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Mathematical proof for “physicist-friendly” reformulation

• f Atiyah-Patodi-Singer index

Hidenori Fukaya (Osaka U.)

HF,. T Onogi, S. Yamaguchi PRD96(2017) no. 12, 125004 [arXiv:1710.03379]

• M. Furuta (U. Tokyo), S. Matsuo (Nagoya

U.),T. Onogi (Osaka U.), S. Yamaguchi (Osaka U.), M. Yamashita (U.Tokyo) [arXiv: 19xx.xxxxx]

My talk today

In 2017 we proposed “A physicist-friendly reformulation

• f the Atiyah-Patodi-Singer(APS)

index theorem.”

F,. Onogi, Yamaguchi PRD96(2017) no.12, 125004 [arXiv: 1710.033379]

Recently, we invited 3 mathematicians and succeeded in a mathematical proof.

F, Furuta, Matuso, Onogi, Yamaguchi, Yamashita, in progress

My talk today

In 2017 we proposed “A physicist-friendly reformulation

• f the Atiyah-Patodi-Singer(APS)

index theorem.”

F,. Onogi, Yamaguchi PRD96(2017) no.12, 125004 [arXiv: 1710.033379]

Recently, we invited 3 mathematicians and succeeded in a mathematical proof.

F, Furuta, Matuso, Onogi, Yamaguchi, Yamashita, in progress

index on a manifold with boundary,

non-integer non-integer integer

Atiyah-Patodi-Singer index theorem

[Atiyah-Patodi-Singer 1975]

Witten 2015 : APS index is a key to understand bulk-edge correspondence in symmetry protected topological insulator:

T is protected ! T-anomalous T-anomalous

[Related works: Metlitski 15, Seiberg-Witten 16, Tachikawa-Yonekura 16&18, Freed-Hopkins 16, Witten 16, Yonekura 16&19 …]

APS index in topological insulator

fermion path integrals

= (−1)−I

What puzzled us

What puzzled us

1. APS boundary condition is non-local, while that of topological matter is local.

What puzzled us

1. APS boundary condition is non-local, while that of topological matter is local. 2. APS is for massless fermion but bulk fermion of topological insulator is massive (gapped).

What puzzled us

1. APS boundary condition is non-local, while that of topological matter is local. 2. APS is for massless fermion but bulk fermion of topological insulator is massive (gapped). 3. No edge-localized modes allowed.

What puzzled us

1. APS boundary condition is non-local, while that of topological matter is local. 2. APS is for massless fermion but bulk fermion of topological insulator is massive (gapped). 3. No edge-localized modes allowed. 4. No “physicist-friendly” description in the literature

[except for Alvarez-Gaume et al. 1985 but boundary condition is obscure.]

What puzzled us

1. APS boundary condition is non-local, while that of topological matter is local. 2. APS is for massless fermion but bulk fermion of topological insulator is massive (gapped). 3. No edge-localized modes allowed. 4. No “physicist-friendly” description in the literature

[except for Alvarez-Gaume et al. 1985 but boundary condition is obscure.]

→ We launched a study group reading original APS paper and it took 3 months to translate it into “physics language”, and we reached an alternative expression.

If we impose local and Lorentz (rotation) invariant boundary condition, + and – chirality sectors do not decouple any more.

+

• angular momentum is

conserved

and the index do not make sense.

Difficulty with boundary

Atiyah-Patodi-Singer boundary condition

Gives up the locality and rotational symmetry but keeps the chirality.

• Eg. 4 dim

boundary

gauge

They impose a non-local b.c. [Atiyah, Patodi, Singer 75]

index = n+ − n−

Beautiful! But physicist- unfriendly.

Locality >> chirality for physicists

Locality (=causality) is essential. We cannot accept APS condition even if it is beautiful.

Locality >> chirality for physicists

Locality (=causality) is essential. We cannot accept APS condition even if it is beautiful.

non-local boundary

hit!

Locality >> chirality for physicists

Locality (=causality) is essential. We cannot accept APS condition even if it is beautiful.

non-local boundary

hit! information

Locality >> chirality for physicists

Locality (=causality) is essential. We cannot accept APS condition even if it is beautiful.

non-local boundary

hit! information information propagates faster than speed of light.

Locality >> chirality for physicists

Locality (=causality) is essential. We cannot accept APS condition even if it is beautiful. → need to give up chirality and consider L/R mixing (massive case)

n+ − n− = 1 32⇡2 Z

x4>0

d4x✏µνρσtr[F µνF ρσ]−⌘(iD3D) 2

Locality >> chirality for physicists

Locality (=causality) is essential. We cannot accept APS condition even if it is beautiful. → need to give up chirality and consider L/R mixing (massive case) Can we still make a fermionic integer (even if it is ugly)?

n+ − n− = 1 32⇡2 Z

x4>0

d4x✏µνρσtr[F µνF ρσ]−⌘(iD3D) 2

Locality >> chirality for physicists

Locality (=causality) is essential. We cannot accept APS condition even if it is beautiful. → need to give up chirality and consider L/R mixing (massive case) Can we still make a fermionic integer (even if it is ugly)? Our answer is “Yes, we can”.

n+ − n− = 1 32⇡2 Z

x4>0

d4x✏µνρσtr[F µνF ρσ]−⌘(iD3D) 2

Contents

✔ 1. Introduction

APS b.c. is unphysical. Let us consider massive case.

• 2. Massive Dirac index without boundary
• 3. New index with boundary
• 4. Mathematical proof
• 5. Discussion
• 6. Summary

Atiyah-Singer(AS) index from massive Dirac operator

Zero-modes of D = still eigenstates of H: Non-zero modes make ± pairs

H = γ5(D + M)

Hφ0 = γ5Mφ0 = ±Mφ0.

Hφi = λiφi

HDφi = −DHφi = −λiDφi

η(H) = X

i

sgnλi = # of +M − # of −M

= AS index!

always jumps by 2.

To increase + modes, we have to borrow

• ne from - (UV) modes.

Good regularizations (e.g. Pauli-Villars, lattice) respect this fact.

H = γ5(D + M)

+M

−M

… … … …

I = 0

I = 1

η(H)

Index(D) = 1 2η(H).

unpaired paired

Perturbative “proof” (in physics sense)

using Pauli-Villars regulator

HP V = γ5(D + Λ), Λ M

H = γ5(D + M)

Fujikawa-method

does not contribute.

1 2η(H)reg = 1 2 [η(H) − η(HP V )] .

⌘(H) = lim

s!0 Tr

H ( √ H2)1+s = 1 √⇡ Z 1 dtt1/2TrHetH2 = 1 √⇡ Z 1 dt0t01/2Tr5 ✓ M + D M ◆ et0D†D/M 2et0, = 1 32⇡2 Z d4x ✏µνρσtrcF µνF ρσ + O(1/M 2).

Contents

✔ 1. Introduction APS b.c. is unphysical. Let us consider massive case.

• 2. Massive Dirac index without boundary

coincides with the AS index.

• 3. New index with boundary
• 4. Mathematical proof
• 5. Discussion
• 6. Summary

✔ I = η(γ5(D + M))reg/2

More physical set-up?

More physical set-up?

In physics,

• 1. Any boundary has “outside”: manifold +

boundary → domain-wall.

• utside

More physical set-up?

In physics,

• 1. Any boundary has “outside”: manifold +

boundary → domain-wall.

• 2. Boundary should not preserve helicity but keep

angular-mom: massless → massive (in bulk)

More physical set-up?

In physics,

• 1. Any boundary has “outside”: manifold +

boundary → domain-wall.

• 2. Boundary should not preserve helicity but keep

angular-mom: massless → massive (in bulk)

• 3. Boundary condition should not be put by hand

→ but automatically chosen.

More physical set-up?

In physics,

• 1. Any boundary has “outside”: manifold +

boundary → domain-wall.

• 2. Boundary should not preserve helicity but keep

angular-mom: massless → massive (in bulk)

• 3. Boundary condition should not be put by hand

→ but automatically chosen.

• 4. Edge-localized modes play the key role.

Domain-wall Dirac operator

Let us consider

• n a closed manifold

with sign flipping mass, without assuming any boundary condition

(we expect it dynamically given.).

[Jackiw-Rebbi 1976, Callan-Harvery 1985, Kaplan 1992 ]

“new” APS index [F-Onogi-Yamaguchi 2017]

= AS index

1 2⌘(5(D + M✏(x4)))reg

1 2η(γ5(D + M))reg

which can be shown by Fujikawa-method. See our paper or my talk slide in 2007.

= 1 32⇡2 Z

x4>0

d4x✏µνρσtr[F µνF ρσ]−⌘(iD3D) 2

Complete set in the free case

Solutions to are where Here, and

Edge mode appears !

ϕ(x4) ⊗ eip·x

{γ5(Dfree + Mε(x4))}2φ = ⇥ −∂2

µ + M 2−2Mγ4δ(x4)

⇤ φ = λ2φ

“Automatic” boundary condition

We didn’t put any boundary condition by hand. But is automatically satisfied due to the domain-

• wall. This condition is LOCAL and PRESERVES

angular-momentum in x4 direction but DOES NOT keep chirality.

Contents

✔ 1.

Introduction

APS b.c. is unphysical. Let us consider massive case.

• 2. Massive Dirac index without boundary

coincides with the AS index.

• 3. New index with boundary domain-wall

coincides with the APS index.

4. Mathematical proof 5. Discussion 6. Summary

✔ I = η(γ5(D + M))reg/2

I = ⌘(5(D + M✏(x4)))reg/2

✔

Overview

with physicist-unfriendly boundary condition [APS 1975]

Ind(DAPS)

= =

with physicist-friendly set-up (topological insulator) [FOY 2017]

1 2η(HDW )

CONJECTURE from perturbation theory in 4D flat space

=

This work = THEOREM (on any even-dim. curved manifold)

[F, Furuta, Matsuo, Onogi, Yamaguchi, Yamashita, 2019]

Theorem 1: APS index = index with infinite cylinder In original APS paper, they showed Index w/ APS b.c. = Index with infinite cylinder attached to the original boundary (w.r.t. square integrable modes).

* On cylinder, gauge fields are constant in the extra-direction.

Theorem 2: Localization (& product formula)

By giving position-dependent “mass”, we can localize the zero modes to “massless” lower- dimensional surface and the index is given by the product:

= generalization of domain-wall fermion m=0 surface

Ind(γs(Dd + ∂s + iγsM(s))) = Ind(Dd) × Ind(γs∂s + M(s))

Theorem 3: In odd-dim, APS index = boundary eta-invariant

exists only in even-dim.

Z F ∧ F ∧ · · ·

Ind(Dodd−dim

APS

) = 1 2 ⇥ η(Dboundary1) − η(Dboundary2) ⇤

5-dimensional Dirac operator

we consider where and is independent of .

Aµ

D5D = ✓ ∂5 + γ5(D4D + m(x4, x5)) −∂5 + γ5(D4D + m(x4, x5)) ◆

m(x4, x5) =    M for x4 > 0 & x5 > 0 for x4 = 0 & x5 = 0 −M2

• therwise

x5

* The following proof is valid for any 2n+1 dimension.

On X4D x R,

s = x5

x4

we compute in two different ways:

• 1. localization
• 2. eta-inv. at

Ind(D5D)

X4D

x5 = ±1.

Localization

Theorem 2 tells us and on the massless surface theorem 1 indicates

s = x5

x4

Ind(D5D)|M,M2→∞ = Ind(D4D

m=0surface) × IndD1D normal

| {z }

=1

= Ind(D4D

m=0surface) = Ind(D X4D

x4>0

APS

)

X4D

x4>0

X4D

x4>0

Boundary eta invariants

Theorem 1 tells us and from theorem 3, we obtain therefore,

s = x5

x4

Ind(D5D) = Ind(D5D APS b.c.ats=±1)

Ind(D5D APS b.c.ats=±1) = 1 2 ⇥ η(D4D

s=1) − η(D4D s=−1)

⇤

= 1 2 ⇥ ⌘(5(D4D + M✏(x4)) − ⌘(5(D4D − M2) ⇤ = 1 2⌘P V reg.(5(D4D + M✏(x4))

Q.E.D.

Ind(D5D) = Ind(DAPS) = 1 2η(HDW )

Contents

✔ 1. Introduction

APS b.c. is unphysical. Let us consider massive case.

• 2. Massive Dirac index without boundary

coincides with the AS index.

• 3. New index with boundary domain-wall

coincides with the APS index.

• 4. Mathematical proof
• 5. Discussion
• 6. Summary

✔ I = η(γ5(D + M))reg/2

I = ⌘(5(D + M✏(x4)))reg/2

✔

Ind(D5D) = Ind(DAPS) = ⌘(5(D + M✏(x4)))reg/2

✔

We don’t need any boundary condition by hand.

The kink structure automatically chooses a local and rotationally symmetric boundary condition, and extension from AS index is simple:

1 2⌘(5(D + M)) → 1 2⌘(5(D + M✏(x)))

massive fermion = chiral symmetry is NOT important.

The lattice fermion “knew” this fact: If the original AS index were given by we should have known the lattice index theorem much before Hasenfratz or Neuberger 1998.

Ind(Dov) = 1 2Trγ5 ✓ 1 − aDov 2 ◆

Dov = 1 a 1 + γ5 HW p H2

W

!

= −1 2Tr HW p H2

W

= −1 2η(γ5(DW − M))!

−1 2η(γ5(D − M))

Massless vs. massive

index theorem with massless Dirac op. index theorem with massive Dirac op.

Trγ5e−D2/M2

Trγ5(1 − aDov/2)

Trγ5e−D2/M2w/ APS b.c.

continuum lattice AS APS not known.

−1 2η(γ5(D − M))

−1 2η(γ5(DW − M))

−1 2⌘(5(D − M✏(x)))

continuum lattice AS APS

Massless vs. massive

index theorem with massless Dirac op. index theorem with massive Dirac op.

Trγ5e−D2/M2

Trγ5(1 − aDov/2)

Trγ5e−D2/M2w/ APS b.c.

continuum lattice AS APS not known.

−1 2η(γ5(D − M))

−1 2η(γ5(DW − M))

−1 2⌘(5(D − M✏(x)))

continuum lattice AS APS

−1 2⌘(5(DW − M✏(x)))?

Massless vs. massive

index theorem with massless Dirac op. index theorem with massive Dirac op.

Trγ5e−D2/M2

Trγ5(1 − aDov/2)

Trγ5e−D2/M2w/ APS b.c.

continuum lattice AS APS not known.

−1 2η(γ5(D − M))

−1 2η(γ5(DW − M))

−1 2⌘(5(D − M✏(x)))

continuum lattice AS APS

Next talk by Kawai

−1 2⌘(5(DW − M✏(x)))?

Contents

✔ 1.

Introduction

APS b.c. is unphysical. Let us consider massive case.

• 2. Massive Dirac index without boundary

coincides with the AS index.

• 3. New index with boundary domain-wall

coincides with the APS index. 4. Mathematical proof 5. Discussion

For index theorems, chiral sym. is NOT important.

6. Summary

✔ I = η(γ5(D + M))reg/2

I = ⌘(5(D + M✏(x4)))reg/2

✔

Ind(D5D) = Ind(DAPS) = ⌘(5(D + M✏(x4)))reg/2

✔ ✔

Summary

• 1. APS index describes bulk-edge

correspondence of topological insulators.

• 2. APS (as well as AS) index can be reformulated


by the eta-inv. of massive domain-wall operator.

• 3. We have given a mathematical proof for

general cases through the 5D index.

• 4. eta-invariant of massive operator unifies the

index theorems (including their lattice version).

Ind(DAPS) = 1 2η(HDW )

Backup slides

Example : 1+1d bulk + 0+1d edge Majorana fermion coupled to gravity

APS index tells consistent with Z8 classification

• f Kitaev’s interacting Majorana

chain.

Z ∝ exp ⇣ 2πin 8 ⌘

Eta invariant = Chern Simons term + integer (non-local effect)

η(iD3D) 2 = CS 2π + integer

= surface term.