Boundary value problems on Riemannian and Lorentzian manifolds - PowerPoint PPT Presentation

Boundary value problems on Riemannian and Lorentzian manifolds Christian Bär (joint with W. Ballmann, S. Hannes, A. Strohmaier) Institut für Mathematik Universität Potsdam AQFT: Where operator algebra meets microlocal analysis Cortona, June 6, 2018

Outline Riemannian manifolds and elliptic operators 1 The Atiyah-Patodi-Singer index theorem General elliptic boundary conditions Lorentzian manifolds and hyperbolic operators 2 Dirac operator on Lorentzian manifolds Fredholm pairs The Lorentzian index theorem The chiral anomaly More general boundary conditions

1. Riemannian manifolds and elliptic operators

Setup M Riemannian manifold, compact, with boundary ∂ M spin structure � spinor bundle SM → M n = dim ( M ) even � splitting SM = S R M ⊕ S L M � Dirac operator D : C ∞ ( M , S R M ) → C ∞ ( M , S L M ) 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 (M. Atiyah, V. Patodi, I. Singer, 1975) Under APS-boundary conditions D is Fredholm and � � ind ( D APS ) = A ( M ) ∧ ch ( E ) M � A ( M ) ∧ ch ( E )) − h ( A 0 ) + η ( A 0 ) T ( � + 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

Which boundary conditions other than APS will work?

Warning APS-boundary conditions cannot be replaced by anti-Atiyah-Patodi-Singer boundary conditions, P − ( f | ∂ M ) = χ ( −∞ , 0 ) ( A 0 )( f | ∂ M ) = 0 Example M = unit disk ⊂ C ∂ D = ∂ = ∂ z Taylor expansion: u = � ∞ n = 0 α n z n A 0 = i d d θ Fourier expansion: u | ∂ M = � n ∈ Z α n e in θ APS-boundary conditions: α n = 0 for n ≥ 0 ⇒ ker ( D ) = { 0 } aAPS-boundary conditions: α n = 0 for n < 0 ⇒ ker ( D ) = infinite dimensional

Generalize APS conditions Notation For an interval J ⊂ R write � � � � � L 2 u ∈ L 2 ( ∂ M ) J ( ∂ M ) = � u = a λ ϕ λ λ ∈ J ∩ spec ( A 0 ) where A 0 ϕ λ = λϕ λ . Similarly for H s J ( ∂ M ) . APS-boundary conditions 1 f | ∂ M ∈ B = H ( −∞ , 0 ) ( ∂ M ) 2 1. Generalization Replace ( −∞ , 0 ) by ( −∞ , a ] for some a ∈ R : 1 B = H ( −∞ , a ] ( ∂ M ) 2

Generalize APS conditions 2. Generalization Deform 1 B = { v + gv | v ∈ H ( −∞ , a ] ( ∂ M ) } 2 1 1 where g : H ( −∞ , a ] ( ∂ M ) → H 2 ( a , ∞ ) ( ∂ M ) is bounded linear. 2 3. Generalization Finite-dimensional modification 1 B = W + ⊕ { v + gv | v ∈ H ( −∞ , a ] ( ∂ M ) } 2 where W + ⊂ C ∞ ( ∂ M ) is finite-dimensional.

Elliptic boundary conditions Definition 1 2 ( ∂ M ) is said to be an elliptic A linear subspace B ⊂ H boundary condition if there is an L 2 -orthogonal decomposition L 2 ( ∂ M ) = V − ⊕ W − ⊕ V + ⊕ W + such that 1 2 } B = W + ⊕ { v + gv | v ∈ V − ∩ H where 1) W ± ⊂ C ∞ ( ∂ M ) finite-dimensional; 2) V − ⊕ W − ⊂ L 2 ( −∞ , a ] ( ∂ M ) and V + ⊕ W + ⊂ L 2 [ − a , ∞ ) ( ∂ M ) , for some a ∈ R ; 3) g : V − → V + and g ∗ : V + → V − are operators of order 0.

Fredholm property and boundary regularity Theorem (Ballmann-B. 2012) Let B be an elliptic boundary condition. Then D B : { f ∈ H 1 ( M , S R ) | f | ∂ M ∈ B } → L 2 ( M , S L ) is Fredholm. Theorem (Ballmann-B. 2012) Let B be an elliptic boundary condition. Then f ∈ H k + 1 ( M , S R ) ⇐ ⇒ D B f ∈ H k ( M , S L ) , for all f ∈ dom D B and k ≥ 0. In particular, f ∈ dom D B is smooth up to the boundary iff D B f is smooth up to the boundary.

Examples 1) Generalized APS: V − = L 2 ( −∞ , a ) ( A 0 ) , V + = L 2 [ a , ∞ ) ( A 0 ) , W − = W + = { 0 } , g = 0. Then 1 B = H ( −∞ , a ) ( A 0 ) . 2 2) Classical local elliptic boundary conditions in the sense of Lopatinsky-Schapiro.

Examples 3) “Transmission” condition � � 1 1 1 2 ( N 1 , S R ) ⊕ H 2 ( N 2 , S R ) | φ ∈ H 2 ( N , S R ) B = ( φ, φ ) ∈ H Here V + = L 2 ( 0 , ∞ ) ( A 0 ⊕ − A 0 ) = L 2 ( 0 , ∞ ) ( A 0 ) ⊕ L 2 ( −∞ , 0 ) ( A 0 ) V − = L 2 ( −∞ , 0 ) ( A 0 ⊕ − A 0 ) = L 2 ( −∞ , 0 ) ( A 0 ) ⊕ L 2 ( 0 , ∞ ) ( A 0 ) W + = { ( φ, φ ) ∈ ker ( A 0 ) ⊕ ker ( A 0 ) } W − = { ( φ, − φ ) ∈ ker ( A 0 ) ⊕ ker ( A 0 ) } � 0 � id g : V − → V + , g = id 0

A deformation argument Replace B by B s where g is replaced by g s with g s = s · g . Then B 1 = transmission condition and B 0 = APS-condition. Hence ind ( D M ) = ind ( D M ′ transm . ) = ind ( D M ′ APS ) . Holds also if M is complete noncompact and D satisfies a coercivity condition at infinity. Implies relative index theorem by Gromov and Lawson (1983).

2. Lorentzian manifolds and hyperbolic operators

Globally hyperbolic spacetimes Let M be a globally hyperbolic Lorentzian manifold with boundary ∂ M = Σ 0 ⊔ Σ 1 Σ j compact smooth spacelike Cauchy hypersurfaces

� � The Cauchy problem Well-posedness of Cauchy problem The map D ⊕ res Σ : C ∞ ( M ; S R ) → C ∞ ( M ; S L ) ⊕ C ∞ (Σ; S R ) is an isomorphism of topological vector spaces. Wave propagator U : { v ∈ C ∞ ( M ; S R ) | Dv = 0 } res Σ 0 res Σ 1 ∼ ∼ = = U � C ∞ (Σ 1 , S R ) C ∞ (Σ 0 , S R ) U extends to unitary operator L 2 (Σ 0 ; S R ) → L 2 (Σ 1 ; S R ) .

Fredholm pairs Definition Let H be a Hilbert space and B 0 , B 1 ⊂ H closed linear subspaces. Then ( B 0 , B 1 ) is called a Fredholm pair if B 0 ∩ B 1 is finite dimensional and B 0 + B 1 is closed and has finite codimension. The number ind ( B 0 , B 1 ) = dim ( B 0 ∩ B 1 ) − dim ( H / ( B 0 + B 1 )) is called the index of the pair ( B 0 , B 1 ) . Elementary properties: 1.) ind ( B 0 , B 1 ) = ind ( B 1 , B 0 ) 2.) ind ( B 0 , B 1 ) = − ind ( B ⊥ 0 , B ⊥ 1 ) 3.) Let B 0 ⊂ B ′ 0 with dim ( B ′ 0 / B 0 ) < ∞ . Then ind ( B ′ 0 , B 1 ) = ind ( B 0 , B 1 ) + dim ( B ′ 0 / B 0 ) .

Fredholm pairs and the Dirac operator Let B 0 ⊂ L 2 (Σ 0 , S R ) and B 1 ⊂ L 2 (Σ 1 , S R ) be closed subspaces. Proposition (B.-Hannes 2017) The following are equivalent: (i) The pair ( B 0 , U − 1 B 1 ) is Fredholm of index k ; (ii) The pair ( UB 0 , B 1 ) is Fredholm of index k ; (iii) The restriction D : { f ∈ FE ( M , S R ) | f | Σ i ∈ B i } → L 2 ( M , S L ) is a Fredholm operator of index k .

Trivial example Let dim ( B 0 ) < ∞ and codim ( B 1 ) < ∞ . Then D with these boundary conditions is Fredholm with index dim ( B 0 ) − codim ( B 1 )

The Lorentzian index theorem Theorem (B.-Strohmaier 2015) Under APS-boundary conditions D is Fredholm. The kernel consists of smooth spinor fields and � � � T ( � ind ( D APS ) = A ( M ) ∧ ch ( E ) + A ( M ) ∧ ch ( E )) M ∂ M − h ( A 0 ) + h ( A 1 ) + η ( A 0 ) − η ( A 1 ) 2 ind ( D APS ) = dim ker [ D : C ∞ APS ( M ; S R ) → C ∞ ( M ; S L )] − dim ker [ D : C ∞ aAPS ( M ; S R ) → C ∞ ( M ; S L )] aAPS conditions are as good as APS-boundary conditions.

The chiral anomaly Want to quantize classical Dirac current J ( X ) = � ψ, X · ψ � Fix a Cauchy hypersurface Σ and try ˙ µ ( p ) = ω Σ (¯ J Σ A ( p )( γ µ ) B Ψ A Ψ B ( p )) ˙ Here ω Σ is the vacuum state associated with Σ . Problem: singularities of two-point function. Need regularization procedure. But: relative current does exist J Σ 0 , Σ 1 = J Σ 0 − J Σ 1

Charge creation and index Theorem (B.-Strohmaier 2015) The relative current J Σ 0 , Σ 1 is coclosed and � J Σ 0 , Σ 1 ( ν Σ ) d Σ = ind ( D APS ) . Q R := Σ Hence � A ( M ) ∧ ch ( E ) − h ( A 0 ) − h ( A 1 ) + η ( A 0 ) − η ( A 1 ) � Q R = . 2 M Similarly � A ( M ) ∧ ch ( E ) + h ( A 0 ) − h ( A 1 ) + η ( A 0 ) − η ( A 1 ) � Q L = − . 2 M Total charge Q = Q R + Q L is zero. Chiral charge Q chir = Q R − Q L is not!

Example • Spacetime M = R × S 4 k − 1 with metric − dt 2 + g t where g t are Berger metrics. • Flat connection on trivial bundle E . • Chiral anomaly: � 2 k � Q Σ 0 , Σ 1 = ( − 1 ) k 2 chir k • See Gibbons 1979 for k = 1.

Boundary conditions in graph form A pair ( B 0 , B 1 ) of closed subspaces B i ⊂ L 2 (Σ i , S R ) form elliptic boundary conditions if there are L 2 -orthogonal decompositions L 2 (Σ i , S R ) = S − i ⊕ W − ⊕ V + ⊕ W + i , i = 0 , 1 , i i such that (i) W + i , W − are finite dimensional; i (ii) W − ⊕ V − ⊂ L 2 ( −∞ , a i ] ( ∂ M ) and W + ⊕ V + ⊂ L 2 [ − a i , ∞ ) ( ∂ M ) i i i i for some a i ∈ R ; 0 → V + (iii) There are bounded linear maps g 0 : V − 0 and g 1 : V + 1 → V − 1 such that B 0 = W + 0 ⊕ Γ( g 0 ) , B 1 = W − 1 ⊕ Γ( g 1 ) , where Γ( g 0 / 1 ) = { v + g 0 / 1 v | v ∈ V ∓ 0 / 1 } .

Boundary conditions in graph form Theorem (B.-Hannes 2017) The pair ( B 0 , B 1 ) is Fredholm provided (A) g 0 or g 1 is compact or (B) � g 0 � · � g 1 � is small enough. 1.) Applies if g 0 = 0 or g 1 = 0. 2.) Conditions (A) and (B) cannot both be dropped (counterexamples).