The Lorentzian index theorem and its local version Christian Br - PowerPoint PPT Presentation

The Lorentzian index theorem and its local version Christian Bär (joint with A. Strohmaier) Institut für Mathematik Universität Potsdam Quantum Physics meets Mathematics Workshop on the occasion of Klaus Fredenhagen’s 70th birthday Hamburg, 9. December 2017

Outline 1 Index theory on Riemannian manifolds The (local) Atiyah-Singer index theorem The Atiyah-Patodi-Singer index theorem Index theory on Lorentzian manifolds 2 Dirac operator on Lorentzian manifolds The Lorentzian index theorem The local Lorentzian index theorem

1. Index theory on Riemannian manifolds

Setup M Riemannian manifold, compact, without boundary spin structure � spinor bundle SM → M n = dim ( M ) even � splitting SM = S L M ⊕ S R M Hermitian vector bundle E → M with connection � twisted spinor bundle V L / R = S L / R M ⊗ E Dirac operator D : C ∞ ( M , V ) → C ∞ ( M , V ) � 0 � D R D = D L 0 Properties of D : linear differential operator of first order elliptic essentially self-adjoint D L is Fredholm, i.e. index is defined ind ( D L ) = dim ker ( D L ) − dim coker ( D L )

Heat kernel expansion D R D L is of Laplace type � heat kernel expansion ∞ � t ց 0 e − tD R D L ( x , x ) ∼ ( 4 π t ) − n / 2 a j ( x ) t j j = 0 ⇒ � ∞ � t ց 0 Tr ( e − tD R D L ) ∼ ( 4 π t ) − n / 2 tr ( a j ( x )) t j M j = 0 Similarly: � ∞ � t ց 0 Tr ( e − tD L D R ) ∼ ( 4 π t ) − n / 2 a j ( x )) t j tr (˜ M j = 0

Index computation ind ( D L ) = dim ker ( D L ) − dim ker ( D R ) = dim ker ( D R D L ) − dim ker ( D L D R ) � � e − t λ k ( D R D L ) − e − t λ k ( D L D R ) = k k = Tr ( e − tD R D L ) − Tr ( e − tD L D R ) � ∞ � ∼ ( 4 π t ) − n / 2 a j ( x ))] t j [ tr ( a j ( x )) − tr (˜ M j = 0 ⇒ � [ tr ( a n / 2 ( x )) − tr (˜ ind ( D L ) = a n / 2 ( x ))] , M � [ tr ( a j ( x )) − tr (˜ a j ( x ))] = 0 for j < n / 2 M

The index theorem Local index theorem The following holds pointwise: � for j < n / 2 0 ( 4 π ) − n / 2 [ tr ( a j ( x )) − tr (˜ a j ( x ))] = ˆ A ( M ) ∧ ch ( E ) | x for j = n / 2 Corollary (Atiyah-Singer 1968) � ˆ ind ( D L ) = A ( M ) ∧ ch ( E ) M

Setup for manifolds with boundary M Riemannian manifold, compact, with boundary ∂ M spin structure � spinor bundle SM → M n = dim ( M ) even � splitting SM = S L M ⊕ S R M Hermitian vector bundle E → M with connection � twisted spinor bundle V L / R = S L / R M ⊗ E Dirac operator D : C ∞ ( M , V ) → C ∞ ( M , V ) � 0 � D R D = D L 0 Need boundary conditions: Let A 0 be the Dirac operator on ∂ M . P + = χ [ 0 , ∞ ) ( A 0 ) = spectral projector APS-boundary conditions: P + ( f | ∂ M ) = 0

Atiyah-Patodi-Singer index theorem Theorem (Atiyah-Patodi-Singer 1975) Under APS-boundary conditions D L is Fredholm and � ˆ ind ( D APS ) = A ( M ) ∧ ch ( E ) L M � A ( M ) ∧ ch ( E )) − h ( A 0 ) + η ( A 0 ) T (ˆ + 2 ∂ M Here h ( A ) = dim ker ( A ) � sign ( λ ) · | λ | − s η ( A ) = η A ( 0 ) where η A ( s ) = λ ∈ spec ( A ) λ � = 0

2. Index theory on Lorentzian manifolds

Index theory in Lorentzian signature? Problem 1: Let D be a differential operator of order k over a closed manifold. Then D : H k → L 2 is Fredholm ⇔ D is elliptic. ⇒ no Lorentzian analog to Atiyah-Singer index theorem Problem 2: Hyperbolic PDEs behave badly on closed manifolds Problem 3: Closed Lorentzian manifolds violate causality conditions ⇒ useless as models in General Relativity But: There exists a Lorentzian analog to the Atiyah-Patodi-Singer index theorem!

Setup for Lorentzian manifolds M globally hyperbolic Lorentzian manifold with boundary ∂ M = S 1 ⊔ S 2 S j smooth compact spacelike Cauchy hypersurfaces spin structure � spinor bundle SM → M n = dim ( M ) even � splitting SM = S L M ⊕ S R M Hermitian vector bundle E → M with connection � twisted spinor bundles V L / R = S L / R M ⊗ E Dirac operator D : C ∞ ( M , V ) → C ∞ ( M , V ) (hyperbolic!) A j Dirac operator on S j (elliptic!)

The Lorentzian index theorem Theorem (B.-Strohmaier 2015) Under APS-boundary conditions D L is Fredholm. The kernel consists of smooth spinor fields and � � � T ( � ind ( D APS ) = A ( M ) ∧ ch ( E ) + A ( M ) ∧ ch ( E )) L M ∂ M − h ( A 1 ) + h ( A 2 ) + η ( A 1 ) − η ( A 2 ) 2

Original proof of the index theorem Step 1: Show that D APS is Fredholm L (microlocal analysis, FIOs) Step 2: Compute index introduce auxiliary Riemannian metric on M use spectral flow to relate the Lorentzian and the Riemannian indices apply classical APS theorem Aim: Replace step 2 by local index theorem

Hadamard type expansion On Minkowski space ( R n , �· , ·� ) define distributions ( k ∈ N 0 ): � ( � x , x � − i ε x 0 ) k + 1 − n / 2 if k + 1 < n / 2 H k = C ( k , n ) · lim � x , x � k + 1 − n / 2 log ( � x , x � − i ε x 0 ) otherwise ε ց 0

Hadamard type expansion consider as distribution on manifold near x using normal coordinates about x vary x � distribution H k defined on neighborhood U of diagonal in M × M � formal bisolution of ( D R D L ) x u ( x , y ) = ( D R D L ) ∗ y u ( x , y ) = 0 on U : ∞ � u = V k ( x , y ) H k k = 0 The Hadamard coefficients V k are recursively defined and C ∞ on U . Formally: V k ( x , x ) = a k ( x )

Hadamard solutions Schwartz kernel theorem 1 : 1 ω ∈ C −∞ ( M × M ) ω : C ∞ c ( M ) → C −∞ ( M ) ← → ˆ ω ( u )( v ) = ω ( u ⊗ v ) ˆ Definition A bidistribution ω ∈ C −∞ ( M × M ) is called bisolution if D R D L ◦ ˆ ω = ˆ ω ◦ D R D L = 0 . It is called to be of Hadamard form if n / 2 − 1 + ℓ � V k H k ∈ C ℓ ( U ) ω − ∀ ℓ ∈ N 0 . k = 0

Hadamard regularization For ω ∈ C 0 ( U , V ⊠ V ∗ ) write [ˆ ω ]( x ) := tr ( ω ( x , x )) For differential operators Q 1 , Q 2 of order m 1 , m 2 put [ Q 1 , ω, Q 2 ] reg := [ Q 1 ◦ ( ω − � N � k = 0 V k H k ) ◦ Q 2 ] with N = n / 2 − 1 + m 1 + m 2 . Proposition 1 For Hadamard bisolutions ω we have: [ D R D L , ω, 1 ] reg = [ 1 , ω, D R D L ] reg = [ V n / 2 ] , [ D L , ω, D R ] reg = [ ˜ V n / 2 ] .

Product manifolds Let M = I × S with metric − dt 2 + g . Put 2 e i ∆ 1 / 2 t ∆ 1 / 2 ( 1 − χ { 0 } (∆)) − it χ { 0 } (∆) f ( t ) = 1 where ∆ = D 2 S . Now � (ˆ ω S u )( t , · ) = f ( t − s ) u ( s , · ) ds R defines a distinguished Hadamard bisolution ω S .

Product manifolds Localize h and η : h x = [ χ { 0 } ( D S )]( x ) η x = η x ( 0 ) where � sign ( λ ) | λ | − s | Φ λ ( x ) | 2 η x ( s ) = λ � = 0 Then � � h x dx = h ( D S ) and η x dx = η ( D S ) M M Proposition 2 In the product case, for the distinguished Hadamard bisolution: ν S ] reg ( t , x ) = 1 [ D L , ω S , / 2 ( η x + h x )

Regularized Dirac current For any Hadamard bisolution ω define regularized Dirac current J ω reg ( ξ ) = [ D L , ω, / ξ ] reg J ω reg is a smooth 1-form on M . In product case, by Proposition 2: J ω S ν S ] reg = 1 reg ( ν S ) = [ D L , ω S , / 2 ( η x + h x )

Crucial computation Assume M has product metric near boundary ∂ M = S 1 ⊔ S 2 ω 1 : distinguished Hadamard bisolution near S 1 , extended to M propagation of singularities ⇒ ω 1 is Hadamard on all of M similarly for ω 2

Crucial computation � � � � � J ω 1 reg − J ω 2 D APS − ind = ( ν S ) dS (Chiral Anomaly) reg L S � � � J ω 1 reg − J ω 2 = ( ν S 1 ) dS 1 reg S 1 � � � J ω 1 J ω 2 δ J ω 2 = reg ( ν S 1 ) dS 1 − reg ( ν S 2 ) dS 2 + reg S 1 S 2 M � δ J ω 2 = 1 2 ( η ( D S 1 ) + h ( D S 1 )) − 1 2 ( η ( D S 2 ) + h ( D S 2 )) + reg M Thus δ J ω 2 reg is the (negative of) the index density.

Local index theorem Theorem (B.-Strohmaier, 2017) δ J ω 2 reg = − ˆ A ( M ) ∧ ch ( E ) pointwise. c (˚ Proof: For all f ∈ C ∞ M ) we have: � � � f · δ J ω 2 J ω 2 [ D L , ω 2 , / reg = reg ( ∇ f ) = ∇ f ] reg M M M � � [ / = ∇ f ◦ D L , ω 2 , 1 ] reg = [( D R ◦ f − f ◦ D R ) ◦ D L , ω 2 , 1 ] reg M M � � � = f · [ D L , ω 2 , D R ] reg − [ D R D L , ω 2 , 1 ] reg M � � � [ ˜ = f · V n / 2 ] − [ V n / 2 ] M ⇒ δ J ω 2 reg = [ ˜ V n / 2 − V n / 2 ] = − ˆ A ( M ) ∧ ch ( E ) .

References • C. Bär and A. Strohmaier: An index theorem for Lorentzian manifolds with compact spacelike Cauchy boundary arXiv:1506.00959 , to appear in Amer. J. Math. • C. Bär and A. Strohmaier: A rigorous geometric derivation of the chiral anomaly in curved backgrounds arXiv:1508.05345 , Commun. Math. Phys. 347 (2016), 703–721 • C. Bär and S. Hannes: Boundary value problems for the Lorentzian Dirac operator arXiv:1704.03224 Thank you for your attention!