Quaternionic geometry in 8 dimensions Simon Salamon Differential - PowerPoint PPT Presentation

Quaternionic geometry in 8 dimensions Simon Salamon Differential Geometry in the Large In honor of Wolfgang Meyer, Florence, 14 July 2016

Four categories of manifolds 1/20 . . . all equipped with an action of I , J , K on each tangent space T x M 4 n ( n � 2) and a torsion-free G connection: Hyperk¨ ahler Hypercomplex Sp ( n ) GL ( n , H ) Quaternion-k¨ ahler Quaternionic Sp ( n ) Sp (1) GL ( n , H ) Sp (1) ‘Hypercomplex’ implies that I , J , K are complex structures. ‘Quaternionic’ implies that the tautological complex structure on the 2-sphere bundle Z ( → M ) is integrable, and the (e.g.) Fueter operator can be defined. One could also add SL ( n , H ) U (1) structures to the 2nd column.

Hyperk¨ ahler manifolds 2/20 We shall soon focus on non-integrable Sp (2) Sp (1) structures, but by way of introduction: For a hyperk¨ ahler manifold, the holonomy of the Levi-Civita connection lies in Sp ( n ), and the Ricci tensor vanishes. Calabi gave explicit complete examples on (e.g.) T ∗ CP n . Many HK metrics can be constructed via the HKLR quotient construction, and abound on moduli spaces (e.g. 8 k − 3 � 8 k ). Any K3 surface admits a HK metric by Yau’s theorem. Beauville described two families K 4 n and A 4 n of compact HK manifolds, arising from Hilbert schemes of points on a K3 or Abelian surface. They satisfy 24 | ( n χ ).

Quaternion-k¨ ahler manifolds 3/20 This time, the holonomy of the Levi-Civita connection lies in Sp ( n ) Sp (1). QK manifolds are Einstein, we assume not Ricci-flat. Curvature-wise, they are ‘nearly hyperk¨ ahler’. Wolf showed that there is a QK symmetric space (and its dual) for each compact simple Lie group G , and that its twistor spaces has a complex contact structure. This talk will focus on G 2 / SO (4) . These spaces are the only known complete QK manifolds with s > 0, but there is an incomplete metric defined by any pair su (2) ⊂ g , combining work of Kronheimer and Swann (next slide). Alekseevsky and Cort´ es have constructed families of complete non symmetric/homogeneous examples with s < 0. LeBrun had shown that there is an infinite-dimensional moduli space.

The miraculous case of G = SU(3) 4/20 Up to conjugacy, su (3) has two TDA’s: su (2) and so (3). SU (3) The first gives rise to the Wolf space CP 2 = S ( U (1) × U (2)) . The second gives rise to the Grassmannian L of special Lagrangian subspaces R 3 ⊂ C 3 , and there are SU (3)-equivariant maps: G 2 ∼ SO (4) \ CP 2 = V   obvious VB   with fibre R 3 � � SU (3) / SO (3) = L Now, Z 3 acts freely on G 2 / SO (4) \ CP 2 , and the quotient is a submanifold U of G r 3 ( su (3)) invariant under a Nahm flow. Its Swann bundle is N = { A ∈ sl (3 , C ) : A 3 = 0 , A 2 � = 0 } .

Coassociative submanifolds 5/20 The Wolf space G 2 / SO (4) parametrizes coassociative subspaces i : R 4 ⊂ R 7 . These are subspaces for which i ∗ ϕ = 0, where ϕ = e 125 − e 345 + e 136 − e 426 + e 147 − e 237 + e 567 is the standard 3-form with stabilizer G 2 . The space L parametrizes some special coassociative submanifolds L ⊥ ⊂ π − 1 ( RP 2 ) of the 7-dimensional total space Λ 2 − T ∗ CP 2   � π RP 2 ⊂ CP 2 with the Bryant-S metric with holonomy G 2 [Karigiannis-MinOo]. Moreover, G 2 / SO (4) is intimately connected with this total space.

Groups containing Sp(2) 6/20 On R 8 = H 2 ∋ ( p , q ) , define a ‘hyperk¨ ahler triple’ 1 2 ( dp ∧ dp + dq ∧ dq ) = ω 1 i + ω 2 j + ω 3 k of 2-forms  ω 1 = e 12 + e 34 + e 56 + e 78  ω 2 = e 13 + e 42 + e 57 + e 86  ω 3 = e 14 + e 23 + e 58 + e 67 . The stabilizer of Ω λ = 1 2 ( λω 1 ∧ ω 1 + ω 2 ∧ ω 2 + ω 3 ∧ ω 3 ) is Sp (2) U (1) ⊂ SU (4) except that: ◮ stab (Ω 1 ) = Sp (2) Sp (1). ◮ stab (Ω − 1 ) = Spin (7).

Closed versus parallel 7/20 A holonomy reduction occurs when ∇ Ω = 0. For the Levi-Civita connection, obviously ∇ Ω=0 ⇒ d Ω=0. ◮ If Ω=Ω − 1 has stabilizer Spin (7) then d Ω=0 ⇒ ∇ Ω = 0 [Fern´ andez-Gray]. In this case, ∇ Ω ∈ Λ 1 ⊗ g ⊥ ∼ = Λ 3 ∼ = Λ 5 . ◮ If Ω=Ω 1 has stabilizer Sp (2) Sp (1), by contrast, d Ω does not determine ∇ Ω [Swann]. It is therefore natural to generalize the class of QK manifolds to those almost-QH ones with Ω ∈ Λ 4 + closed (so harmonic) but not parallel.

Instrinic torsion for Sp(2)Sp(1) 8/20 The space Λ 1 ⊗ ( sp (2)+ sp (1)) ⊥ has 4 components: If ∇ Ω lies in. . . blue then d Ω = 0 red then ‘ideal’: d ω i = � α j i ∧ ω j , work by Mac´ ıa green then quaternionic. Corollary. (Ideal or quaternionic) and d Ω = 0 ⇒ ∇ Ω = 0

Harmonic Sp(2)Sp(1) reductions 9/20 A first example was found on M 8 = M 6 × T 2 where M 6 = Γ \ N is a symplectic nilmanifold with a pair of simple closed 3-forms, defining a ‘tri-Lagrangian geometry’. The structure group of M 6 reduces to a diagonal SO (3). There are many more examples of the form M 7 × S 1 obtained by setting Ω = α ∧ e 8 + β and using the fact that Sp (2) Sp (1) ∩ SO (7) = G ∗ 2 α ∩ G ∗ 2 β = SO (4) . [Conti-Madsen classify 11 nilmanifolds and find solvmanifolds]. Are there simply-connected examples? Theorem [CMS]. The parallel QK 4-form on G 2 / SO (4) can be ‘freely’ deformed to a closed form with stabilizer Sp (2) Sp (1) invariant by the cohomogenous-one action by SU (3).

Symmetric spaces in 8 dimensions 10/20 Apart from CP 4 (whose holonomy U (4) is not so special), there are 4 compact models which all admit a cohomogenous-one action, with principal orbits SU (3) / U (1) 1 , − 1 and two ends chosen from S 5 , CP 2 , L = SU (3) / SO (3) [Gambioli]. The first three are quaternion-k¨ ahler: CP 2 , CP 2 G r 2 ( C 4 ) SU (4) / U (2) Sp (1) HP 2 CP 2 , S 5 Sp (3) / Sp (2) Sp (1) CP 2 , L G 2 / SO (4) G 2 / SU (2) Sp (1) SU (3) 2 / ∆ S 5 , L SU (3)

Topological remarks 11/20 The Wolf spaces G r 2 ( C 4 ) , HP 2 , G 2 / SO (4) are all spin with s > 0. They satisfy � A 2 =0 and 8 χ = 4 p 2 − p 2 1 . The latter is also valid for any 8-manifold whose structure group reduces to Spin (7) , and the Wolf spaces all have such structures, but not holonomy equal to Spin (7) as this would require � A 2 =1. Nonetheless one can search for closed non-parallel 4-forms on the Wolf spaces [with motivation from Foscolo-Haskins’ construction ahler metrics on S 6 and S 3 × S 3 ]. of new nearly-k¨

Quest for the QK 4-form on V 12/20 The construction of exceptional metrics on vector bundles over 3- and 4-manifolds made use of ‘dictionaries’ of tautological differential forms. It was natural to use similar techniques to identity the parallel 4-form Ω over V , but this took a few years: Proposition [CM]. The parallel QK 4-form Ω can be expressed SU (3)-equivariantly on V as 3 sin 2 ( r ) cos 2 ( r ) √ β + sin 2 ( r ) cos 2 ( r ) 3 sin(2 r ) b ˜ a ˜ βǫ − − 5 sin(2 r )+sin(6 r )+4 r cos(2 r ) bb β + γǫǫ √ r 2 r 2 3 r 3 r 128 + sin 4 ( r )(cos(2 r )+cos(4 r )+1) √ 3(2 r cos(2 r ) − sin(2 r )) bbb a ǫ + b β a ǫ √ 8 r 3 3 r 4 2 ab ab β + sin 2 ( r )(5 r − 6 sin(2 r ) − 3 sin(4 r )+ r (13 cos(2 r )+5 cos(4 r )+cos(6 r ))) + 3(2 r sin(4 r )+cos(4 r ) − 1) ab ǫǫǫ √ 4 r 4 3 r 5 96 + sin 3 (2 r )(sin(2 r ) − 2 r cos(2 r )) abb a ǫǫ − sin 3 (2 r ) cos(2 r ) a γ a γ 32 r 6 8 r 3 √ 3 b ˜ and equals 3 bb β + 2 β when r = 0.

Letters, syllables and words 13/20 The SU (3)-invariant differential forms on V arise from forms defined on the fibre with values in the exterior algebra of the base, everything invariant by SO (3). Syllables arise by contracting letters using the inner product or the volume form on R 3 . Examples: ◮ the syllable aa equals r = � ( a i ) 2 ; = R 3 ⊕ R 7 , and the value of the syllable a β x L ) ∼ = so (5) ∼ ◮ Λ 2 ( T ∗ is the pullback of the 2-form in R 3 it represents: √ √ a 1 ( − e 12 +2 e 34 ) + a 2 ( e 13 − e 24 − 3 e 25 ) + a 3 ( e 14 + 3 e 15 − e 56 ); ◮ differentiating the a i gives b i = da i + connection forms , then bbb = b 1 ∧ b 2 ∧ b 3 and bb β = S b i ∧ b j ∧ β k ; ◮ words like bb β and bbb a ǫ of degree 4 can be formed by wedging 1 or 2 syllables together.

More invariant 4-forms 14/20 A generic SU (3)-invariant 4-form on V is k 1 bb β + k 2 b ˜ β + k 3 ab ˜ β + k 4 b γǫ + k 5 a ˜ βǫ + k 6 γǫǫ + k 7 bbb a ǫ + k 8 b β a ǫ + k 9 ab ab β + k 10 ab a γǫ + k 11 ab ǫǫǫ + k 12 abb a ǫǫ + k 13 ab β a ǫ + k 14 a γ a γ . It extends smoothly across L iff k i are smooth even functions of r . It is closed if and only if  k ′ k ′ 1 = rk 9 , 2 = 8 rk 8 ,     k 5 = 1 k 3 = k 13 = 0 , 3 k 1 , 2 rk ′ 12 + 12 k 12 − 1 r k ′  14 = 0 ,    24 1 6 + 1 r k ′ r k ′ 8 − 72 k 11 − 6 k 7 = 0

Group parameters 15/20 In order to express the parallel 4-form relative to the standard basis ( e i , b j ), we need to express the k i in terms of parameters � λ 8 � � λ 4 � λ 9 λ 5 e i λ 1 , e i λ 2 , e i λ 13 , e i λ 14 , , e i λ 3 , , λ 12 λ 10 λ 11 λ 6 λ 7 for the group U (1) 4 ⋊ GL (2 , R ) × U (1) × GL (2 , R ) × R ∗ that commutes with the U (1) stabilizer of each SU (3) orbit. Closure imposes ODE’s on the λ i ( but not λ 7 ), and we find a solution λ 1 = λ 2 = λ 3 = λ 4 = λ 9 = λ 10 = 0 , λ 12 = − 1 , √ 3 , λ 8 = 1 2 (3 − 2 cos 2 r ) cos r , λ 5 = − cos(2 r ) , λ 6 = √ , λ 13 = (1+2 cos 2 r ) sin r λ 11 = cos(2 r ) 3 sin r , 2 r 2 r √ 2 ( − 1 + 2 cos 2 r ) cos r . 3 λ 14 =