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).