Generalised Chern-Simons Theory and ${\rm G}_2$-Instantons over Associative Fibrations

0.4 MB·

by  Henrique N. Sá Earp

Symmetry, Integrability and Geometry: Methods and Applications SIGMA 10 (2014), 083, 11 pages
Generalised Chern–Simons Theory
and G -Instantons over Associative Fibrations
2
Henrique N. SA´ EARP
Imecc - Institute of Mathematics, Statistics and Scientific Computing, Unicamp, Brazil
E-mail: [email protected]
URL: http://www.ime.unicamp.br/~hqsaearp/
Received January 29, 2014, in final form August 07, 2014; Published online August 11, 2014
4 http://dx.doi.org/10.3842/SIGMA.2014.083
1
0
2 Abstract. Adjusting conventional Chern–Simons theory to G2-manifolds, one describes
G -instantons on bundles over a certain class of 7-dimensional flat tori which fiber non-
g 2
trivially over T4, by a pullback argument. Moreover, if c (cid:54)=0, any (generic) deformation of
u 2
A the G2-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 :

2 Y7 →X4 relates G -instantons on pullback bundles f∗E →Y and self-dual connections on
2
1
the bundle E →X over the base, a fact which may be of independent interest.

]
Key words: Chern–Simons; Yang–Mills; G -manifolds; associative fibrations
G 2
D
2010 Mathematics Subject Classification: 53C07; 53C38; 58J28
.
h
t
a
1 Introduction
m
[
This article fits in the context of gauge theory in higher dimensions, following the seminal

3
works of S. Donaldson & R. Thomas, G. Tian and others [4, 16]. The common thread to such
v
2 generalisations is the presence of a closed differential form on the base manifold Y, inducing
6 an analogous notion of anti-self-dual connections, or instantons, on bundles over Y. In the
4
case at hand, G -manifolds are 7-dimensional Riemannian manifolds with holonomy in the Lie
5 2
1. group G2, which implies the existence of precisely such a structure. This allows one to make
sense of G -instantons as the energy-minimising gauge classes of connections, solutions to the
0 2
4 corresponding Yang–Mills equation.
1
Heuristically, G -instantons are somewhat analogous to flat connections in dimension 3.
: 2
v Given a bundle over a compact 3-manifold, with space of connections A and gauge group G, the
i
X Chern–Simons functional is a multi-valued real function on the quotient B = A/G, with integer
r periods, whose critical points are precisely the flat connections [3, § 2.5]. Similar theories can
a
be formulated in higher dimensions in the presence of a suitable closed differential form [4, 15];
e.g. on a G -manifold (Y,ϕ), the coassociative 4-form ∗ϕ allows for the definition of a functional
2
of Chern–Simons type1. Its ‘gradient’, the Chern–Simons 1-form, vanishes precisely at the 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 G -manifolds as a particular case of the rich theory
2
of calibrated geometries [6], for which the G -structure ϕ is a calibration 3-form. Then a 3-
2
dimensional submanifold P is said to be associative if it is calibrated by ϕ, i.e., if ϕ| =
P
dVol| . The deformation theory of associative submanifolds is known to be obstructed [9], so
P
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
1Infactonlytheconditiond∗ϕ=0isrequired,sothediscussionextendstocasesinwhichtheG -structureϕ
2
is not necessarily torsion-free.
2 H.N. S´a Earp
fibrations f : Y7 → X4. Given a bundle E → X, a connection A on its pullback E is locally of
the form
3
A l=oc A (x)+(cid:88)σ (x,t)dti,
t i
i=1
where {A } is a family of connections on E parametrised by the associative fibers P := f−1(x)
t x
and σ ∈ Ω0(Y,f∗g ). In Section 3.1 I prove the following relation between G -instantons and
i E 2
families of self-dual connections over the base:
Theorem 1. Let f : Y → X define an associative fibration and E → Y be the pullback from an
indecomposable vector bundle E → X.
(i) If a connection A on E is a G -instanton, then {A } is a family of self-dual connections
2 t
on E, satisfying
∂A
t
= d σ , i = 1,2,3.
∂ti At i
(ii) If, moreover, the family A ≡ A is constant, then A = f∗A is a pullback.
t t0 t0
NB: We denote henceforth by M4 the moduli space of SD connections on the base and
+
by M7 the moduli space of G -instantons relative to G -structure ϕ.
ϕ 2 2
Finally, over the remaining of Section 3, these ideas are applied to a concrete example of
certain T3-fibrations over T4, topologically equivalent to the 7-torus, which I will call G -torus
2
fibrations [11]. Deforming the metric (i.e. the lattice) on T4 induces a change on the fibration
map and hence on the G -structure, and one can use Chern–Simons formalism to see how this
2
affects the moduli of G -instantons:
2
Theorem2. Letf : T → T4 beaG -torusfibration, E → Tbethepullbackofanindecomposable
2
vector bundle E → T4 and ϕ denote the G -structure of T; then
2
(i) every SD connection on E lifts to a G -instanton on E, i.e.,
2
f∗M4 ⊂ M7;
+ ϕ
(ii) if, moreover, c (E) (cid:54)= 0, thenanyperturbationϕ+φawayfromtheclassoffibredstructures
2
causes the moduli space of G -instantons to vanish, i.e.,
2
M7 = ∅.
ϕ+φ
The construction of G -instantons is a recent and active research area. Indeed Theorem 2
2
yieldsnontrivial,albeitnongeneric,examplesofG -instantonmoduli,wheneveracomplexvector
2
bundle E → T4 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 G -structures with torsion have been found
2
eg. for cylinders over nearly-K¨ahler 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 G -torus fibration as
2
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 E → T, or indeed if that space coincides with our
definition of M7.
Generalised Chern–Simons Theory and G -instantons over Associative Fibrations 3
2
2 Gauge theory over G -manifolds
2
IwillconciselyrecalltheessentialsofgaugetheoryonG -manifolds,whilereferringtheinterested
2
reader to a more detailed exposition in [12].
Let Y be an oriented smooth 7-manifold; a G -structure is a smooth 3-form ϕ ∈ Ω3(Y) such
2
that, at every point p ∈ Y, one has ϕ = f∗(ϕ ) for some frame f : T Y → R7 and (adopting
p p 0 p p
the conventions of [14])
ϕ = e567+ω ∧e5+ω ∧e6+ω ∧e7 (1)
0 1 2 3
with
ω = e12−e34, ω = e13−e42 and ω = e14−e23.
1 2 3
Moreover, ϕ determines a Riemannian metric g(ϕ) induced by the pointwise inner-product
1
(cid:104)u,v(cid:105)e1...7 := (u(cid:121)ϕ )∧(v(cid:121)ϕ )∧ϕ , (2)
0 0 0
6
under which ∗ ϕ is given pointwise by
ϕ
∗ϕ = e1234−ω ∧e67−ω ∧e75−ω ∧e56. (3)
0 1 2 3
Such a pair (Y,ϕ) is a G -manifold if dϕ = 0 and d∗ ϕ = 0.
2 ϕ
2.1 The G -instanton equation
2
TheG -structureallowsfora7-dimensionalanalogueofconventionalYang–Millstheory,yielding
2
a notion of (anti-)self-duality for 2-forms. Under the usual identification between 2-forms and
matrices, we have g ⊂ so(7) (cid:39) Λ2, so we denote Λ2 := g and Λ2 its orthogonal complement
2 14 2 7
in Λ2:
Λ2 = Λ2⊕Λ2 . (4)
7 14
It is easy to check that Λ2 = (cid:104)e (cid:121)ϕ ,...,e (cid:121)ϕ (cid:105), hence the orthogonal projection onto Λ2 in (4)
7 1 0 7 0 7
is given by
L : Λ2 → Λ6,
∗ϕ0
η (cid:55)→ η∧∗ϕ
0
in the sense that [1, p. 541]
L | : Λ2 →˜ Λ6 and L | = 0.
∗ϕ0 Λ27 7 ∗ϕ0 Λ214
Furthermore, since (4) splits Λ2 into irreducible representations of G , a little inspection on
2
generators reveals that (Λ2) is respectively the −2-eigenspace of the G -equivariant linear map
7 +1 2
14
T : Λ2 → Λ2,
ϕ0
η (cid:55)→ T η := ∗(η∧ϕ ).
ϕ0 0
Consider now a vector bundle E → Y over a compact G -manifold (Y,ϕ); the curvature
2
F := F of some connection A decomposes according to the splitting (4):
A
F = F ⊕F , F ∈ Ω2(EndE), i = 7,14.
A 7 14 i i
4 H.N. S´a Earp
The L2-norm of F is the Yang–Mills functional:
A
YM(A) := (cid:107)F (cid:107)2 = (cid:107)F (cid:107)2+(cid:107)F (cid:107)2. (5)
A 7 14
It is well-known that the values of YM(A) can be related to a certain characteristic class of
the bundle E, given (up to choice of orientation) by
(cid:90)
κ(E) := − tr(cid:0)F2(cid:1)∧ϕ.
A
Y
Using the property dϕ = 0, a standard argument of Chern–Weil theory [10] shows that the
deRhamclass[tr(F2)∧ϕ]isindependentofA,thustheintegralisindeedatopologicalinvariant.
A
The eigenspace decomposition of T implies (up to a sign)
ϕ
κ(E) = −2(cid:107)F (cid:107)2+(cid:107)F (cid:107)2,
7 14
and combining with (5) we get
1 3
YM(A) = − κ(E)+ (cid:107)F (cid:107)2 = κ(E)+3(cid:107)F (cid:107)2.
14 7
2 2
Hence YM(A) attains its absolute minimum at a connection whose curvature lies either in Λ2
7
or in Λ2 . Moreover, since YM ≥ 0, the sign of κ(E) obstructs the existence of one type or the
14
other, so we fix κ(E) ≥ 0 and define G -instantons as connections with F ∈ Λ2 , i.e., such that
2 14
YM(A) = κ(E). These are precisely the solutions of the G -instanton equation:
2
F ∧∗ϕ = 0 (6a)
A
or, equivalently,
F −∗(F ∧ϕ) = 0. (6b)
A A
If instead κ(E) ≤ 0, we may still reverse orientation and consider F ∈ Λ2 , but then the above
14
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 ϑ
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 G -manifolds, which we now describe, was first examined in the author’s thesis [11].
2
Let E → Y be a vector bundle; the space A is an affine space modelled on Ω1(g ) so, fixing
E
a reference connection A ∈ A,
0
A = A +Ω1(g )
0 E
and, accordingly, vectors at A ∈ A are 1-forms a,b,... ∈ T A (cid:39) Ω1(g ) and vector fields are
A E
maps α,β,... : A → Ω1(g ). In this notation we define the Chern–Simons functional by
E
(cid:90) (cid:18) (cid:19)
2
ϑ(A) := 1 tr d a∧a+ a∧a∧a ∧∗ϕ,
2 A0 3
Y
fixing ϑ(A ) = 0. This function is obtained by integration of the Chern–Simons 1-form
0
(cid:90)
ρ(β) = ρ (β ) := tr(F ∧β )∧∗ϕ. (7)
A A A A A
Y
Generalised Chern–Simons Theory and G -instantons over Associative Fibrations 5
2
We find ϑ explicitly by integrating ρ over paths A(t) = A +ta, from A to any A = A +a:
0 0 0
(cid:90) 1 (cid:90) 1(cid:18)(cid:90) (cid:19)
ϑ(A)−ϑ(A ) = ρ (A˙(t))dt = tr(cid:0)(cid:0)F +td a+t2a∧a(cid:1)∧a(cid:1)∧∗ϕ dt
0 A(t) A0 A0
0 0 Y
(cid:90) (cid:18) (cid:19)
1 2
= tr d a∧a+ a∧a∧a ∧∗ϕ+K,
2 A0 3
Y
where K = K(A ,a) is a constant and vanishes if A is an instanton.
0 0
The co-closedness condition d∗ϕ = 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 ∈ Ω1(g ) at A, the leading
E
term in the expansion of ρ,
(cid:90)
ρ (b)−ρ (b) = tr(d a∧b)∧∗ϕ+O(cid:0)|b|2(cid:1),
A+a A A
Y
is symmetric by Stokes’ theorem:
(cid:90) (cid:90)
tr(d a∧b−a∧d b)∧∗ϕ = d(tr(b∧a)∧∗ϕ) = 0.
A A
Y Y
We conclude that
ρ (b)−ρ (b) = ρ (a)−ρ (a)+O(cid:0)|b|2(cid:1)
A+a A A+b A
and, comparing reciprocal Lie derivatives on parallel vector fields α ≡ a, β ≡ b near a point A,
we have:
1(cid:8) (cid:9)
dρ(α,β) = (L ρ) (a)−(L ρ) (b) = lim ρ (a)−ρ (a))−(ρ (b)−ρ (b))
A b A a A A+hb A A+ha A
h→0 h
1 (cid:8) (cid:9)
= lim (ρ (ha)−ρ (ha))−(ρ (hb)−ρ (hb)) = 0.
h→0 h2(cid:124) A+hb A (cid:123)(cid:122) A+ha A (cid:125)
(cid:0) (cid:1)
O |h|3
Since A is contractible, by the Poincar´e lemma ρ is the derivative of some function ϑ. Again by
Stokes, ρ vanishes along G-orbits im(d ) (cid:39) T {G.A}. Thus ρ descends to the quotient B and so
A A
does ϑ, locally.
2.3 Periodicity of ϑ
Consider the gauge action of g ∈ G and some path {A(t)} ⊂ A connecting an instanton A
t∈[0,1]
to g.A. The natural projection p : Y ×[0,1] → Y induces a bundle
1
E −p(cid:101)→1 E
g
↓ ↓
Y ×[0,1] −p→1 Y
g
and, using g to identify the fibres (E ) (cid:39) (E ) , one may think of E as a bundle over
g 0 g 1 g
Y × S1. Moreover, in some local trivialisation, the path A(t) = A (t)dxi gives a connection
i
A = A dt+A dxi on E :
0 i g
(A ) = 0, (A ) = A (t) .
0 (t,p) i (t,p) i p
The corresponding curvature 2-form is F = (F ) dxi∧dt+(F ) dxj ∧dxk, where
A A i0 A jk
(F ) = A˙ (t), (F ) = (F ) .
A i0 i A jk A jk
6 H.N. S´a Earp
The periods of ϑ are then of the form
(cid:90) 1 (cid:90)
ϑ(g.A)−ϑ(A) = ρ (A˙(t))dt = tr(F ∧A˙ (t)dxi)∧dt∧∗ϕ
A(t) A(t) i
0 Y×[0,1]
(cid:90)
= trF ∧F ∧∗ϕ = 1 (cid:10)c (E ) (cid:96) [∗ϕ],Y ×S1(cid:11).
A A 8π2 2 g
Y×S1
The Ku¨nneth formula for Y ×S1 gives
H4(cid:0)Y ×S1,R(cid:1) = H4(Y,R)⊕H3(Y,R)⊗H1(cid:0)S1,R(cid:1)
(cid:124) (cid:123)(cid:122) (cid:125)
Z
and obviously H4(Y) (cid:96) [∗ϕ] = 0 so, denoting by c(cid:48)(E ) the component lying in H3(Y) and by
2 g
S := [ 1 c(cid:48)(E )]PD its normalised Poincar´e dual, we are left with
g 8π2 2 g
ϑ(g.A)−ϑ(A) = (cid:104)[∗ϕ],S (cid:105).
g
Consequently, the periods of ϑ lie in the set
(cid:40)(cid:90) (cid:12) (cid:41)
∗ϕ(cid:12)(cid:12)Sg ∈ H4(Y,R) .
(cid:12)
Sg
That may seem odd at first, because ∗ϕ 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 M =
Crit(ρ) of G -instantons, there is not much to worry, for the gradient ρ = dϑ is unambiguously
2
defined on B.
3 Instantons over G -torus fibrations
2
Instances of G -manifolds fibred by associative submanifolds in the literature are relatively
2
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 T7 = T4 ×T3
and K3×T3 and also CY3 ×S1 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 G -instantons to
2
their gauge theory over the base; in particular we do not address the possibility of singular
fibres.
Definition 1. A G -manifold (Y7,ϕ) is called an associative fibration over a compact oriented
2
Riemannianfour-manifold(X4,η)ifitisthetotalspaceofaRiemanniansubmersionf : Y → X
such that each fibre P := f−1(x) ⊂ Y is a smooth associative submanifold.
x
SinceeachfibreP is3-dimensionalandorientable, itstangentbundleisdifferentiablytrivial
x
and we may choose global coordinates t = (t1,t2,t3) induced respectively by a global coframe
{e ,e ,e } := {dt1,dt2,dt3}. Thus near each y ∈ P we may complete the triplet into a local
5 6 7 x
orthogonal coframe {e ,...,e } of T∗Y such that ϕ has the form (1), and the point y is
1 7 y
unambiguously described by (x,t(y)).
Lemma 1. Let f : Y → X define an associative fibration and E → Y be the pullback from
a vector bundle E → X; then a connection A on E is self-dual if, and only if, f∗A is a G -
2
instanton on E.
Generalised Chern–Simons Theory and G -instantons over Associative Fibrations 7
2
Proof. Let F := (F ) be the curvature 2-form at y ∈ P ; then
f∗A y x
∗ (F ∧ϕ) l=oc ∗ (cid:2)F ∧(cid:0)ϕ| +ω ∧e5+ω ∧e6+ω ∧e7(cid:1)(cid:3)=∗ F+∗ (cid:2)O(F−)∧f∗dVol (cid:3),
ϕ ϕ Px 1 2 3 η ϕ η
where O(F−) := (F − F )e5 + (F − F )e6 + (F − F )e7 vanishes precisely when A is
34 12 42 13 23 14
self-dual, i.e., when F = ∗ F satisfies the G -instanton equation (6b). (cid:4)
η 2
WearenowinpositiontoproveTheorem1. LetusexaminethegeneralformofaG -instanton
2
on E. An arbitrary connection A on E is locally of the form
3
A(y) l=oc A (x)+(cid:88)σ (x,t)dti,
t i
i=1
where {A } is a family of connections on E and σ ∈ Ω0(Y,f∗g ). The curvature of A is
t t∈t(Px) i E
3 (cid:18) (cid:19)
F = F +(cid:88) d σ − ∂At ∧dti+F
A At At i ∂ti σ
i=1
with
3 (cid:18) (cid:19)
F := (cid:88) ∂σi − ∂σj + 1[σ ,σ ] dti∧dtj.
σ ∂tj ∂ti 2 i j
i,j=1
Replacing F into the G -instanton equation (6a) and using the expression (3) of ∗ϕ in the
A 2
natural frame {e ,...,e }, we have
1 7
(cid:32) 3 (cid:33)
F +(cid:88)(d σ − ∂At)∧e4+i+F ∧(cid:0)e1234−ω ∧e67−ω ∧e75−ω ∧e56(cid:1) = 0.
At At i ∂ti σ 1 2 3
i=1
Using the following elementary properties
F ∧e1234 = 0, F ∧ω ∧e67 = [(F ) −(F ) ](cid:0)∗e5(cid:1),
At At 1 At 34 At 12
F ∧ω ∧e75 = [(F ) −(F ) ](cid:0)∗e6(cid:1), F ∧ω ∧e56 = [(F ) −(F ) ](cid:0)∗e7(cid:1),
At 2 At 42 At 13 At 3 At 23 At 14
F ∧e4+i∧e4+j = 0, F ∧e1234 = (F ) (cid:0)∗e5(cid:1)+(F ) (cid:0)∗e6(cid:1)+(F ) (cid:0)∗e7(cid:1),
σ σ σ 23 σ 31 σ 12
and the fact that each d σ and ∂At are locally 1-forms on the base, hence their wedge product
At i ∂ti
with e1234 = dVol vanishes, the equation simplifies to
η
3 (cid:18) (cid:19)
(cid:88) d σ − ∂At ∧ω = 0 and F− −Q(F ) = 0,
At i ∂ti i At σ
i=1
where Q is the linear map on 2-forms defined by
3
Q(cid:0)dti∧dtj(cid:1) = Q(cid:0)e4+i∧e4+j(cid:1) := (cid:88)(cid:15)ijkω .
k
k=1
(cid:80)
On the other hand, if A = A + σ is a G -instanton, then it minimises the Yang–Mills
t i 2
functional (5). This implies
(cid:88)(cid:13)(cid:13)(cid:13)(cid:13)dAtσi− ∂∂Atit(cid:13)(cid:13)(cid:13)(cid:13)2+(cid:107)Fσ(cid:107)2 = 0,
8 H.N. S´a Earp
since otherwise the pullback component A alone would violate the minimum energy:
t
YM(A ) = (cid:107)F (cid:107)2 < (cid:107)F (cid:107)2 = YM(A).
t At A
In particular F ≡ 0 and so every A must be SD. Finally, if the family A ≡ A is constant,
σ t t t0
then d σ = 0 implies σ ≡ 0, since by assumption E is indecomposable and therefore does not
At0 i
admit nontrivial parallel sections. This concludes the proof of Theorem 1.
Remark 1. If M4 is discrete, then by continuity the family {A } is contained in a gauge orbit;
+ t
if the family is constant, then A is a pullback.
3.2 G -torus fibrations
2
A 7-torus T7 = R7/Λ naturally inherits the G -structure ϕ from R7. Recall from Section 2.2
2
that a connection A on some bundle over T7 is a G -instanton if and only if it is a zero of the
2
Chern–Simons 1-form (7):
(cid:90)
ρ (b) = tr(F ∧b)∧∗ϕ. (8)
A A
T7
One asks what is the behaviour of the moduli space of G -instantons under perturbations
2
ϕ → ϕ+φ of the G -structure. More precisely, given suitable assumptions, one asks whether
2
(ϕ+φ)-instantons exist at all once we deform the lattice. As a working example, we consider
the following class of flat T3-fibred 7-tori:
Definition 2. A G -torus fibration structure is a triplet (η,L,α) in which:
2
• η is a metric on R4;
• L is a lattice on the subspace Λ2(R4,η) of η-self-dual 2-forms;
+
• α : R4 → Λ2(R4,η) is a linear map.
+
.
Given the above data, set V = R4⊕Λ2 and form the torus T = V/L˜, with the lattice
+
L˜ =. (cid:8)(µ,ν +αµ)|µ ∈ Z4, ν ∈ L(cid:9) ⊂ V.
Then T inherits from V the G -structure ϕ which makes the generators of L˜ orthonormal with
2
respect to the induced inner-product (2). It is straightforward to check that T is an associative
fibrationasinDefinition1: denotingbye5,e6,e7 the(ν+αµ)-orthonormalbasisofthefibreΛ2,
+
the flat G -structure (1) simplifies to ϕ| = e567 = dVol | ; moreover the lattice L˜ on every
2 Λ2 ϕ Λ2
+ +
tangent subspace normal to the fibre is just the lattice µ from the base, so the corresponding
metrics are the same. Although T fibres over the 4-torus R4/µ, the induced metric g(ϕ) is not,
in general, a Riemannian product.
Suppose the moduli space M4 of self-dual connections on E → T4 is nonempty; then we
+
have trivial solutions to the G -instanton equation on the pullback E → T simply by lifting M4
2 +
as in Lemma 1, which proves the first part of Theorem 2:
Corollary 1. If A is a self-dual connection on E → T4, then its pullback f∗A by the fibration
map f : T → T4 is a G -instanton on E.
2
For future reference, I denote the set of such ϕ-instantons obtained by lifts from M4 by
+
M(cid:103)4 := f∗M4 ⊂ B7. (9)
+ +
We know from 4-dimensional gauge theory that SD connections on a complex vector bundle
E → T4 correspond to stable holomorphic structures on E, thus in such cases we have examples
of G -instantons on bundles over T.
2
Generalised Chern–Simons Theory and G -instantons over Associative Fibrations 9
2
3.3 Deformations of T
WorkingonabundleE → TwithcompactstructuregroupoverafixedG -torusfibration,letus
2
ponder in generality about the behaviour of instantons under a deformation of the G -structure:
2
ϕ → ϕ+φ, ∗ ϕ → ∗ ϕ+ξ , ξ := ∗ (ϕ+φ)−∗ ϕ ∈ Ω4(T).
ϕ ϕ φ φ ϕ+φ ϕ
An arbitrary deformation φ does not in general preserve the fibred structure of T:
Proposition 1. A deformation ξ ∈ Λ4(T) of the coassociative 4-form ∗ ϕ has four orthogonal
φ ϕ
components, with the following significance:
Λ4(cid:0)R4⊕Λ2(cid:1) = Λ4(cid:0)R4(cid:1) ⊕ Λ3(cid:0)R4(cid:1)⊗Λ1(cid:0)Λ2(cid:1) ⊕ Λ2(cid:0)R4(cid:1)⊗Λ2(cid:0)Λ2(cid:1) ⊕ Λ1(cid:0)R4(cid:1)⊗Λ3(cid:0)Λ2(cid:1),
+ + + +
(cid:124) (cid:123)(cid:122) (cid:125) (cid:124) (cid:123)(cid:122) (cid:125) (cid:124) (cid:123)(cid:122) (cid:125) (cid:124) (cid:123)(cid:122) (cid:125)
(I) (II) (III) (IV)
(I) corresponds to a rescaling of the metric η on R4;
(II) redefines the map α;
(III) splits as Hom(Λ2,Λ2)⊕Hom(Λ2,Λ2), where the first factor modifies the lattice L and
+ + − +
the second one affects the conformal class of η;
(IV) parametrises deformations transverse to the fibred structures.
Proof. Let us examine the four cases.
(I) If ξ ∈ Λ4(R4) (cid:39) R, then it must be a multiple of ∗ϕ| = e1234 = dVol .
φ R4 η
(II) Since Λ3(R4)⊗Λ1(Λ2) (cid:39) R4 ⊗(Λ2)∗ (cid:39) Hom(R4,Λ2), such deformations are precisely
+ + +
linear maps R4 → Λ2.
+
(III) Clearly Λ2(R4) = Λ2 ⊕Λ2 and Λ2(Λ2) (cid:39) (Λ2)∗, so the product decomposes as
+ − + +
(cid:0)Λ2 ⊗(cid:0)Λ2(cid:1)∗(cid:1)⊕(cid:0)Λ2 ⊗(cid:0)Λ2(cid:1)∗(cid:1) (cid:39) Hom(cid:0)Λ2,Λ2(cid:1)⊕Hom(cid:0)Λ2,Λ2(cid:1).
+ + − + + + − +
Now, on one hand, acting with an endomorphism on Λ2 is equivalent to redefining the triplet
+
{e5,e6,e7}, hence the lattice L ⊂ Λ2. On the other hand, since the orthogonal split Λ2 =
+
Λ2 ⊕Λ2 is conformally invariant, a map Λ2 → Λ2 redefines the orthogonal complement of Λ2
− + − + −
and hence the conformal class.
(IV) Since Λ3(Λ2) (cid:39) R, this component is just Λ1(R4), which is irreducible in the sense
+
that T has no distinguished subspaces in R4. Then either every 7-torus is a G -fibration, which
2
is obviously false, or these are precisely the deformations away from said structures. (cid:4)
Wewillnowdescribewhathappenstothezeroesof (8)underthecorrespondingperturbation
of the Chern–Simons 1-form:
(cid:90)
ρ → ρ := ρ+r , (r ) (b) = tr(F ∧b)∧ξ .
φ φ φ A A φ
T
Clearly a ϕ-instanton A is also a (ϕ + φ)-instanton if and only if (r ) ≡ 0. There is little
φ A
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.
DenotinghenceforthbyAthespaceofconnectionsoverthe7-manifoldT,letusbrieflydigress
into the translation action of some vector v ∈ T on some A ∈ A. The first order variation is
given by the bundle-valued 1-form
(β ) := v(cid:121)F ,
v A A
which we interpret as a vector in T A. Notice first that in the direction β the value of the
A v
Chern–Simons 1-form is independent of the base-point:
10 H.N. S´a Earp
Lemma 2. The function ρ(β ) : A → R is constant.
v
Proof. The computation is straightforward:
(cid:90) (cid:90)
1
ρ(β ) = trF ∧v(cid:121)F ∧∗ϕ = − trF ∧F ∧(v(cid:121)∗ϕ)
v A+ha A+ha A+ha A+ha A+ha
2
T T
(cid:90) (cid:90)
1 1
= − (trF ∧F +dχ)∧(v(cid:121)∗ϕ) = − trF ∧F ∧(v(cid:121)∗ϕ) = ρ(β ) ,
A A A A v A
2 2
T T
where dχ is the exact differential given by Chern–Weil theory and we use Stokes’ theorem and
Cartan’s identity d(v(cid:121)∗ϕ) = L (∗ϕ) = 0, since ϕ is constant on the flat torus. (cid:4)
v
Similarly, evaluating r on β gives
φ v
(cid:90) (cid:90)
r (β ) = tr(F ∧(β ) )∧ξ = −1 tr(F ∧F )∧(v(cid:121)ξ ) = (cid:104)c (E),S (v)(cid:105),
φ v A A v A φ 2 A A φ 2 φ
T T
.
where S (v) = −1[v(cid:121)ξ ]PD, and this depends only on the topology of E, not on the point A.
φ 2 φ
Remark 2. Hence we may interpret φ as defining a linear functional
N : R7 → R,
φ
v (cid:55)→ (cid:104)c (E),S (v)(cid:105),
2 φ
such that N (cid:54)= 0 implies no ϕ-instanton is still a (ϕ+φ)-instanton. This is, however, a rather
φ
weak obstruction, since the map φ (cid:55)→ N has kernel of dimension at least 28 and thus, in
φ
principle, leaves plenty of possibilities for instantons of perturbed G -structures.
2
Now consider specifically a translation vector on the base v ∈ T4. Notice that for deforma-
tions φ of types (I), (II) or (III) the contraction of ξ with such v gives S (v) = 0, so φ only
φ φ
effectivelycontributestothefunctionρ(β )whenξ ∈ Λ1(R4),whichmeanstheperturbedtorus
v φ
is no longer a fibred structure (Proposition 1). Moreover, either the bundle E is flat and β
v
vanishes identically, or c (E) (cid:54)= 0 and the following holds:
2
Lemma 3. If c (E) (cid:54)= 0 and φ is of type (IV), then there exists v ∈ T4 such that r (β ) is
2 φ v
a non-zero constant.
Proof. Denoting T3 the typical fibre of f (and setting Vol(T3) = 1), we may assume
ξ = −2ε∧dVol
φ T3
for some 0 (cid:54)= ε ∈ Λ1(T4). One can always choose v ∈ T4 such that ε(v) (cid:54)= 0, and consider
(β ) = v(cid:121)F . Then
v A A
(cid:90) (cid:90)
r (β ) = −2 tr(F ∧v(cid:121)F )∧ε∧dVol = −2 tr(F ∧v(cid:121)F )∧ε = ε(v)·c (E),
φ v A A A T3 A A 2
T T4
which is nonzero by assumption. (cid:4)
So far we know from Corollary 1 that the set M4 of self-dual connections (modulo gauge)
+
over T4 lifts to instantons (cf. (9)) of the original G -structure ϕ (i.e. to zeroes of ρ). However,
2
for bundles with non-trivial c , this generic case degenerates precisely under deformations of
2
type (IV):
Proposition 2. Let E → (T,ϕ) be the pullback of a stable SU(n)-bundle E over T4 with
c (E) (cid:54)= 0; then E admits no (ϕ + φ)-instantons, for any perturbation φ away from a fibred
2
structure (i.e. of type (IV) in Proposition 1).