A flow approach to special holonomy Hartmut Wei LMU M unchen - PowerPoint PPT Presentation

A flow approach to special holonomy Hartmut Weiß LMU M¨ unchen EMS/DMF Joint Mathematical Weekend, Aarhus

Talk based on ◮ A heat flow for special metrics joint with F. Witt, Adv. Math. 231, 2012, no. 6 ◮ Energy functionals and soliton equations for G 2 -forms joint with F. Witt, Ann. Global Anal. Geom. 42, 2012, no. 4 ◮ A spinorial energy functional: critical points and gradient flow joint with B. Ammann and F. Witt, arXiv:1207.3529 ◮ The spinor flow on surfaces joint with B. Ammann and F. Witt, in preparation

The holonomy group of a Riemannian manifold Definition Let ( M , g ) be a Riemannian manifold. The Riemannian metric g determines the Levi-Civita connection ∇ g : Γ( M , TM ) → Γ( M , T ∗ M ⊗ TM ) , Y �→ ( X �→ ∇ g X Y ) and hence a linear isometry P γ : T x 0 M → T x 1 M for any path γ from x 0 to x 1 , the parallel transport along γ . Then Hol( M , g ) := { P γ ∈ O ( T x 0 M ) : γ loop in x 0 } is the holonomy group of ( M , g ) and Hol 0 ( M , g ) := { P γ ∈ SO ( T x 0 M ) : γ nullhomotopic } ⊂ Hol( M , g ) the reduced holonomy group.

The holonomy group of a Riemannian manifold Berger’s list If ( M , g ) irreducible, simply-connected and non-symmetric, then Hol( M , g ) is one of the following: Hol( M , g ) dim M geometry SO ( n ) n generic U ( m ) 2 m K¨ ahler SU ( m ) 2 m Calabi-Yau (Ricci-flat) Sp ( k ) 4 k hyperk¨ ahler (Ricci-flat) Sp (1) Sp ( k ) 4 k quaternion-K¨ ahler (Einstein) 7 G 2 (Ricci-flat) G 2 Spin 7 8 Spin 7 (Ricci-flat) Hitchin: ∃ parallel unit spinor ⇔ g Ricci-flat and of special holonomy ⇔ Hol( M , g ) = SU ( m ) , Sp ( k ) , G 2 or Spin 7

The spinorial energy functional Definition Let M be a compact spin manifold, n = dim M ≥ 2. ◮ g a Riemannian metric � Σ g M → M , the complex g -spinor bundle, typical fiber: the complex spinor module Σ n ◮ Σ M → M the universal spinor bundle , typical fiber: the vector + + n / Spin n ∼ bundle ( � n × Σ n ) / Spin n → � = ⊙ 2 + R n ∗ , which GL GL carries a connection, the Bourguignon-Gauduchon connection ◮ 1 : 1 Correspondence → g ∈ Γ( ⊙ 2 + T ∗ M ) , ϕ ∈ Γ(Σ g M ) Φ ∈ Γ(Σ M ) ← ◮ g t path of Riemannian metrics � horizontal lift Φ t = ( g t , ϕ t ) ∈ Γ(Σ M ) using Bourguignon-Gauduchon ◮ Geometric interpretation of parallel transport provided by generalized cylinder construction of B¨ ar-Gauduchon-Moroianu

The spinorial energy functional Definition Let M be a compact spin manifold, n = dim M ≥ 2. ◮ �· , ·� = Re h ( · , · ) real inner product on spinors ◮ S (Σ M ) = { Φ x = ( g x , ϕ x ) ∈ Σ M : � ϕ x , ϕ x � = 1 } ◮ N = Γ( S (Σ M )), the space of unit spinors We consider the energy functional E : N − → R ≥ 0 � → 1 |∇ g ϕ | 2 Φ �− g dv g 2 M where Φ = ( g , ϕ ) as above

The spinorial energy functional Symmetries ◮ Diffeomorphism Invariance F spin-diffeomorphism ⇒ E ( F ∗ Φ) = E (Φ) ◮ Scaling λ ∈ R ⇒ E ( λ 2 g , ϕ ) = λ n − 2 E ( g , ϕ ) ◮ Representation theory L : Σ n → Σ n Spin n -equivariant isometry ⇒ E ( g , L ( ϕ )) = E ( g , ϕ ) Example: Σ n = Σ R n ⊗ R C ⇔ ∃ real structure J : Σ n → Σ n

The spinorial energy functional The gradient For ( g , ϕ ) ∈ N consider the subbundle ϕ ⊥ = { ˙ ϕ x ∈ Σ g M : � ϕ x , ˙ ϕ x � = 0 } Using the Gauduchon-Bourguignon connection we split T ( g ,ϕ ) N = Γ( ⊙ 2 T ∗ M ) ⊕ Γ( ϕ ⊥ ) Consider negative gradient of E : N → R in L 2 -sense − grad E ( g , ϕ ) =: Q ( g , ϕ ) = ( Q 1 ( g , ϕ ) , Q 2 ( g , ϕ )) with Q 1 ( g , ϕ ) ∈ Γ( ⊙ 2 T ∗ M ) and Q 2 ( g , ϕ ) ∈ Γ( ϕ ⊥ ), i.e. � − D g ,ϕ E ( ˙ g ) g + � Q 2 ( g , ϕ ) , ˙ ϕ � dv g g , ˙ ϕ ) = ( Q 1 ( g , ϕ ) g , ˙ M

The spinorial energy functional The gradient Theorem (Ammann-W.-Witt) 4 |∇ g ϕ | 2 Q 1 ( g , ϕ ) = − 1 g g − 1 4 div g T g ,ϕ + 1 2 �∇ g ϕ ⊗ ∇ g ϕ � Q 2 ( g , ϕ ) = −∇ g ∗ ∇ g ϕ + |∇ g ϕ | 2 g ϕ where ◮ T g ,ϕ ∈ Γ( T ∗ M ⊗ ⊙ 2 T ∗ M ) is the symmetrization in the second and third component of the 3 -tensor ( X , Y , Z ) �→ � ( X ∧ Y ) · ϕ, ∇ g Z ϕ � ◮ �∇ g ϕ ⊗ ∇ g ϕ � is the symmetric 2 -tensor defined by ( X , Y ) �→ �∇ g X ϕ, ∇ g Y ϕ �

The spinorial energy functional Critical points ( n ≥ 3) Taking the trace of the first component yields − 4 Tr g Q 1 ( g , ϕ ) = Tr g div g T g ,ϕ + ( n − 2) |∇ g ϕ | 2 g , in particular � � |∇ g ϕ | 2 − 4 Tr g Q 1 ( g , ϕ ) dv g = ( n − 2) g dv g . M M Corollary Let n ≥ 3 . Then ( g , ϕ ) is critical ⇔ ∇ g ϕ = 0 , in particular g is Ricci-flat and of special holonomy. ϕ is a g -Killing spinor with constant λ ∈ R if ∇ g X ϕ = λ X · ϕ for all X ∈ Γ( TM ). Killing spinors are critical points under the constraint vol ( M , g ) = 1.

The spinorial energy functional Critical points ( n = 2) The functional is scale invariant in this dimension. Hence ( g , ϕ ) critical point ⇔ ( g , ϕ ) constrained critical point Theorem (Ammann-W.-Witt) Let n = 2 . Then ◮ χ ( M ) > 0 : ( g , ϕ ) is critical ⇔ ( g , ϕ ) is a global minimum ⇔ ϕ = cos ϑ ψ + sin ϑ ω · ψ for a g-Killing spinor ψ ( ω the real volume element, ϑ ∈ R ) ◮ χ ( M ) = 0 : ( g , ϕ ) is a global minimum ⇔ ∇ g ϕ = 0 ◮ χ ( M ) < 0 : ( g , ϕ ) is a global minimum ⇔ D g ϕ = 0

The spinor flow Short-time existence and uniqueness Consider the spinor flow with initial condition Φ ∈ N ∂ t Φ t = Q (Φ t ) , Φ 0 = Φ for time-dependent family Φ t = ( g t , ϕ t ) ∈ N , t ≥ 0. Theorem (Ammann-W.-Witt) The spinor flow has a unique short-time solution. Uniqueness implies: All symmetries are preserved under the flow. Ingredients of proof: ◮ σ ξ ( D Ω Q ) ≥ 0 for all ξ ∈ T ∗ M ◮ ker σ ξ ( D Ω Q ) precisely coming from diffeomorphism invariance ◮ DeTurck trick: ˜ Q (Φ) := Q (Φ) + L X (Φ) Φ for X (Φ) a cleverly chosen vector field

G 2 -geometry Locally Let Ω 0 := e 127 + e 347 + e 567 + e 135 − e 146 − e 236 − e 245 ∈ Λ 3 R 7 ∗ where e ijk := e i ∧ e j ∧ e k . Then G 2 := { A ∈ GL 7 : A ∗ Ω 0 = Ω 0 } ⊂ SO (7) i.e. G 2 preserves Euclidean metric and standard orientation on R 7 . + R 7 ∗ := GL + Λ 3 7 -orbit of Ω 0 ∼ = GL + 7 / G 2 + R 7 ∗ ⊂ Λ 3 R 7 ∗ is The orbit Λ 3 ◮ open (dim Λ 3 R 7 ∗ = 35 = 49 − 14 = dim GL + 7 − dim G 2 ) ◮ a positive cone (Ω ∈ Λ 3 + R 7 ∗ , λ > 0 ⇒ λ Ω ∈ Λ 3 + R 7 ∗ )

G 2 -geometry Globally Let M 7 be compact and oriented. Set Λ 3 + T ∗ M := P GL + 7 Λ 3 + R 7 ∗ 7 × GL + A section Ω ∈ Γ( M , Λ 3 + T ∗ M ) =: Ω 3 + ( M ) is called positive 3-form. Ω ∈ Ω 3 + ( M ) ← → reduction of structure group of TM from GL + 7 to G 2 ⊂ SO (7) In particular: Ω � metric quantities g Ω , ⋆ Ω , vol Ω , . . . Hol( M , g Ω ) ⊂ G 2 ⇐ = = = = ⇒ d Ω = d ⋆ Ω Ω = 0 Fernandez , Gray Ric g Ω = 0 ⇒ = = = = = Bonan Ω ∈ Ω 3 + ( M ) satisfying d Ω = d ⋆ Ω Ω = 0 is called torsion-free .

The G 2 -flow Definition Let M be a compact, oriented 7-manifold and Ω 3 + ( M ) the space of positive 3-forms on M . Consider D : Ω 3 + ( M ) − → R ≥ 0 � {| d Ω | 2 Ω + | d ⋆ Ω Ω | 2 → 1 Ω �− Ω } vol Ω 2 M Properties of D : ◮ Diff + ( M )-invariant ◮ positively homogenous ( D ( λ Ω) = λ 5 / 3 D (Ω) for λ > 0) ◮ Ω critical w.r.t. D ⇔ Ω torsion-free Let Q (Ω) := − grad D (Ω) be the negative L 2 -gradient of D .

The G 2 -flow Short-time existence Consider the G 2 -flow with initial condition Ω ∈ Ω 3 + ( M ) ∂ t Ω t = Q (Ω t ) , Ω 0 = Ω for time-dependent family Ω t ∈ Ω 3 + ( M ), t ≥ 0. Theorem (W.-Witt) The G 2 -flow has a unique short-time solution. Ingredients of proof: ◮ σ ξ ( D Ω Q ) ≥ 0 for all ξ ∈ T ∗ M ◮ ker σ ξ ( D Ω Q ) precisely coming from diffeomorphism invariance ◮ DeTurck trick: ˜ Q (Ω) := Q (Ω) + L X (Ω) Ω for X (Ω) a cleverly chosen vector field

The G 2 -flow Stability The G 2 -flow is stable near a critical point. More precisely: Theorem (W.-Witt) Let ¯ Ω ∈ Ω 3 + ( M ) be torsion-free. Then for any initial condition sufficiently close to ¯ Ω in the C ∞ -topology the G 2 -flow exists for all times and converges modulo diffeomorphisms to a torsion-free positive 3 -form on M. Ingredients of proof: ◮ linear stability ◮ integrability of infinitesimal deformations ◮ compare nonlinear evolution with solution of linearized equation (estimates!)

The G 2 -flow Spinorial reformulation Let Σ n be the complex spin representation of Spin n . Representation theory: Σ 7 = Σ R 7 ⊗ R C , dim R Σ R 7 = 8. Basic facts: 7 ) ∼ = S 7 ◮ Spin 7 acts transitively on S (Σ R 7 ) ∼ ◮ S (Σ R = Spin 7 / G 2 + R 7 ∗ ∼ ◮ Λ 3 = GL + 7 / G 2 ≃ R P 7 1 : 1 Correspondence Ω ← → spin structure , g , {± ϕ } Then � � |∇ g ϕ | 2 vol g + scal g vol g D (Ω) = 8 M M