instantons over associative fibrations

We consider pullback bundles over smooth associative fibrations, and relate
${\rm G}_{2}$-instantons to their gauge theory over the base; in particular we
do not address the possibility of singular fibres.

######  Definition 3.1.

A ${\rm G}_{2}$-manifold $(Y^{7},\varphi)$ is called an _associative
fibration_ over a compact oriented Riemannian four-manifold $(X^{4},\eta)$ if
it is the total space of a Riemannian submersion $f:Y\to X$ such that each
fibre $P_{x}:=f^{-1}(x)\subset Y$ is a smooth associative submanifold.

Since each fibre $P_{x}$ is $3$-dimensional and orientable, its tangent bundle
is differentiably trivial and we may choose global coordinates
$t=(t^{1},t^{2},t^{3})$ induced respectively by a global coframe
$\\{e_{5},e_{6},e_{7}\\}:=\\{dt^{1},dt^{2},dt^{3}\\}$. Thus near each $y\in
P_{x}$ we may complete the triplet into a local orthogonal coframe
$\\{e_{1},\dots,e_{7}\\}$ of $T^{*}Y$ such that $\varphi_{y}$ has the form
(1), and the point $y$ is unambiguously described by $(x,t(y))$.

######  Lemma 3.2.

Let $f:Y\to X$ define an associative fibration and $\mathbf{E}\to Y$ be the
pullback from a vector bundle $E\to X$; then a connection $A$ on $E$ is self-
dual if, and only if, $f^{*}A$ is a ${\rm G}_{2}$-instanton on $\mathbf{E}$.

######  Proof 3.3.

Let $F:=(F_{f^{*}A})_{y}$ be the curvature 2-form at $y\in P_{x}$; then

$\displaystyle\ast_{\varphi}(F\wedge\varphi)\overset{\rm
loc}{=}\ast_{\varphi}\big{[}F\wedge\big{(}\varphi|_{P_{x}}+\omega_{1}\wedge
e^{5}+\omega_{2}\wedge e^{6}+\omega_{3}\wedge
e^{7}\big{)}\big{]}\\!=\\!\ast_{\eta}F\\!+\ast_{\varphi}\big{[}O(F^{-})\wedge
f^{*}d\operatorname{Vol}_{\eta}\big{]},\\!$

where
$O(F^{-}):=(F_{34}-F_{12})e^{5}+(F_{42}-F_{13})e^{6}+(F_{23}-F_{14})e^{7}$
vanishes precisely when $A$ is self-dual, i.e., when $F=\ast_{\eta}F$
satisfies the ${\rm G}_{2}$-instanton equation (6b).

We are now in position to prove Theorem 1.1. Let us examine the general form
of a ${\rm G}_{2}$-instanton on $\mathbf{E}$. An arbitrary connection
$\mathbf{A}$ on $\mathbf{E}$ is locally of the form

$\displaystyle\mathbf{A}(y)\overset{\rm
loc}{=}A_{t}(x)+\sum\limits_{i=1}^{3}\sigma_{i}(x,t)dt^{i},$

where $\\{A_{t}\\}_{t\in t(P_{x})}$ is a family of connections on $E$ and
$\sigma_{i}\in\Omega^{0}(Y,f^{*}\mathfrak{g}_{E})$. The curvature of
$\mathbf{A}$ is

$\displaystyle
F_{\mathbf{A}}=F_{A_{t}}+\sum\limits_{i=1}^{3}\left(d_{A_{t}}\sigma_{i}-\frac{\partial
A_{t}}{\partial t^{i}}\right)\wedge dt^{i}+F_{\sigma}$

with

$\displaystyle
F_{\sigma}:=\sum\limits_{i,j=1}^{3}\left(\frac{\partial\sigma_{i}}{\partial
t^{j}}-\frac{\partial\sigma_{j}}{\partial
t^{i}}+\frac{1}{2}[\sigma_{i},\sigma_{j}]\right)dt^{i}\wedge dt^{j}.$

Replacing $F_{\mathbf{A}}$ into the ${\rm G}_{2}$-instanton equation (6a) and
using the expression (3) of $\ast\varphi$ in the natural frame
$\\{e_{1},\dots,e_{7}\\}$, we have

$\displaystyle\left(F_{A_{t}}+\sum\limits_{i=1}^{3}(d_{A_{t}}\sigma_{i}-\frac{\partial
A_{t}}{\partial t^{i}})\wedge
e^{4+i}+F_{\sigma}\right)\wedge\big{(}e^{1234}-\omega_{1}\wedge
e^{67}-\omega_{2}\wedge e^{75}-\omega_{3}\wedge e^{56}\big{)}=0.$

Using the following elementary properties

$\displaystyle F_{A_{t}}\wedge e^{1234}=0,\qquad
F_{A_{t}}\wedge\omega_{1}\wedge
e^{67}=[(F_{A_{t}})_{34}-(F_{A_{t}})_{12}]\big{(}{\ast}e^{5}\big{)},$
$\displaystyle F_{A_{t}}\wedge\omega_{2}\wedge
e^{75}=[(F_{A_{t}})_{42}-(F_{A_{t}})_{13}]\big{(}{\ast}e^{6}\big{)},\qquad
F_{A_{t}}\wedge\omega_{3}\wedge
e^{56}=[(F_{A_{t}})_{23}-(F_{A_{t}})_{14}]\big{(}{\ast}e^{7}\big{)},$
$\displaystyle F_{\sigma}\wedge e^{4+i}\wedge e^{4+j}=0,\qquad
F_{\sigma}\wedge
e^{1234}=(F_{\sigma})_{23}\big{(}{\ast}e^{5}\big{)}+(F_{\sigma})_{31}\big{(}{\ast}e^{6}\big{)}+(F_{\sigma})_{12}\big{(}{\ast}e^{7}\big{)},$

and the fact that each $d_{A_{t}}\sigma_{i}$ and $\frac{\partial
A_{t}}{\partial t^{i}}$ are locally $1$-forms on the base, hence their wedge
product with $e^{1234}=d\operatorname{Vol}_{\eta}$ vanishes, the equation
simplifies to

$\displaystyle\sum\limits_{i=1}^{3}\left(d_{A_{t}}\sigma_{i}-\frac{\partial
A_{t}}{\partial t^{i}}\right)\wedge\omega_{i}=0\qquad\text{and}\qquad
F_{A_{t}}^{-}-Q(F_{\sigma})=0,$

where $Q$ is the linear map on $2$-forms defined by

$\displaystyle Q\big{(}dt^{i}\wedge dt^{j}\big{)}=Q\big{(}e^{4+i}\wedge
e^{4+j}\big{)}:=\sum\limits_{k=1}^{3}\epsilon^{ijk}\omega_{k}.$

On the other hand, if $\mathbf{A}=A_{t}+\sum\sigma_{i}$ is a ${\rm
G}_{2}$-instanton, then it minimises the Yang–Mills functional (5). This
implies

$\displaystyle\sum\left\|d_{A_{t}}\sigma_{i}-\frac{\partial A_{t}}{\partial
t^{i}}\right\|^{2}+\|F_{\sigma}\|^{2}=0,$

since otherwise the pullback component $A_{t}$ alone would violate the minimum
energy:

$\displaystyle{\rm YM}(A_{t})=\|F_{A_{t}}\|^{2}<\|F_{\mathbf{A}}\|^{2}={\rm
YM}(\mathbf{A}).$

In particular $F_{\sigma}\equiv 0$ and so every $A_{t}$ must be SD. Finally,
if the family $A_{t}\equiv A_{t_{0}}$ is constant, then
$d_{A_{t_{0}}}\sigma_{i}=0$ implies $\sigma\equiv 0$, since by assumption
$\mathbf{E}$ is indecomposable and therefore does not admit nontrivial
parallel sections. This concludes the proof of Theorem 1.1.

######  Remark 3.4.

If $\mathcal{M}^{4}_{+}$ is discrete, then by continuity the family
$\\{A_{t}\\}$ is contained in a gauge orbit; if the family is constant, then
$\mathbf{A}$ is a pullback.

###  3.2 $\boldsymbol{{\rm G}_{2}}$-torus fibrations

A $7$-torus $T^{7}=\mathbb{R}^{7}/\Lambda$ naturally inherits the ${\rm
G}_{2}$-structure $\varphi$ from $\mathbb{R}^{7}$. Recall from Section 2.2
that a connection $A$ on some bundle over $T^{7}$ is a ${\rm G}_{2}$-instanton
if and only if it is a zero of the Chern–Simons $1$-form (7):

$\displaystyle\rho_{A}(b)=\int_{T^{7}}\operatorname{tr}(F_{A}\wedge
b)\wedge\ast\varphi.$ (8)

One asks what is the behaviour of the moduli space of ${\rm G}_{2}$-instantons
under perturbations $\varphi\rightarrow\varphi+\phi$ of the ${\rm
G}_{2}$-structure. More precisely, given suitable assumptions, one asks
whether $(\varphi+\phi)$-instantons exist at all once we deform the lattice.
As a working example, we consider the following class of flat $T^{3}$-fibred
7-tori:

######  Definition 3.5.

A  _${\rm G}_{2}$ -torus fibration structure_ is a triplet $(\eta,L,\alpha)$
in which:

  * •

$\eta$ is a metric on $\mathbb{R}^{4}$;

  * •

$L$ is a lattice on the subspace $\Lambda^{2}_{+}(\mathbb{R}^{4},\eta)$ of
$\eta$-self-dual $2$-forms;

  * •

$\alpha:\mathbb{R}^{4}\rightarrow\Lambda^{2}_{+}(\mathbb{R}^{4},\eta)$ is a
linear map.

Given the above data, set $V\doteq\mathbb{R}^{4}\oplus\Lambda^{2}_{+}$ and
form the torus $\mathbb{T}=V/\tilde{L}$, with the lattice

$\displaystyle\tilde{L}\doteq\big{\\{}(\mu,\nu+\alpha\mu)\,|\,\mu\in\mathbb{Z}^{4},\,\nu\in
L\big{\\}}\subset V.$

Then $\mathbb{T}$ inherits from $V$ the ${\rm G}_{2}$-structure $\varphi$
which makes the generators of $\tilde{L}$ orthonormal with respect to the
induced inner-product (2). It is straightforward to check that $\mathbb{T}$ is
an associative fibration as in Definition 3.1: denoting by $e^{5}$, $e^{6}$,
$e^{7}$ the $(\nu+\alpha\mu)$-orthonormal basis of the fibre
$\Lambda^{2}_{+}$, the flat ${\rm G}_{2}$-structure (1) simplifies to
$\varphi|_{\Lambda^{2}_{+}}=e^{567}=d\operatorname{Vol}_{\varphi}|_{\Lambda^{2}_{+}}$;
moreover the lattice $\tilde{L}$ on every tangent subspace normal to the fibre
is just the lattice $\mu$ from the base, so the corresponding metrics are the
same. Although $\mathbb{T}$ fibres over the 4-torus $\mathbb{R}^{4}/\mu$, the
induced metric $g(\varphi)$ is _not_ , in general, a Riemannian product.

Suppose the moduli space $\mathcal{M}^{4}_{+}$ of self-dual connections on
$E\to T^{4}$ is nonempty; then we have trivial solutions to the ${\rm
G}_{2}$-instanton equation on the pullback $\mathbf{E}\to\mathbb{T}$ simply by
lifting $\mathcal{M}^{4}_{+}$ as in Lemma 3.2, which proves the first part of
Theorem 1.2:

######  Corollary 3.6.

If $A$ is a self-dual connection on $E\to T^{4}$, then its pullback $f^{*}A$
by the fibration map $f:\mathbb{T}\rightarrow T^{4}$ is a ${\rm
G}_{2}$-instanton on $\mathbf{E}$.

For future reference, I denote the set of such $\varphi$-instantons obtained
by lifts from $\mathcal{M}^{4}_{+}$ by

$\displaystyle\widetilde{\mathcal{M}^{4}_{+}}:=f^{*}\mathcal{M}^{4}_{+}\subset\mathcal{B}^{7}.$
(9)

We know from $4$-dimensional gauge theory that SD connections on a complex
vector bundle $E\to T^{4}$ correspond to stable holomorphic structures on $E$,
thus in such cases we have examples of ${\rm G}_{2}$-instantons on bundles
over $\mathbb{T}$.

###  3.3 Deformations of $\boldsymbol{\mathbb{T}}$

Working on a bundle $\mathbf{E}\rightarrow\mathbb{T}$ with compact structure
group over a fixed ${\rm G}_{2}$-torus fibration, let us ponder in generality
about the behaviour of instantons under a deformation of the ${\rm
G}_{2}$-structure:

$\displaystyle\varphi\rightarrow\varphi+\phi,\qquad*_{\varphi}\varphi\rightarrow*_{\varphi}\varphi+\xi_{\phi},\qquad\xi_{\phi}:=*_{\varphi+\phi}(\varphi+\phi)-*_{\varphi}\varphi\in\Omega^{4}(\mathbb{T}).$

An arbitrary deformation $\phi$ does not in general preserve the fibred
structure of $\mathbb{T}$:

######  Proposition 3.7.

A deformation $\xi_{\phi}\in\Lambda^{4}(\mathbb{T})$ of the coassociative
$4$-form $*_{\varphi}\varphi$ has four orthogonal components, with the
following significance:

$\displaystyle\Lambda^{4}\big{(}\mathbb{R}^{4}\oplus\Lambda^{2}_{+}\big{)}=\underset{\rm(I)}{\underbrace{\Lambda^{4}\big{(}\mathbb{R}^{4}\big{)}}}\,\oplus\,\underset{\rm(II)}{\underbrace{\Lambda^{3}\big{(}\mathbb{R}^{4}\big{)}\otimes\Lambda^{1}\big{(}\Lambda^{2}_{+}\big{)}}}\,\oplus\,\underset{\rm(III)}{\underbrace{\Lambda^{2}\big{(}\mathbb{R}^{4}\big{)}\otimes\Lambda^{2}\big{(}\Lambda^{2}_{+}\big{)}}}\,\oplus\,\underset{\rm(IV)}{\underbrace{\Lambda^{1}\big{(}\mathbb{R}^{4}\big{)}\otimes\Lambda^{3}\big{(}\Lambda^{2}_{+}\big{)}}},$

  1. (I)

corresponds to a rescaling of the metric $\eta$ on $\mathbb{R}^{4}$;

  2. (II)

redefines the map $\alpha$;

  3. (III)

splits as
$\operatorname{Hom}(\Lambda^{2}_{+},\Lambda^{2}_{+})\oplus\operatorname{Hom}(\Lambda^{2}_{-},\Lambda^{2}_{+})$,
where the first factor modifies the lattice $L$ and the second one affects the
conformal class of $\eta$;

  4. (IV)

parametrises deformations transverse to the fibred structures.

######  Proof 3.8.

Let us examine the four cases.

(I) If $\xi_{\phi}\in\Lambda^{4}(\mathbb{R}^{4})\simeq\mathbb{R}$, then it
must be a multiple of
$\ast\varphi|_{\mathbb{R}^{4}}=e^{1234}=d\operatorname{Vol}_{\eta}$.

(II) Since
$\Lambda^{3}(\mathbb{R}^{4})\otimes\Lambda^{1}(\Lambda^{2}_{+})\simeq\mathbb{R}^{4}\otimes(\Lambda^{2}_{+})^{*}\simeq\operatorname{Hom}(\mathbb{R}^{4},\Lambda^{2}_{+})$,
such deformations are precisely linear maps
$\mathbb{R}^{4}\rightarrow\Lambda^{2}_{+}$.

(III) Clearly
$\Lambda^{2}(\mathbb{R}^{4})=\Lambda^{2}_{+}\oplus\Lambda^{2}_{-}$ and
$\Lambda^{2}(\Lambda^{2}_{+})\simeq(\Lambda^{2}_{+})^{*}$, so the product
decomposes as

$\displaystyle\big{(}\Lambda^{2}_{+}\otimes\big{(}\Lambda^{2}_{+}\big{)}^{*}\big{)}\oplus\big{(}\Lambda^{2}_{-}\otimes\big{(}\Lambda^{2}_{+}\big{)}^{*}\big{)}\simeq\operatorname{Hom}\big{(}\Lambda^{2}_{+},\Lambda^{2}_{+}\big{)}\oplus\operatorname{Hom}\big{(}\Lambda^{2}_{-},\Lambda^{2}_{+}\big{)}.$

Now, on one hand, acting with an endomorphism on $\Lambda^{2}_{+}$ is
equivalent to redefining the triplet $\\{e^{5},e^{6},e^{7}\\}$, hence the
lattice $L\subset\Lambda^{2}_{+}$. On the other hand, since the orthogonal
split $\Lambda^{2}=\Lambda^{2}_{-}\oplus\Lambda^{2}_{+}$ is conformally
invariant, a map $\Lambda^{2}_{-}\to\Lambda^{2}_{+}$ redefines the orthogonal
complement of $\Lambda^{2}_{-}$ and hence the conformal class.

(IV) Since $\Lambda^{3}(\Lambda^{2}_{+})\simeq\mathbb{R}$, this component is
just $\Lambda^{1}(\mathbb{R}^{4})$, which is irreducible in the sense that
$\mathbb{T}$ has no distinguished subspaces in $\mathbb{R}^{4}$. Then either
every $7$-torus is a ${\rm G}_{2}$-fibration, which is obviously false, or
these are precisely the deformations away from said structures.

We will now describe what happens to the zeroes of (8) under the corresponding
perturbation of the Chern–Simons $1$-form:

$\displaystyle\rho\rightarrow\rho_{\phi}:=\rho+r_{\phi},\qquad(r_{\phi})_{A}(b)=\int_{\mathbb{T}}\operatorname{tr}(F_{A}\wedge
b)\wedge\xi_{\phi}.$

Clearly a $\varphi$-instanton $A$ is also a $(\varphi+\phi)$-instanton if and
only if $(r_{\phi})_{A}\equiv 0$. There is little reason, however, to expect
such a coincidence; as we will see, the topology of the bundle may constrain
the existence of instantons under certain – indeed most – deformations.

Denoting henceforth by $\mathcal{A}$ the space of connections over the
$7$-manifold $\mathbb{T}$, let us briefly digress into the translation action
of some vector $v\in\mathbb{T}$ on some $A\in\mathcal{A}$. The first order
variation is given by the bundle-valued 1-form

$\displaystyle(\beta_{v})_{A}:=v\lrcorner F_{A},$

which we interpret as a vector in $T_{A}\mathcal{A}$. Notice first that in the
direction $\beta_{v}$ the value of the Chern–Simons 1-form is independent of
the base-point:

######  Lemma 3.9.

The function $\rho(\beta_{v}):\mathcal{A}\to\mathbb{R}$ is constant.

######  Proof 3.10.

The computation is straightforward:

$\displaystyle\rho(\beta_{v})_{A+ha}=\int_{\mathbb{T}}\operatorname{tr}F_{A+ha}\wedge
v\lrcorner
F_{A+ha}\wedge\ast\varphi=-\frac{1}{2}\int_{\mathbb{T}}\operatorname{tr}F_{A+ha}\wedge
F_{A+ha}\wedge(v\lrcorner\ast\varphi)$
$\displaystyle\phantom{\rho(\beta_{v})_{A+ha}}=-\frac{1}{2}\int_{\mathbb{T}}(\operatorname{tr}F_{A}\wedge
F_{A}+d\chi)\wedge(v\lrcorner\ast\varphi)=-\frac{1}{2}\int_{\mathbb{T}}\operatorname{tr}F_{A}\wedge
F_{A}\wedge(v\lrcorner\ast\varphi)=\rho(\beta_{v})_{A},$

where $d\chi$ is the exact differential given by Chern–Weil theory and we use
Stokes’ theorem and Cartan’s identity
$d(v\lrcorner\ast\varphi)=\mathcal{L}_{v}(\ast\varphi)=0$, since $\varphi$ is
constant on the flat torus.

Similarly, evaluating $r_{\phi}$ on $\beta_{v}$ gives

$\displaystyle
r_{\phi}(\beta_{v})_{A}=\int_{\mathbb{T}}\operatorname{tr}(F_{A}\wedge(\beta_{v})_{A})\wedge\xi_{\phi}=-\tfrac{1}{2}\int_{\mathbb{T}}\operatorname{tr}(F_{A}\wedge
F_{A})\wedge(v\lrcorner\xi_{\phi})=\langle c_{2}(E),S_{\phi}(v)\rangle,$

where $S_{\phi}(v)\doteq-\tfrac{1}{2}[v\lrcorner\xi_{\phi}]^{PD}$, and this
depends only on the topology of $E$, not on the point $A$.

######  Remark 3.11.

Hence we may interpret $\phi$ as defining a linear functional

$\displaystyle N_{\phi}:\ \ $
$\displaystyle\mathbb{R}^{7}\rightarrow\mathbb{R},$ $\displaystyle
v\mapsto\langle c_{2}(E),S_{\phi}(v)\rangle,$

such that $N_{\phi}\neq 0$ implies no $\varphi$-instanton is still a
$(\varphi+\phi)$-instanton. This is, however, a rather weak obstruction, since
the map $\phi\mapsto N_{\phi}$ has kernel of dimension at least $28$ and thus,
in principle, leaves plenty of possibilities for instantons of perturbed ${\rm
G}_{2}$-structures.

Now consider specifically a translation vector on the base $v\in T^{4}$.
Notice that for deformations $\phi$ of types (I), (II) or (III) the
contraction of $\xi_{\phi}$ with such $v$ gives $S_{\phi}(v)=0$, so $\phi$
only effectively contributes to the function $\rho(\beta_{v})$ when
$\xi_{\phi}\in\Lambda^{1}(\mathbb{R}^{4})$, which means the perturbed torus is
no longer a fibred structure (Proposition 3.7). Moreover, either the bundle
$E$ is flat and $\beta_{v}$ vanishes identically, or $c_{2}(E)\neq 0$ and the
following holds:

######  Lemma 3.12.

If $c_{2}(E)\neq 0$ and $\phi$ is of type (IV), then there exists $v\in T^{4}$
such that $r_{\phi}(\beta_{v})$ is a non-zero constant.

######  Proof 3.13.

Denoting $T^{3}$ the typical fibre of $f$ (and setting
$\operatorname{Vol}(T^{3})=1$), we may assume

$\displaystyle\xi_{\phi}=-2\varepsilon\wedge d\operatorname{Vol}_{T^{3}}$

for some $0\neq\varepsilon\in\Lambda^{1}(T^{4})$. One can always choose $v\in
T^{4}$ such that $\varepsilon(v)\neq 0$, and consider
$(\beta_{v})_{A}=v\lrcorner F_{A}$. Then

$\displaystyle
r_{\phi}(\beta_{v})_{A}=-2\int_{\mathbb{T}}\operatorname{tr}(F_{A}\wedge
v\lrcorner F_{A})\wedge\varepsilon\wedge
d\operatorname{Vol}_{T^{3}}=-2\int_{T^{4}}\operatorname{tr}(F_{A}\wedge
v\lrcorner F_{A})\wedge\varepsilon=\varepsilon(v)\cdot c_{2}(E),$

which is nonzero by assumption.

So far we know from Corollary 3.6 that the set $\mathcal{M}^{4}_{+}$ of self-
dual connections (modulo gauge) over $T^{4}$ lifts to instantons (cf. (9)) of
the original ${\rm G}_{2}$-structure $\varphi$ (i.e. to zeroes of $\rho$).
However, for bundles with non-trivial $c_{2}$, this generic case degenerates
precisely under deformations of type (IV):

######  Proposition 3.14.

Let $\mathbf{E}\rightarrow(\mathbb{T},\varphi)$ be the pullback of a stable
${\rm SU}(n)$-bundle $E$ over $T^{4}$ with $c_{2}(E)\neq 0$; then $E$ admits
no $(\varphi+\phi)$-instantons, for any perturbation $\phi$ away from a fibred
structure $($i.e. of type (IV) in Proposition 3.7).

######  Proof 3.15.

Fix a lifted $\varphi$-instanton $A\in\widetilde{\mathcal{M}^{4}_{+}}$; for
any $A+ha\in\mathcal{A}$, Lemma 3.9 gives
$\rho_{A+ha}(\beta_{v})\equiv\rho_{A}(\beta_{v})=0$. Taking $v\in T^{4}$ as in
Lemma 3.12 we have

$\displaystyle\rho_{\phi}(\beta_{v})_{A+ha}=r_{\phi}(\beta_{v})_{A+ha}+\rho(\beta_{v})_{A+ha}=\underset{\neq
0}{\underbrace{\varepsilon(v)\cdot
c_{2}(E)}}+\underset{0}{\underbrace{\rho(\beta_{v})_{A}}},$

hence $A+ha$ is not a $(\varphi+\phi)$-instanton.

Combining Corollary 3.6 and Proposition 3.14 we obtain Theorem 1.2.

### Acknowledgements

The author thanks Simon Donaldson for suggesting the matter of this paper,
Thomas Walpuski for sharing some unpublished notes and Marcos Jardim for
several useful discussions. Special thanks also to the anonymous referees for
numerous mathematical and reference contributions.

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