The Atiyah-Patodi-Singer index and domain-wall fermion Dirac - PowerPoint PPT Presentation

The Atiyah-Patodi-Singer index and domain-wall fermion Dirac operators Shinichiroh Matsuo 2020-08-07 01:10–02:00 UTC Nagoya University, Japan

This talk is based on a joint work arXiv:1910.01987 (to appear in CMP) of three mathematicians and three physicists: • Mikio Furuta • Hidenori Fukaya • Mayuko Yamashita • Tetsuya Onogi • Shinichiroh Matsuo • Satoshi Yamaguchi 1/28

Main theorem Theorem (FFMOYY arXiv:1910.01987) For m ≫ 0 , we have a formula Ind APS ( D | X + ) = η ( D + m κγ ) − η ( D − m γ ) . 2 • The Atiyah-Patodi-Singer index is expressed in terms of the η -invariant of domain-wall fermion Dirac operators. • The original motivation comes from the bulk-edge correspondence of topological insulators in condensed matter physics. • The proof is based on a Witten localisation argument. 2/28

Plan of the talk 1. Reviews of the Atiyah-Singer index and the eta invariant 2. The Atiyah-Patodi-Singer index 3. Domain-wall fermion Dirac operators 4. Main theorem 5. The proof of a toy model 6. The proof of the main theorem: Witten localisation 3/28

Index and Eta

Let X be a closed manifold and S → X a hermitian bundle. Assume dim X is even. Assume S is Z / 2-graded: there exists γ : Γ( S ) → Γ( S ) such that γ 2 = id S . � � 1 0 γ = . 0 − 1 Let D : Γ( S ) → Γ( S ) be a 1st order elliptic differential operator. Assume D is odd and self-adjoint: � � 0 D − D = and D − = ( D + ) ∗ . D + 0 Defjnition Ind D := dim Ker D + − dim Ker D − = dim Ker D + − dim Coker D + 4/28

Fix m ̸ = 0 and consider � � m D − D + m γ = : Γ( S ) → Γ( S ) . D + − m This is self-adjoint but no longer odd; thus, its spectrum is real but not symmetric around 0. For Re( z ) ≫ 0, let � sign λ j η ( D + m γ )( z ) := | λ j | z , λ j where { λ j } = Spec( D + m γ ) . Note that λ j ̸ = 0 for any j . Defjnition η ( D + m γ ) := η ( D + m γ )( 0 ) . The eta invariant describes the overall asymmetry of the spectrum of a self-adjoint operator. 5/28

Proposition For any m > 0 , we have a formula Ind( D ) = η ( D + m γ ) − η ( D − m γ ) . 2 This formula might be unfamiliar; however, we can prove it easily, for example, by diagonalising D 2 and γ simultaneously. We will explain another proof later. We will generalise this formula to handle compact manifolds with boundary and the Atiyah-Patodi-Singer index. 6/28

Proposition For any m > 0 , we have a formula Ind( D ) = η ( D + m γ ) − η ( D − m γ ) . 2 We will generalise this formula to handle compact manifolds with boundary and the Atiyah-Patodi-Singer index by using domain-wall fermion Dirac operators. Theorem (FFMOYY arXiv:1910.01987) For m ≫ 0 , we have a formula Ind APS ( D | X + ) = η ( D + m κγ ) − η ( D − m γ ) . 2 Next, we review the Atiyah-Patodi-Singer index. 7/28

The Atiyah-Patodi-Singer index

Let Y ⊂ X be a separating submanifold that decomposes X into two compact manifolds X + and X − with common boundary Y . Assume Y has a collar neighbourhood isometric to ( − 4 , 4 ) × Y . � ( − 4 , 4 ) × Y ⊂ X = X − X + Y X − X + Y 8/28

Assume S → X and D : Γ( S ) → Γ( S ) are standard on ( − 4 , 4 ) × Y in the sense that there exists a hermitian bundle E → Y and a self-adjoint elliptic operator A : Γ( E ) → Γ( E ) such that S = C 2 ⊗ E and � � � � 0 D ∗ 0 ∂ u + A D = + = D + 0 − ∂ u + A 0 on ( − 4 , 4 ) × Y . X − X + Y Assume also A has no zero eigenvalues. 9/28

Let � X + := ( −∞ , 0 ] × Y ∪ X + . ( −∞ , 0 ) × Y X + We assumed D is translation invariant on ( − 4 , 4 ) × Y : � � � � 0 D ∗ 0 ∂ u + A D = + = . D + 0 − ∂ u + A 0 Thus, D | X + naturally extends to � X + , which is denoted by � D . This is Fredholm if A has no zero eigenvalues. Defjnition (Atiyah-Patodi-Singer index) Ind APS ( D | X + ) := Ind( � D ) 10/28

Domain-wall fermion Dirac operators

Let κ : X → R be a step function such that κ ≡ ± 1 on X ± . κ X − X + Y Defjnition For m > 0, D + m κγ : Γ( S ) → Γ( S ) is called a domain-wall fermion Dirac operator. 11/28

D + m κγ is self-adjoint but not odd. κ X − X + Y � � 0 ∂ u + A D = on ( − 4 , 4 ) × Y − ∂ u + A 0 Proposition If Ker A = { 0 } , then Ker( D + m κγ ) = { 0 } for m ≫ 0 . Next we will defjne η ( D + m κγ ) . 12/28

The eta invariant of domain-wall fermion Dirac operators Since Ker( D + m κγ ) = { 0 } , there exists a constant C m > 0 such that Ker( D + m κγ + f ) = { 0 } if ∥ f ∥ 2 < C m . Corollary of the variational formula of the eta invariant Assume both m κγ + f 1 and m κγ + f 2 are smooth with ∥ f 1 ∥ 2 < C m and ∥ f 2 ∥ 2 < C m . Then, we have η ( D + m κγ + f 1 ) = η ( D + m κγ + f 2 ) . Defjnition For any f with ∥ f ∥ 2 < C m and m κγ + f smooth, we set η ( D + m κγ ) := η ( D + m κγ + f ) . 13/28

Main theorem

Main theorem Theorem (FFMOYY arXiv:1910.01987) For m ≫ 0 , we have a formula Ind APS ( D | X + ) = η ( D + m κγ ) − η ( D − m γ ) . 2 κ X − X + Y • The Atiyah-Patodi-Singer index is expressed in terms of the η -invariant of domain-wall fermion Dirac operators. • The original motivation comes from physics. 14/28 • The proof is based on a Witten localisation argument.

The proof of a toy model

Toy model Proposition For any m > 0 , we have a formula Ind( D ) = η ( D + m γ ) − η ( D − m γ ) . 2 As a warm-up, we will prove this formula in the spirit of our proof of the main theorem. 15/28

Let � κ AS : R × X → R be a step function such that � κ AS ≡ 1 on ( 0 , ∞ ) × X and � κ AS ≡ − 1 on ( −∞ , 0 ) × X . κ AS ≡ + 1 � κ AS ≡ − 1 � We consider � D m : L 2 ( R × X ; S ⊕ S ) → L 2 ( R × X ; S ⊕ S ) defjned by � � 0 ( D + m � κ AS γ ) + ∂ t D m := � . ( D + m � 0 κ AS γ ) − ∂ t This is a Fredholm operator. 16/28

Model case: the Jackiw-Rebbi solution on R For any m > 0, we have d dte − m | t | = − m sgn e − m | t | , where sgn( ± t ) = ± 1. As m → ∞ , the solution concentrates at 0. e − m | t | m sgn t O � � � � � � 0 ∂ t + m sgn 0 0 = . e − m | t | − ∂ t + m sgn 0 0 17/28

κ AS ≡ + 1 � κ AS ≡ − 1 � � � 0 ( D + m � κ AS γ ) + ∂ t � D m := ( D + m � 0 κ AS γ ) − ∂ t ( e − m | t | ) ′ = − m sgn e − m | t | Proposition (Product formula) Ind( D ) = Ind( � D m ) Assume D φ = 0. Set φ ± := ( φ ± γφ ) / 2. Then, we have � � � � e − m | t | φ − 0 ( D + m � κ AS γ ) + ∂ t = 0 . e − m | t | φ + ( D + m � 0 κ AS γ ) − ∂ t 18/28

κ AS ≡ + 1 � κ AS ≡ − 1 � � � 0 ( D + m � κ AS γ ) + ∂ t � D m := ( D + m � 0 κ AS γ ) − ∂ t Proposition (APS formula) D m ) = η ( D + m γ ) − η ( D − m γ ) Ind( � 2 • Note that D + m � κ AS ( ± 1 , · ) γ = D ± m γ . • Perturb � κ AS slightly near { 0 } × X to get a smooth operator. • Use the Atiyah-Patodi-Singer index theorem on R × X . • Since dim R × X is odd, the constant term in the asymptotic expansion of the heat kernel vanishes. 19/28

Proposition Ind( D ) = η ( D + m γ ) − η ( D − m γ ) . 2 � � 0 ( D + m � κ AS γ ) + ∂ t � D m := ( D + m � 0 κ AS γ ) − ∂ t By the product formula, we have Ind( D ) = Ind( � D m ) . By the APS formula, we have D m ) = η ( D + m γ ) − η ( D − m γ ) Ind( � . 2 20/28

The proof of the main theorem

Outline of the proof Theorem (FFMOYY arXiv:1910.01987) For m ≫ 0 , we have a formula Ind APS ( D | X + ) = η ( D + m κγ ) − η ( D − m γ ) . 2 The proof is modelled on the original embedding proof of the Atiyah-Singer index theorem. 1. Embed � X + into R × X . 2. Extend both � D on � X + and D + m κγ on { 10 } × X to R × X . 3. Use the product formula, the APS formula, and a Witten localisation argument. 21/28

Embedding of � X + into R × X � X + := ( −∞ , 0 ] × Y ∪ X + . ( −∞ , 0 ) × Y X + We can embed � X + into R × X as follows: � X + 22/28

Extension of � D and D + m κγ to R × X ( R × X ) \ � X + has two connected components. We denote by ( R × X ) − the one containing {− 10 } × X + and by ( R × X ) + the other half. Let � κ APS : R × X → [ − 1 , 1 ] be a step function such that � κ APS ≡ ± 1 on ( R × X ) ± . κ APS ≡ + 1 � κ APS ≡ − 1 � � X + We consider � � 0 ( D + m � κ APS γ ) + ∂ t � D m := . ( D + m � 0 κ APS γ ) − ∂ t 23/28

κ APS ≡ + 1 � κ APS ≡ − 1 � � X + � � 0 ( D + m � κ APS γ ) + ∂ t � D m := . ( D + m � 0 κ APS γ ) − ∂ t κ APS ≡ κ on { 10 } × X . � Proposition (APS formula) D m ) = η ( D + m κγ ) − η ( D − m γ ) Ind( � 2 24/28

κ APS ≡ + 1 � κ APS ≡ − 1 � � X + � � 0 ( D + m � κ APS γ ) + ∂ t � D m := . ( D + m � 0 κ APS γ ) − ∂ t The restriction of � D m to a tubular neighbourhood of � X + is isomorphic to � � ( � 0 D + m sgn γ ) + ∂ t ( � D + m sgn γ ) − ∂ t 0 on R × � X + near { 0 }× � X + , where � D is the extension of D | X + to � X + . 25/28