Overview Background Setting Examples Deformation theory Functional Questions Deformed G2-instantons Jason D. Lotay Oxford 3 September 2020 (Joint work with Goncalo Oliveira) Overview Background Setting Examples Deformation theory Functional Questions Overview Deformed G2-instantons special connections in 7 dimensions \mirror" to calibrated cycles enumerative invariants? critical points of Chern{Simons-type functional F Results first non-trivial examples examples detect different G2-structures (including nearly parallel and isometric) deformation theory: obstructions and topology of moduli space relation to F Overview Background Setting Examples Deformation theory Functional Questions Key example: the 7-sphere Hopf fibration: S3 !S7 !S4 ts round metric g = gS3 + gS4 2 \canonical variation" gt = t gS3 + gS4 for t > 0 ts np p gt Einstein , g = g1 or g = g1= 5 7 Octonions: S ⊆ O cross product × 3-form ts ts ' (u; v; w) = g (u × v; w)G 2-structure Fact: 'ts determines g ts d'ts = λ ∗ 'ts for λ > 0 constant $ nearly parallel Note: d ∗ 'ts = 0 Overview Background Setting Examples Deformation theory Functional Questions Deformed G2-instantons 7 X , ' G2-structure, d ∗ ' = 0, connection A on bundle over X Definition (J.-H. Lee{N.C. Leung) A deformed G2-instanton (dG2) , curvature FA satisfies 1 3 FA ^ ∗' + 6 FA = 0 7 3 4 3 4 R = R ⊕ R , x = (x1; x2; x3) 2 R , y = (y0; y1; y2; y3) 2 R , '(u; v; w) = gR7 (u × v; w) \mirror branes" A = i(aj (x)dxj + uk (x)dyk ) dG2-instanton , L = Graph(u) associative 'jL = volL and i(aj dxj ) flat A = i(vj (y)dxj + bk (y)dyk ) dG2-instanton , M = Graph(v) coassociative ∗'jM = volM and B = i(bk dyk ) anti-self-dual (ASD) FB = − ∗ FB primary interest in U(1)-connections Overview Background Setting Examples Deformation theory Functional Questions Lower dimensions 4 dimensions: π : X 7 ! Z 4, Z ASD Einstein, connection B on Z Lemma ∗ π B dG2-instanton , B ASD 4 7 ASD-instantons on S dG2-instantons on S 6 dimensions: π : X 7 ! Y 6, Y Calabi{Yau 3-fold Hol ⊆ SU(3) ! K¨ahlerform on Y , connection B on Y Lemma ∗ π B dG2-instanton , B deformed Hermitian{Yang{Mills (0;2) 3 FB = 0 and Im (! + FB ) = 0: Conjecture: existence of dHYM $ stability condition Overview Background Setting Examples Deformation theory Functional Questions G2-structures on the 7-sphere Recall: S3 !S7 !S4, 3-form 'ts inducing round metric g ts 3 S = SU(2) left-invariant coframe η1; η2; η3 4 !1;!2;!3 orthogonal self-dual 2-forms on S with length 2 two 1-parameter families of 3-forms for t > 0: ± 3 't = ±t η1 ^ η2 ^ η3 − tη1 ^ !1 − tη2 ^ !2 ∓ tη3 ^ !3 ± 2 + − 't induces gt = t gS3 + gS4 ) 't and 't isometric Lemma ± d ∗ 't = 0 '± nearly parallel , 'ts = '− or 'np = '+ p t 1 1= 5 Note: 'np induces \squashed" Einstein metric g np Overview Background Setting Examples Deformation theory Functional Questions 3-Sasakian 7-manifolds Definition 7 ts + 7 2 2 ts (X ; g ) 3-Sasakian , cone (R × X ; g = dr + r g ) 7 ts hyperk¨ahler Hol(g) ⊆ Sp(2) generalizes (S ; g ) Fact: 9 infinitely many 3-Sasakian 7-manifolds V 3 ! X 7 ! Z 4, V = SU(2)=Γ, Z ASD Einstein 2 gt = t gV + gZ and ± 3 't = ±t η1 ^ η2 ^ η3 − tη1 ^ !1 − tη2 ^ !2 ∓ tη3 ^ !3 ts − ts ' = '1 nearly parallel inducing g 'np = '+ p nearly parallel inducing g np \squashed" Einstein 1= 5 metric, cone has Hol = Spin(7) Example: Aloff–Wallach space SU(3) × SU(2)= U(1) × SU(2) Overview Background Setting Examples Deformation theory Functional Questions Non-trivial examples 7 ts ± (X ; g ) 3-Sasakian, 't inducing gt 3 a = (a1; a2; a3) 2 R A = i(a1η1 + a2η2 + a3η3) connection on trivial line bundle, r = jaj \distance to trivial connection" Theorem (L.{Oliveira) 7 ± A dG2-instanton on (X ;'t ) , r r + − 't 't 'np 'ts S1 S2 2 points point point p1 t p1 t 2 2 Overview Background Setting Examples Deformation theory Functional Questions Observations Proof: explicit calculation solve quadratic equations r r + − 't 't 'np 'ts S1 S2 2 points point point p1 t p1 t 2 2 + − distinct solution spaces for isometric 't and 't distinct solution spaces for nearly parallel 'ts and 'np Can construct examples on non-trivial line bundles on Aloff–Wallach space with similar behaviour Overview Background Setting Examples Deformation theory Functional Questions Deformation theory: obstructions Key question: is deformation theory unobstructed or obstructed? Unobstructed: moduli space locally smooth manifold of expected dimension, i.e. linearised dG2-instanton operator L surjective 1 2 1 5 L = ( 2 FA + ∗') ^ d :Ω ! dΩ infinitesimal deformations guaranteed to extend to deformations Obstructed: L not surjective some infinitesimal deformations may not extend Theorem (L.{Oliveira) ts np Non-trivial dG2-instantons constructed for ' and ' are obstructed ts np Trivial dG2-instanton unobstructed for ' and ' but obstructed for '± p 1= 2 Overview Background Setting Examples Deformation theory Functional Questions Deformation theory: moduli space Proof: (Kawai{Yamamoto) unobstructed , rigid and isolated r r + − 't 't 'np 'ts S1 S2 2 points point point p1 t p1 t 2 2 At t = p1 : 9 infinitesimal deformation of trivial dG -instanton ) 2 2 obstructed Corollary ts Moduli space of dG2-instantons on trivial line bundle for ' and 'np contains at least two components of different dimensions Overview Background Setting Examples Deformation theory Functional Questions Chern{Simons-type functional 7 A0 reference connection on line bundle L on (X ;') A connection on L A = A0 + s(A − A0) connection on L over X × [0; 1] curvature F Proposition (Karigiannis{N.C. Leung) A dG2-instanton , A critical point of functional Z F(A) = eF+∗' X ×[0;1] Recall: our examples A = i(a1η1 + a2η2 + a3η3) restriction of F is function of two variables x and y: 2 2 2 x = a3 and y = a1 + a2 Overview Background Setting Examples Deformation theory Functional Questions + ts − F for '1 and ' = '1 with level sets trivial connection only critical point ) local maximum otherwise trivial connection saddle point non-trivial dG2-instantons are local maxima Overview Background Setting Examples Deformation theory Functional Questions F for 'np = '+ p and '− p with level sets 1= 5 1= 5 trivial connection is local minimum continuous families of dG2-instantons are local maxima two isolated examples are saddle points Overview Background Setting Examples Deformation theory Functional Questions Questions non-trivial dG2-instantons for holonomy G2-manifolds? (adiabatic) limits of dG2-instantons? dependence of moduli space on G2-structure? \mirror count"? applications of F to compactness or deformation theory? Spin(7) version?.