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Domain-wall fermion and index theorem Hidenori Fukaya (Osaka U.) work in progress with M. Furuta (U. Tokyo), S. Matsuo (Nagoya U.), T. Onogi(Osaka U.), S. Yamaguchi (Osaka U.), M. Yamashita (U.Tokyo) Cf. HF,. T Onogi, S. Yamaguchi PRD96(2017) no.12, 125004 [arXiv:1710.03379] Supported by Kobo-Kenkyu budget 18H04484 in JSPS Grant-in-Aid for Scientific Research on Innovative Areas “Discrete Geometric Analysis for Materials Design” Grant Number 17H06460

Domain-wall fermion and index theorem › ... › Fukaya-APSDW.pdf · 1.Introduction 2.Physicistʼs view of index theorems 3.Atiyah-Singer index with massive fermion operator 4.Index

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Page 1: Domain-wall fermion and index theorem › ... › Fukaya-APSDW.pdf · 1.Introduction 2.Physicistʼs view of index theorems 3.Atiyah-Singer index with massive fermion operator 4.Index

Domain-wall fermion and index theorem Hidenori Fukaya (Osaka U.) work in progress with M. Furuta (U. Tokyo), S. Matsuo (Nagoya U.), T. Onogi(Osaka U.), S. Yamaguchi (Osaka U.), M. Yamashita (U.Tokyo) Cf. HF,. T Onogi, S. Yamaguchi PRD96(2017) no.12, 125004 [arXiv:1710.03379]

Supported by Kobo-Kenkyu budget 18H04484 in JSPS Grant-in-Aid for Scientific Research on Innovative Areas “Discrete Geometric Analysis for Materials Design” Grant Number 17H06460

Page 2: Domain-wall fermion and index theorem › ... › Fukaya-APSDW.pdf · 1.Introduction 2.Physicistʼs view of index theorems 3.Atiyah-Singer index with massive fermion operator 4.Index

What’s new in this talk In 2017 Dec 8, I gave a seminar here, about “A physicist-friendly reformulation of the Atiyah-Patodi-Singer index theorem.” F,. Onogi, Yamaguchi PRD96(2017) no.12, 125004 [arXiv:1710.03379]

Today, I will talk about its mathematical proof. F, Furuta, Matuso, Onogi, Yamaguchi, Yamashita, in progress

Page 3: Domain-wall fermion and index theorem › ... › Fukaya-APSDW.pdf · 1.Introduction 2.Physicistʼs view of index theorems 3.Atiyah-Singer index with massive fermion operator 4.Index

Atiyah-Patodi-Singer (APS) index theorem [1975]

boundary bulk

* Here we (mainly) consider 4-dimensional flat Euclidean space with boundary at x4=0.

Ind(DAPS) =1

32⇡2

Z

x4>0d

4x✏

µ⌫⇢�

tr[Fµ⌫

F

⇢�]�⌘(iD3D)

2<latexit sha1_base64="Du+4nfOKg/InS16du6RqKPyJono=">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</latexit>

curvature

Page 4: Domain-wall fermion and index theorem › ... › Fukaya-APSDW.pdf · 1.Introduction 2.Physicistʼs view of index theorems 3.Atiyah-Singer index with massive fermion operator 4.Index

Witten 2015 : APS index is a key to understand bulk-edge correspondence in (Time-reversal (T)) symmetry protected topological insulator:

T is protected !

T-anomalous

T-anomalous

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

APS index in topological insulator

fermion path integrals

= (�1)�I<latexit sha1_base64="ZkAQC2onPO8oJv+ZIqufXzFuF68=">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</latexit>

Page 5: Domain-wall fermion and index theorem › ... › Fukaya-APSDW.pdf · 1.Introduction 2.Physicistʼs view of index theorems 3.Atiyah-Singer index with massive fermion operator 4.Index

What puzzled us

Page 6: Domain-wall fermion and index theorem › ... › Fukaya-APSDW.pdf · 1.Introduction 2.Physicistʼs view of index theorems 3.Atiyah-Singer index with massive fermion operator 4.Index

What puzzled us1. APS boundary condition is non-local, while that of

topological matter is local.

Page 7: Domain-wall fermion and index theorem › ... › Fukaya-APSDW.pdf · 1.Introduction 2.Physicistʼs view of index theorems 3.Atiyah-Singer index with massive fermion operator 4.Index

What puzzled us1. 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).

Page 8: Domain-wall fermion and index theorem › ... › Fukaya-APSDW.pdf · 1.Introduction 2.Physicistʼs view of index theorems 3.Atiyah-Singer index with massive fermion operator 4.Index

What puzzled us1. 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 “physicist-friendly” description in the literature [except for Alvarez-Gaume et al. 1985 (but boundary condition is obscure.)]

Page 9: Domain-wall fermion and index theorem › ... › Fukaya-APSDW.pdf · 1.Introduction 2.Physicistʼs view of index theorems 3.Atiyah-Singer index with massive fermion operator 4.Index

What puzzled us1. 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 “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 found another fermionic quantity, which coincides with the APS index [FOY 2017].

Page 10: Domain-wall fermion and index theorem › ... › Fukaya-APSDW.pdf · 1.Introduction 2.Physicistʼs view of index theorems 3.Atiyah-Singer index with massive fermion operator 4.Index

A physicist-friendly reformulation using domain-wall fermion [F, Onogi, Yamaguchi 2017]

perturbative computation

coincides with APS index, keeping the features of topological insulator.

I =1

2⌘(HDW )

<latexit sha1_base64="OdwhOw81TEK/2lvie7bBUri9M8g=">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</latexit>

1. massive Dirac in bulk (massless mode at edge)

2. local boundary cond. 3. SO(2) rotational sym.

on boundary is kept.

⌘(H) =regX

��0

�regX

�<0<latexit sha1_base64="/Wo4FR0mfFjpoHBvuUTvFHngC1U=">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</latexit><latexit sha1_base64="/Wo4FR0mfFjpoHBvuUTvFHngC1U=">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</latexit><latexit sha1_base64="/Wo4FR0mfFjpoHBvuUTvFHngC1U=">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</latexit><latexit sha1_base64="/Wo4FR0mfFjpoHBvuUTvFHngC1U=">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</latexit>

Page 11: Domain-wall fermion and index theorem › ... › Fukaya-APSDW.pdf · 1.Introduction 2.Physicistʼs view of index theorems 3.Atiyah-Singer index with massive fermion operator 4.Index

Mathematicians joined us.In August, I gave a talk to Mikio Furuta (U. Tokyo). He said “Interesting!” Moreover, only 1 week later, he proposed a proof of

[F, Furuta, Matsuo, Onogi, Yamaguchi, and Yamashita, in progress.]

1

2⌘(Hreg

DW ) = Ind(DAPS)<latexit sha1_base64="lK9MLCKgfll3VOG4fp3b5qFE1Zg=">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</latexit><latexit sha1_base64="lK9MLCKgfll3VOG4fp3b5qFE1Zg=">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</latexit><latexit sha1_base64="lK9MLCKgfll3VOG4fp3b5qFE1Zg=">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</latexit><latexit sha1_base64="lK9MLCKgfll3VOG4fp3b5qFE1Zg=">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</latexit>

Page 12: Domain-wall fermion and index theorem › ... › Fukaya-APSDW.pdf · 1.Introduction 2.Physicistʼs view of index theorems 3.Atiyah-Singer index with massive fermion operator 4.Index

Overview

with physicist-unfriendly boundary condition [APS 1975]

Ind(DAPS)<latexit sha1_base64="OwUMoHkUjEOLyCemqdI2i/cIoNI=">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</latexit>

= =with physicist-friendly

set-up (topological insulator) [FOY 2017]

1

2⌘(HDW )

<latexit sha1_base64="wgj8OT7RSel3V3JC/ODlv1aMMgs=">AAACnXichVHLSsNAFD2N7/po1U3BhcWi6KZMqqC4Kj6gC5FWrRWslCRONZgmIZkWNNQP8AdcuFJ0IaJbP8CNP+DCTxCXCm5ceJsGREW9w8ycOXPPnTlc1TZ0VzD2GJJaWtvaOzq7wt09vX2RaP/AumtVHY3nNcuwnA1VcbmhmzwvdGHwDdvhSkU1eEHdm2/cF2rccXXLXBP7Nt+qKDumXtY1RRBVisaKZUfRPLnupepFLpTxTMlbKNQn4qVogiWZH/GfQA5AAkFkregtitiGBQ1VVMBhQhA2oMClsQkZDDZxW/CIcwjp/j1HHWHSVimLU4ZC7B6tO3TaDFiTzo2arq/W6BWDpkPKOEbZA7tkL+yeXbEn9v5rLc+v0fjLPu1qU8vtUuQotvr2r6pCu8Dup+oPhUrZf3sSKGPG96KTN9tnGi61Zv3awfHL6uzKqDfGztgz+Ttlj+yOHJq1V+0ix1dOEKYGyd/b8ROsp5LyZDKVm0qk54JWdWIIIxinfkwjjQyyyNO7hzjHNW6kYWlRWpKWm6lSKNAM4ktIhQ9F9Zke</latexit>

in physicist’s sense

=

THIS WORK! [FMOYY 2019]

Page 13: Domain-wall fermion and index theorem › ... › Fukaya-APSDW.pdf · 1.Introduction 2.Physicistʼs view of index theorems 3.Atiyah-Singer index with massive fermion operator 4.Index

Contents✔ 1. Introduction

2. Physicist’s view of index theorems 3. Atiyah-Singer index with massive fermion operator 4. Index with domain-wall fermion Dirac operator [F, Onogi, Yamaguchi

2017]

5. Mathematical proof [this work] 6. Discussion 7. Summary

Page 14: Domain-wall fermion and index theorem › ... › Fukaya-APSDW.pdf · 1.Introduction 2.Physicistʼs view of index theorems 3.Atiyah-Singer index with massive fermion operator 4.Index

A theorem on the number of solutions of Dirac equation

#sol with + chirality #sol with - chirality

Atiyah-Singer index theorem [1968] on a manifold without boundary

we consider U(1) or SU(N) gauge field (connection).

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Page 15: Domain-wall fermion and index theorem › ... › Fukaya-APSDW.pdf · 1.Introduction 2.Physicistʼs view of index theorems 3.Atiyah-Singer index with massive fermion operator 4.Index

Dirac equation = EOM of electronsSchrodinger equation (non-relativisitic)

Klein-Gordon equation (consistent only for bosons) Dirac equation

Dirac operator:

⇥�@2t + @2i +m2

⇤ = 0.

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[�i�µ@µ +m] [i�µ@

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[i�µ@µ +m] = 0.

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D = �µ(@µ + iAµ)<latexit sha1_base64="VHDPLeMhoyNawD/KYqdCNgxnyiw=">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</latexit><latexit sha1_base64="VHDPLeMhoyNawD/KYqdCNgxnyiw=">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</latexit><latexit sha1_base64="VHDPLeMhoyNawD/KYqdCNgxnyiw=">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</latexit><latexit sha1_base64="VHDPLeMhoyNawD/KYqdCNgxnyiw=">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</latexit>

i

@

@t

+1

2m

@

2

@x

2i

� = 0.

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Page 16: Domain-wall fermion and index theorem › ... › Fukaya-APSDW.pdf · 1.Introduction 2.Physicistʼs view of index theorems 3.Atiyah-Singer index with massive fermion operator 4.Index

Gamma matrices and chirality

4x4 gamma matrices in Euclidean 4D space

Chirality operator

Algebra

D = �µ(@µ + iAµ)<latexit sha1_base64="VHDPLeMhoyNawD/KYqdCNgxnyiw=">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</latexit><latexit sha1_base64="VHDPLeMhoyNawD/KYqdCNgxnyiw=">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</latexit><latexit sha1_base64="VHDPLeMhoyNawD/KYqdCNgxnyiw=">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</latexit><latexit sha1_base64="VHDPLeMhoyNawD/KYqdCNgxnyiw=">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</latexit>

�i =

✓0 i�i

�i�i 0

<latexit sha1_base64="f5I8AujvIa+4dt3vOCzy0gGkROU=">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</latexit><latexit sha1_base64="f5I8AujvIa+4dt3vOCzy0gGkROU=">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</latexit><latexit sha1_base64="f5I8AujvIa+4dt3vOCzy0gGkROU=">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</latexit><latexit sha1_base64="f5I8AujvIa+4dt3vOCzy0gGkROU=">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</latexit>

{�µ, �⌫} = 2�µ⌫<latexit sha1_base64="p+yFn6xl9PgHi838zGXGiI/qoSY=">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</latexit><latexit sha1_base64="p+yFn6xl9PgHi838zGXGiI/qoSY=">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</latexit><latexit sha1_base64="p+yFn6xl9PgHi838zGXGiI/qoSY=">AAACr3ichVFNSxtRFD1Oq7XxK+pG6CY0pJQi8kYKLYIgunGZGJMIGR3fTF7iw/li5k3ADvkDrty56KqFLkr3umyhG/9AF/kJpUuFbrrozWRKsWJ6hzfvvHPvue8drhU4MlKM9ce0Bw/HJx5NPs5NTc/MzuXnF+qRH4e2qNm+44d7Fo+EIz1RU1I5Yi8IBXctRzSs461BvtEVYSR9b1edBGLf5R1PtqXNFVFm/oWRGB3uutw03Hj5D/Rio7e+arSEo7iZUIaYnpkvshWWRuEu0DNQRBZlP38JAy34sBHDhYAHRdgBR0RfEzoYAuL2kRAXEpJpXqCHHGljqhJUwYk9pn+HTs2M9eg86BmlaptucWiFpCygxL6xj+yaXbFP7Dv7dW+vJO0xeMsJ7dZQKwJz7nSp+vO/Kpd2haO/qhEKi6pHe1Jo43XqRZK3IGUGLu1h/+6b8+vq2k4pecbesx/k7x3rs6/k0Ove2B8qYuctcjQg/d9x3AX11RWdcOVlcWMzG9UknuApntM8XmED2yijRvee4QKf8UXTtYZ2oB0OS7WxTLOIW6HJ37XfocA=</latexit><latexit sha1_base64="p+yFn6xl9PgHi838zGXGiI/qoSY=">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</latexit>

{�5, �⌫} = 0<latexit sha1_base64="TQg9mnR0WUyPKG5Cw1wd8oowo3w=">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</latexit><latexit sha1_base64="TQg9mnR0WUyPKG5Cw1wd8oowo3w=">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</latexit><latexit sha1_base64="TQg9mnR0WUyPKG5Cw1wd8oowo3w=">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</latexit><latexit sha1_base64="TQg9mnR0WUyPKG5Cw1wd8oowo3w=">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</latexit>

Tr�5�µ�⌫�⇢�� = 4i✏µ⌫⇢�<latexit sha1_base64="p6gr6uK4klINfr07X09QDS4c52o=">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</latexit><latexit sha1_base64="p6gr6uK4klINfr07X09QDS4c52o=">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</latexit><latexit sha1_base64="p6gr6uK4klINfr07X09QDS4c52o=">AAAC43ichVFNT9tAEH2YFtK0hUAvlbhYRKl6QpuKCoRUCcGlx/AREikO0dpdwgqvbdlOBLXyB+BWVeqh4kClHlB/QH9AOfAHOOQntD0mEhcOjB2nFaDSsex5+2berJ/G9GwZhIx1R7TRBw/HxjOPso+fPJ2YzE1NbwVuy7dE2XJt16+aPBC2dEQ5lKEtqp4vuDJtUTH3VuN6pS38QLrOZnjgibriTUfuSIuHRDVy24Yy3f1o0+8YTa4Ub7xOs6FaQ+T8Qf6uO4SBbCquv9HndWkIL5A2TYtiEXXHbUm908jl2RxLQr8LiinII42Sm/sOA+/gwkILCgIOQsI2OAJ6aiiCwSOujog4n5BM6gIdZEnboi5BHZzYPfo26VRLWYfO8cwgUVt0i02vT0odBXbBTlmPnbNv7Ce7+uesKJkR/8sBZXOgFV5j8vD5xuV/VYpyiN2/qnsUJnXf7ynEDhYTL5K8eQkTu7QG89vvP/U2ltYL0Qv2hf0mfyesy36QQ6fdt76uifXPyNKCirfXcRdsvZorEl6bzy+vpKvKYAazeEn7WMAy3qKEMt17hl/ooa8J7Uj7oH0ctGojqeYZboR2fA3nVbgK</latexit><latexit sha1_base64="p6gr6uK4klINfr07X09QDS4c52o=">AAAC43ichVFNT9tAEH2YFtK0hUAvlbhYRKl6QpuKCoRUCcGlx/AREikO0dpdwgqvbdlOBLXyB+BWVeqh4kClHlB/QH9AOfAHOOQntD0mEhcOjB2nFaDSsex5+2berJ/G9GwZhIx1R7TRBw/HxjOPso+fPJ2YzE1NbwVuy7dE2XJt16+aPBC2dEQ5lKEtqp4vuDJtUTH3VuN6pS38QLrOZnjgibriTUfuSIuHRDVy24Yy3f1o0+8YTa4Ub7xOs6FaQ+T8Qf6uO4SBbCquv9HndWkIL5A2TYtiEXXHbUm908jl2RxLQr8LiinII42Sm/sOA+/gwkILCgIOQsI2OAJ6aiiCwSOujog4n5BM6gIdZEnboi5BHZzYPfo26VRLWYfO8cwgUVt0i02vT0odBXbBTlmPnbNv7Ce7+uesKJkR/8sBZXOgFV5j8vD5xuV/VYpyiN2/qnsUJnXf7ynEDhYTL5K8eQkTu7QG89vvP/U2ltYL0Qv2hf0mfyesy36QQ6fdt76uifXPyNKCirfXcRdsvZorEl6bzy+vpKvKYAazeEn7WMAy3qKEMt17hl/ooa8J7Uj7oH0ctGojqeYZboR2fA3nVbgK</latexit>

Tr�5(up to 3 �’s) = 0

<latexit sha1_base64="rtHWdhdzQaaQUGR3oq/xeaZT7TE=">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</latexit><latexit sha1_base64="rtHWdhdzQaaQUGR3oq/xeaZT7TE=">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</latexit><latexit sha1_base64="rtHWdhdzQaaQUGR3oq/xeaZT7TE=">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</latexit><latexit sha1_base64="rtHWdhdzQaaQUGR3oq/xeaZT7TE=">AAACtnichVG7ThtBFD1sEuI4AUzSRKJZxebVoLsJUaxIkaykSekHtpFssHaXwVmxL+2OLYzlH8gPUFARKQXKB1BRRYL8AAWfgFIaiSZFrndXioIF3NHMnDlzz505uoZvW6EkuphQHjx8NPk49ST99NnU9Exm9nkt9DqBKaqmZ3vBuqGHwrZcUZWWtMW6HwjdMWxRN3Y+je7rXRGElueuyZ4vNhy97VrblqlLploZrekY3m5/LRg027rj6K23SzHT8VXpqW/UXMznFsPBsvpBJbWVydIKRaGOAy0BWSRR9DLHaGILHkx04EDAhWRsQ0fIowENBJ+5DfSZCxhZ0b3AAGnWdjhLcIbO7A6vbT41Etbl86hmGKlNfsXmGbBSxTyd0xEN6Rf9oEv6c2utflRj9Jce70asFX5r5uvLyvW9Kod3iS//VHcoDM6+25PENvKRF4u9+REzcmnG9bt7+8PK+/J8f4G+0W/2d0gX9JMdut0r83tJlA+Q5gZpN9sxDmqvVzTGpdVs4WPSqhTm8ApL3I93KOAziqjyu/s4wSnOlLyyqQilHacqE4nmBf4Lxf8LPbWiDw==</latexit>

gamma matrices space-time derivatives EM field (connection)

�4 =

✓0 11 0

<latexit sha1_base64="V1OkV7ZaTAGGSsALYf4hBp0siCI=">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</latexit>

�5 = i�1�2�3�4 =

✓1 00 �1

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�i<latexit sha1_base64="ZoqmMi3HMoQ+OWkMh52J0Mtts5Y=">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</latexit>

Pauli matrices

Page 17: Domain-wall fermion and index theorem › ... › Fukaya-APSDW.pdf · 1.Introduction 2.Physicistʼs view of index theorems 3.Atiyah-Singer index with massive fermion operator 4.Index

Chirality = spin in moving direction

Left-handed fermion has Right-handed has

but this is true only for massless fermion.

�5 = 1<latexit sha1_base64="oj5mElH+MvcysHqsmLDYdzlWFX8=">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</latexit><latexit sha1_base64="oj5mElH+MvcysHqsmLDYdzlWFX8=">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</latexit><latexit sha1_base64="oj5mElH+MvcysHqsmLDYdzlWFX8=">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</latexit><latexit sha1_base64="oj5mElH+MvcysHqsmLDYdzlWFX8=">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</latexit>

�5 = �1<latexit sha1_base64="NbYSM6y9KjR7Kocf33h8U/eJ1fg=">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</latexit><latexit sha1_base64="NbYSM6y9KjR7Kocf33h8U/eJ1fg=">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</latexit><latexit sha1_base64="NbYSM6y9KjR7Kocf33h8U/eJ1fg=">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</latexit><latexit sha1_base64="NbYSM6y9KjR7Kocf33h8U/eJ1fg=">AAACj3ichVG7SgNBFD2u7/iK2gg2waDYGGZFUQQlaKOdGqOCCWF3ncTBfbE7CWjwB/wALSx8gIX4AX6AjT9g4SeIZQQbC+9uFkSD8Q4zc+bMPXfmcHXXFL5k7KVFaW1r7+js6o719Pb1D8QHh7Z9p+wZPGs4puPt6prPTWHzrBTS5LuuxzVLN/mOfrgS3O9UuOcLx96SRy7PW1rJFkVhaJKoXK6kWZZWmF2cUhOFeJKlWBiJRqBGIIko1p34A3LYhwMDZVjgsCEJm9Dg09iDCgaXuDyqxHmERHjPcYIYacuUxSlDI/aQ1hKd9iLWpnNQ0w/VBr1i0vRImcA4e2Z3rMae2D17ZZ9/1qqGNYK/HNGu17XcLQycjmQ+/lVZtEscfKuaKHTKbu5Jooj50Isgb27IBC6Nev3K8Xkts7A5Xp1gN+yN/F2zF/ZIDu3Ku3G7wTcvEKMGqb/b0Qi2p1Mq4Y2ZZHo5alUXRjGGSerHHNJYxTqy9K6LM1ziShlU5pQlJV1PVVoizTB+hLL2BUvuk5M=</latexit>

=

0

BB@

1

2

3

4

1

CCA =

0

BB@

os

1

| "i| #i os

4

1

CCA

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* os=off-shell modes, non-classical, not satisfyingE = mc2

p1 + p2/m2c2

<latexit sha1_base64="6HfkwFv9hNfSG9ycXSQCsXHE09E=">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</latexit><latexit sha1_base64="6HfkwFv9hNfSG9ycXSQCsXHE09E=">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</latexit><latexit sha1_base64="6HfkwFv9hNfSG9ycXSQCsXHE09E=">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</latexit><latexit sha1_base64="6HfkwFv9hNfSG9ycXSQCsXHE09E=">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</latexit>

Page 18: Domain-wall fermion and index theorem › ... › Fukaya-APSDW.pdf · 1.Introduction 2.Physicistʼs view of index theorems 3.Atiyah-Singer index with massive fermion operator 4.Index

Chirality = spin in moving directionFor massive fermion, we can flip the chirality by Lorentz transformation,

but for massless fermion (with speed of light) we cannot. Naively, for the index theorem, fermion needs to be massless.

Page 19: Domain-wall fermion and index theorem › ... › Fukaya-APSDW.pdf · 1.Introduction 2.Physicistʼs view of index theorems 3.Atiyah-Singer index with massive fermion operator 4.Index

Atiyah-Singer index

Eigenmodes make ± pairs except for zero-modes:

D = �µ(@µ + iAµ)<latexit sha1_base64="VHDPLeMhoyNawD/KYqdCNgxnyiw=">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</latexit><latexit sha1_base64="VHDPLeMhoyNawD/KYqdCNgxnyiw=">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</latexit><latexit sha1_base64="VHDPLeMhoyNawD/KYqdCNgxnyiw=">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</latexit><latexit sha1_base64="VHDPLeMhoyNawD/KYqdCNgxnyiw=">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</latexit>

{D, �5} = 0.<latexit sha1_base64="bMa++0fT2CXNiHgwTBoNs+GlVO8=">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</latexit><latexit sha1_base64="bMa++0fT2CXNiHgwTBoNs+GlVO8=">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</latexit><latexit sha1_base64="bMa++0fT2CXNiHgwTBoNs+GlVO8=">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</latexit><latexit sha1_base64="bMa++0fT2CXNiHgwTBoNs+GlVO8=">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</latexit>

n+ � n� = Tr�reg5 .

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�5�(x) = +�(x) ! �5D�(x) = �D�5�(x) = �D�(x)<latexit sha1_base64="2XdrI56m49wOI6Q0AZxgYNEsvHU=">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</latexit><latexit 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#sol with + chirality #sol with - chirality

Page 20: Domain-wall fermion and index theorem › ... › Fukaya-APSDW.pdf · 1.Introduction 2.Physicistʼs view of index theorems 3.Atiyah-Singer index with massive fermion operator 4.Index

Physicist-friendly description (Fujikawa method 1979) 1.Heat-kernel regularization

2. plane-wave complete set

3. perturbative expansion

Tr�reg5 = lim

M!1Tr�5e

D2

M2

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⇣D2 = DµD

µ +g

4[�µ, �⌫ ]Fµ⌫

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Page 21: Domain-wall fermion and index theorem › ... › Fukaya-APSDW.pdf · 1.Introduction 2.Physicistʼs view of index theorems 3.Atiyah-Singer index with massive fermion operator 4.Index

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

Page 22: Domain-wall fermion and index theorem › ... › Fukaya-APSDW.pdf · 1.Introduction 2.Physicistʼs view of index theorems 3.Atiyah-Singer index with massive fermion operator 4.Index

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�<latexit sha1_base64="esrqkDvu22FjxU7ro8/y0gLZqzs=">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</latexit><latexit sha1_base64="esrqkDvu22FjxU7ro8/y0gLZqzs=">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</latexit><latexit sha1_base64="esrqkDvu22FjxU7ro8/y0gLZqzs=">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</latexit><latexit sha1_base64="esrqkDvu22FjxU7ro8/y0gLZqzs=">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</latexit>

Beautiful! But physicist-unfriendly.

[�5, A] = 0.<latexit sha1_base64="LZzK259zFvQ/9XO2GV4Si+lJ/8w=">AAAB/HicZVDLSgMxFM3UV62vsS7dBEvBhQwzVdGNUHXjsoJ9QDsMmTTThiaZIcmIpdRPcSUoiFt/xJV/Y9oOYtsDFw7n3Mu994QJo0q77o+VW1ldW9/Ibxa2tnd29+z9YkPFqcSkjmMWy1aIFGFUkLqmmpFWIgniISPNcHA78ZuPRCoaiwc9TIjPUU/QiGKkjRTYxXanhzhHwfkJvPbhFXSdwC65jjsFXCZeRkogQy2wvzvdGKecCI0ZUqrtuYn2R0hqihkZFzqpIgnCA9QjbUMF4kT5o+ntY1g2ShdGsTQlNJyq/ydGiCs15KHp5Ej31aI3Ef+88twqHV36IyqSVBOBZ5uilEEdw0kSsEslwZoNDUFYUnMsxH0kEdYmr4JJwVv8eZk0Ko536lTuz0rVmyyPPDgER+AYeOACVMEdqIE6wOAJvIA38G49W6/Wh/U5a81Z2cwBmIP19QuFf5Kh</latexit>

Page 23: Domain-wall fermion and index theorem › ... › Fukaya-APSDW.pdf · 1.Introduction 2.Physicistʼs view of index theorems 3.Atiyah-Singer index with massive fermion operator 4.Index

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.

Page 24: Domain-wall fermion and index theorem › ... › Fukaya-APSDW.pdf · 1.Introduction 2.Physicistʼs view of index theorems 3.Atiyah-Singer index with massive fermion operator 4.Index

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>0d

4x✏

µ⌫⇢�

tr[Fµ⌫

F

⇢�]�⌘(iD3D)

2<latexit sha1_base64="I7GfGAPgiGlSFncQPFOz0BiA9qo=">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</latexit><latexit sha1_base64="I7GfGAPgiGlSFncQPFOz0BiA9qo=">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</latexit><latexit sha1_base64="I7GfGAPgiGlSFncQPFOz0BiA9qo=">AAADHnichVHdahNBFD679afGn8Z6I3izGCIVSZxNA4qgFC3iZX9MW8gky+5mkg7dP2ZnQ+owL+ALiHijghfiA/gAgngjXgn2EcTLCt4IenazULRYZ9mZb75zvnPm43hJwFNJyJ5hzhw7fuLk7KnK6TNnz81Vz89vpHEmfNbx4yAWW56bsoBHrCO5DNhWIpgbegHb9Hbu5fHNMRMpj6OHcjdhvdAdRXzIfVci5VSfRs61RuQ0blsVOhSur2ytFls04f2WpjySjpo47TtEW4N+e2JRlqQ8QJ2iYUajjIrtmKZ8FLpaURFaUuju/X4Z1Dk6SOhZVLKJLJ6sBBto1Zh2pEy6C3y5X1RYXNZXtWppXXGqNdIkxbIOA7sENSjXSlx9CxQGEIMPGYTAIAKJOAAXUvy6YAOBBLkeKOQEIl7EGWiooDbDLIYZLrI7uI/w1i3ZCO95zbRQ+9glwF+g0oI6+Uxek33ygbwhX8nPf9ZSRY38Lbt4elMtS5y5xxfXf/xXFeIpYftAdYTCw+yjPUkYws3CC0dvScHkLv1p/fGjJ/vrt9bq6gp5Sb6hvxdkj7xDh9H4u/9qla09g3xA9t/jOAw2Wk2bNO3Vdm3pbjmqWbgEl2EB53EDluABrEAH+/4y6kbTuG4+N9+bH81P01TTKDUX4I9lfvkNYsfMFQ==</latexit><latexit sha1_base64="I7GfGAPgiGlSFncQPFOz0BiA9qo=">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</latexit>

Page 25: Domain-wall fermion and index theorem › ... › Fukaya-APSDW.pdf · 1.Introduction 2.Physicistʼs view of index theorems 3.Atiyah-Singer index with massive fermion operator 4.Index

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>0d

4x✏

µ⌫⇢�

tr[Fµ⌫

F

⇢�]�⌘(iD3D)

2<latexit sha1_base64="I7GfGAPgiGlSFncQPFOz0BiA9qo=">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</latexit><latexit sha1_base64="I7GfGAPgiGlSFncQPFOz0BiA9qo=">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</latexit><latexit sha1_base64="I7GfGAPgiGlSFncQPFOz0BiA9qo=">AAADHnichVHdahNBFD679afGn8Z6I3izGCIVSZxNA4qgFC3iZX9MW8gky+5mkg7dP2ZnQ+owL+ALiHijghfiA/gAgngjXgn2EcTLCt4IenazULRYZ9mZb75zvnPm43hJwFNJyJ5hzhw7fuLk7KnK6TNnz81Vz89vpHEmfNbx4yAWW56bsoBHrCO5DNhWIpgbegHb9Hbu5fHNMRMpj6OHcjdhvdAdRXzIfVci5VSfRs61RuQ0blsVOhSur2ytFls04f2WpjySjpo47TtEW4N+e2JRlqQ8QJ2iYUajjIrtmKZ8FLpaURFaUuju/X4Z1Dk6SOhZVLKJLJ6sBBto1Zh2pEy6C3y5X1RYXNZXtWppXXGqNdIkxbIOA7sENSjXSlx9CxQGEIMPGYTAIAKJOAAXUvy6YAOBBLkeKOQEIl7EGWiooDbDLIYZLrI7uI/w1i3ZCO95zbRQ+9glwF+g0oI6+Uxek33ygbwhX8nPf9ZSRY38Lbt4elMtS5y5xxfXf/xXFeIpYftAdYTCw+yjPUkYws3CC0dvScHkLv1p/fGjJ/vrt9bq6gp5Sb6hvxdkj7xDh9H4u/9qla09g3xA9t/jOAw2Wk2bNO3Vdm3pbjmqWbgEl2EB53EDluABrEAH+/4y6kbTuG4+N9+bH81P01TTKDUX4I9lfvkNYsfMFQ==</latexit><latexit sha1_base64="I7GfGAPgiGlSFncQPFOz0BiA9qo=">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</latexit>

Page 26: Domain-wall fermion and index theorem › ... › Fukaya-APSDW.pdf · 1.Introduction 2.Physicistʼs view of index theorems 3.Atiyah-Singer index with massive fermion operator 4.Index

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>0d

4x✏

µ⌫⇢�

tr[Fµ⌫

F

⇢�]�⌘(iD3D)

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Page 27: Domain-wall fermion and index theorem › ... › Fukaya-APSDW.pdf · 1.Introduction 2.Physicistʼs view of index theorems 3.Atiyah-Singer index with massive fermion operator 4.Index

Contents✔ 1. Introduction

2. Physicist’s view of index theorems AS index is physicist-friendly but APS is not. 3. Atiyah-Singer index with massive fermion operator 4. Index with domain-wall fermion Dirac operator [F, Onogi, Yamaguchi

2017]

5. Mathematical proof [this work] 6. Discussion 7. Summary

Page 28: Domain-wall fermion and index theorem › ... › Fukaya-APSDW.pdf · 1.Introduction 2.Physicistʼs view of index theorems 3.Atiyah-Singer index with massive fermion operator 4.Index

Massive Dirac operator

Let’s consider a Hermitian operator:

on a manifold without boundary.

D +M =

✓M DLR

DRL M

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Anti-Hermitian Hermitian (propotional to identity matrix)

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Zero-modes & non-zero modes

Zero-modes of D = still eigenstates of H:

Non-zero modes make ± pairs

H = �5(D +M)<latexit sha1_base64="p5LMQXIq6ZnPd6BFmqhXjlSY4WQ=">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</latexit><latexit sha1_base64="p5LMQXIq6ZnPd6BFmqhXjlSY4WQ=">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</latexit><latexit sha1_base64="p5LMQXIq6ZnPd6BFmqhXjlSY4WQ=">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</latexit><latexit sha1_base64="p5LMQXIq6ZnPd6BFmqhXjlSY4WQ=">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</latexit>

H�0 = �5M�0 = ±M�0.<latexit sha1_base64="C0aKYE6gmA0VJvi6oqtiXI2qLeg=">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</latexit><latexit sha1_base64="C0aKYE6gmA0VJvi6oqtiXI2qLeg=">AAACwXichVFNS9xQFD1GW3WsdawbwU3oYOlquCktFWlBdONG8GtUcCQk8Tk+zMcjeTNUw/wB/4ALVwpdFH+AP8CNWwUX/gRxacGNC+9kIraV6g3JO+/ce+7N4brKl4kmuuwwOrteve7u6S30vel/O1AcfLeURPXYExUv8qN4xXUS4ctQVLTUvlhRsXAC1xfL7tZUK7/cEHEio3BRbyuxFji1UG5Iz9FM2cVv02ZVbUqbzO9mteYEgWN/mfmD0uKHzqaksVhvplUVNM2HfNkulqhMWZhPgZWDEvKYjYrHqGIdETzUEUAghGbsw0HCzyosEBRza0iZixnJLC/QRIG1da4SXOEwu8XfGt9Wczbke6tnkqk9nuLzG7PSxChd0C+6oVM6oiu6+2+vNOvR+pdtPt22Vih7YHd44fZFVcCnxuaj6hmFy9XPe9LYwFjmRbI3lTEtl167f2Nn72ZhfH40/UCHdM3+DuiSTthh2Pjt/ZwT8/so8IKsf9fxFCx9KltUtuY+lyYm81X1YATv8ZH38RUTmMYsKjz3EKc4w7kxZUhDGXG71OjINUP4K4z0HjUqpps=</latexit><latexit sha1_base64="C0aKYE6gmA0VJvi6oqtiXI2qLeg=">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</latexit><latexit sha1_base64="C0aKYE6gmA0VJvi6oqtiXI2qLeg=">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</latexit>

H�i = �i�i<latexit sha1_base64="0fyxNdpYF1KpaiLvJiR7kuyP0Pw=">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</latexit><latexit sha1_base64="0fyxNdpYF1KpaiLvJiR7kuyP0Pw=">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</latexit><latexit sha1_base64="0fyxNdpYF1KpaiLvJiR7kuyP0Pw=">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</latexit><latexit sha1_base64="0fyxNdpYF1KpaiLvJiR7kuyP0Pw=">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</latexit>

HD�i = �DH�i = ��iD�i<latexit sha1_base64="y6nmTZIz2hON0aR57QBnLXJe84U=">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</latexit><latexit sha1_base64="y6nmTZIz2hON0aR57QBnLXJe84U=">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</latexit><latexit sha1_base64="y6nmTZIz2hON0aR57QBnLXJe84U=">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</latexit><latexit sha1_base64="y6nmTZIz2hON0aR57QBnLXJe84U=">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</latexit>

D = �µ(@µ + iAµ)<latexit sha1_base64="VHDPLeMhoyNawD/KYqdCNgxnyiw=">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</latexit><latexit sha1_base64="VHDPLeMhoyNawD/KYqdCNgxnyiw=">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</latexit><latexit sha1_base64="VHDPLeMhoyNawD/KYqdCNgxnyiw=">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</latexit><latexit sha1_base64="VHDPLeMhoyNawD/KYqdCNgxnyiw=">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</latexit>

Page 30: Domain-wall fermion and index theorem › ... › Fukaya-APSDW.pdf · 1.Introduction 2.Physicistʼs view of index theorems 3.Atiyah-Singer index with massive fermion operator 4.Index

Eta invariant of massive Dirac operator

coincides with the original AS index!

⌘(H) =

X

i

sgn�i

= # of +M �# of �M<latexit sha1_base64="EYJacf+qyQBslO91dxUqFJEVKZ4=">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</latexit><latexit sha1_base64="EYJacf+qyQBslO91dxUqFJEVKZ4=">AAAC2HichVE9axRRFD2ZRI3rRzaxEWwebhIikuWOCAkBIWizjZDdzSYhmbDMTN6uj8wXM2+XJMuCnWhpY2GlYBH8AWJt4x9IkcrWxTKCjYV3ZgdCDMY7vHn3nXvOfe9wnchTiSY6HjFGxy5dvjJ+tXDt+o2bE8XJqbUk7MSubLihF8Ybjp1ITwWyoZX25EYUS9t3PLnu7D5J6+tdGScqDFb1fiS3fbsdqJZybc1Qs1i3pLbnKvfE7CNhJR2/qYTlO+FeL2kHfWF53GnHZrBgWUwZlqySCFti+v7T6b6YF2ew+RRrFktUpizE+cTMkxLyWAmLn2BhByFcdOBDIoDm3IONhL8tmCBEjG2jx1jMmcrqEn0UWNthlmSGzegu/9t82srRgM9pzyRTu3yLxytmpcAMHdEhndBX+kgD+v3PXr2sR/qWfd6doVZGzYlXt+u//qvyedd4dqq6QOEw+2JPGi0sZl4Ue4syJHXpDvt3D96c1JdqM71Zek8/2N87OqYv7DDo/nQ/VGXtLQo8IPPvcZxP1h6UTSqb1Yel5cf5qMZxB3cxx/NYwDIqWEGD7/2Mb/iOgbFpPDdeGC+HVGMk19zCmTBe/wE28azu</latexit><latexit sha1_base64="EYJacf+qyQBslO91dxUqFJEVKZ4=">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</latexit><latexit sha1_base64="EYJacf+qyQBslO91dxUqFJEVKZ4=">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</latexit>

H = �5(D +M)<latexit sha1_base64="p5LMQXIq6ZnPd6BFmqhXjlSY4WQ=">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</latexit><latexit sha1_base64="p5LMQXIq6ZnPd6BFmqhXjlSY4WQ=">AAACmnichVFdKwRRGH6M7/Wxi5TiYrKtSG3viEgp4YKk2LVLoW1mHGsyX83MbrH5A/6AC1eUC/wAP8CNP+DCT5BLyo0L785OCeE9nXOe85z3ec95ejXXNPyA6LFOqm9obGpuaY21tXd0xhNd3XnfKXm6yOmO6XibmuoL07BFLjACU2y6nlAtzRQb2sF89X6jLDzfcOz14NAVO5ZatI09Q1cDpgqJ3kV5Rt4uqpalFibk4QV5VF6RRwqJJKUpDPknUCKQRBSrTuIW29iFAx0lWBCwETA2ocLnsQUFBJe5HVSY8xgZ4b3AMWKsLXGW4AyV2QNei3zailibz9WafqjW+RWTp8dKGSl6oCt6oXu6oSd6/7VWJaxR/csh71pNK9xC/KQv+/avyuI9wP6n6g+Fxtl/ewqwh6nQi8He3JCputRr9ctHpy/Z6UyqMkQX9Mz+zumR7tihXX7VL9dE5gwxbpDyvR0/QX4srVBaWRtPzs5FrWpBPwYxzP2YxCwWsYocv1vBOa5xIw1Ic9KStFxLleoiTQ++hLT+Aaz1lbU=</latexit><latexit sha1_base64="p5LMQXIq6ZnPd6BFmqhXjlSY4WQ=">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</latexit><latexit sha1_base64="p5LMQXIq6ZnPd6BFmqhXjlSY4WQ=">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</latexit>

Page 31: Domain-wall fermion and index theorem › ... › Fukaya-APSDW.pdf · 1.Introduction 2.Physicistʼs view of index theorems 3.Atiyah-Singer index with massive fermion operator 4.Index

Eta invariant of massive Dirac operator

coincides with the original AS index! In fact, we need a factor 1/2.

⌘(H) =

X

i

sgn�i

= # of +M �# of �M<latexit sha1_base64="EYJacf+qyQBslO91dxUqFJEVKZ4=">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</latexit><latexit sha1_base64="EYJacf+qyQBslO91dxUqFJEVKZ4=">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</latexit><latexit sha1_base64="EYJacf+qyQBslO91dxUqFJEVKZ4=">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</latexit><latexit sha1_base64="EYJacf+qyQBslO91dxUqFJEVKZ4=">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</latexit>

H = �5(D +M)<latexit sha1_base64="p5LMQXIq6ZnPd6BFmqhXjlSY4WQ=">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</latexit><latexit sha1_base64="p5LMQXIq6ZnPd6BFmqhXjlSY4WQ=">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</latexit><latexit sha1_base64="p5LMQXIq6ZnPd6BFmqhXjlSY4WQ=">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</latexit><latexit sha1_base64="p5LMQXIq6ZnPd6BFmqhXjlSY4WQ=">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</latexit>

Index(D) =

1

2

⌘(H)

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Page 32: Domain-wall fermion and index theorem › ... › Fukaya-APSDW.pdf · 1.Introduction 2.Physicistʼs view of index theorems 3.Atiyah-Singer index with massive fermion operator 4.Index

always jumps by 2.

To increase + modes, we have to borrow one from - (UV) modes.

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

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�M<latexit sha1_base64="AjbFY+pH/iFUQuZs57UzQAx3wn4=">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</latexit><latexit sha1_base64="AjbFY+pH/iFUQuZs57UzQAx3wn4=">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</latexit><latexit sha1_base64="AjbFY+pH/iFUQuZs57UzQAx3wn4=">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</latexit><latexit sha1_base64="AjbFY+pH/iFUQuZs57UzQAx3wn4=">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</latexit>

…I = 0

<latexit sha1_base64="z3ZENP780TAeKYvoBlrwINmdU6s=">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</latexit><latexit sha1_base64="z3ZENP780TAeKYvoBlrwINmdU6s=">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</latexit><latexit sha1_base64="z3ZENP780TAeKYvoBlrwINmdU6s=">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</latexit><latexit sha1_base64="z3ZENP780TAeKYvoBlrwINmdU6s=">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</latexit>

I = 1<latexit sha1_base64="pGyjXuI5I8EiTAoUfsywzZTstTE=">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</latexit><latexit sha1_base64="pGyjXuI5I8EiTAoUfsywzZTstTE=">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</latexit><latexit sha1_base64="pGyjXuI5I8EiTAoUfsywzZTstTE=">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</latexit><latexit sha1_base64="pGyjXuI5I8EiTAoUfsywzZTstTE=">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</latexit>

⌘(H)<latexit sha1_base64="A89dKLbbcDrqFW1tQ1LXIFhsn3g=">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</latexit><latexit sha1_base64="A89dKLbbcDrqFW1tQ1LXIFhsn3g=">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</latexit><latexit sha1_base64="A89dKLbbcDrqFW1tQ1LXIFhsn3g=">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</latexit><latexit sha1_base64="A89dKLbbcDrqFW1tQ1LXIFhsn3g=">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</latexit>

Index(D) =

1

2

⌘(H).<latexit sha1_base64="CGEzvhiJuRoQzxJu69hjKjylcSU=">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</latexit><latexit sha1_base64="CGEzvhiJuRoQzxJu69hjKjylcSU=">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</latexit><latexit sha1_base64="CGEzvhiJuRoQzxJu69hjKjylcSU=">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</latexit><latexit sha1_base64="CGEzvhiJuRoQzxJu69hjKjylcSU=">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</latexit>

unpairedpaired

Page 33: Domain-wall fermion and index theorem › ... › Fukaya-APSDW.pdf · 1.Introduction 2.Physicistʼs view of index theorems 3.Atiyah-Singer index with massive fermion operator 4.Index

Perturbative “proof” (in physics sense)using Pauli-Villars regulator

HPV = �5(D + ⇤), ⇤ � M<latexit sha1_base64="kHBv7p5qdD4DtpkVhrEwsTuhFWc=">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</latexit><latexit sha1_base64="kHBv7p5qdD4DtpkVhrEwsTuhFWc=">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</latexit><latexit sha1_base64="kHBv7p5qdD4DtpkVhrEwsTuhFWc=">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</latexit><latexit sha1_base64="kHBv7p5qdD4DtpkVhrEwsTuhFWc=">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</latexit>

H = �5(D +M)<latexit sha1_base64="p5LMQXIq6ZnPd6BFmqhXjlSY4WQ=">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</latexit><latexit sha1_base64="p5LMQXIq6ZnPd6BFmqhXjlSY4WQ=">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</latexit><latexit sha1_base64="p5LMQXIq6ZnPd6BFmqhXjlSY4WQ=">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</latexit><latexit sha1_base64="p5LMQXIq6ZnPd6BFmqhXjlSY4WQ=">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</latexit>

Fujikawa-method does not contribute.

⌘(H) = lims!0

TrH

(pH2)1+s

=1p⇡

Z 1

0dtt�1/2TrHe�tH2

=1p⇡

Z 1

0dt0t0�1/2Tr�5

✓sgnM +

D

M

◆e�t0D†D/M2

e�t0 ,

= sgnM1

32⇡2

Zd4x ✏µ⌫⇢�trcF

µ⌫F ⇢� +O(1/M2).<latexit sha1_base64="KTbFE0Ypzrx0UeomhlJoPSgdwAM=">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</latexit><latexit sha1_base64="KTbFE0Ypzrx0UeomhlJoPSgdwAM=">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</latexit><latexit sha1_base64="KTbFE0Ypzrx0UeomhlJoPSgdwAM=">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</latexit><latexit sha1_base64="KTbFE0Ypzrx0UeomhlJoPSgdwAM=">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</latexit>

1

2⌘(H)reg =

1

2[⌘(H)� ⌘(HPV )] .

<latexit sha1_base64="4NuvKzY/UMUm9mnIhEDchEFGH3c=">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</latexit><latexit sha1_base64="4NuvKzY/UMUm9mnIhEDchEFGH3c=">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</latexit><latexit sha1_base64="4NuvKzY/UMUm9mnIhEDchEFGH3c=">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</latexit><latexit sha1_base64="4NuvKzY/UMUm9mnIhEDchEFGH3c=">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</latexit>

Page 34: Domain-wall fermion and index theorem › ... › Fukaya-APSDW.pdf · 1.Introduction 2.Physicistʼs view of index theorems 3.Atiyah-Singer index with massive fermion operator 4.Index

Contents✔ 1. Introduction

2. Physicist’s view of index theorems AS index is physicist-friendly but APS is not. 3. Atiyah-Singer index with massive fermion operator coincides with the AS index. 4. Index with domain-wall fermion Dirac operator [F, Onogi, Yamaguchi

2017]

5. Mathematical proof [this work] 6. Discussion 7. Summary

I = ⌘(�5(D +M))reg/2<latexit sha1_base64="9Hjmz73sFWiddmPGmw/3Dli7i7c=">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</latexit><latexit sha1_base64="9Hjmz73sFWiddmPGmw/3Dli7i7c=">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</latexit><latexit sha1_base64="9Hjmz73sFWiddmPGmw/3Dli7i7c=">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</latexit><latexit sha1_base64="9Hjmz73sFWiddmPGmw/3Dli7i7c=">AAACr3icSyrIySwuMTC4ycjEzMLKxs7BycXNw8vHLyAoFFacX1qUnBqanJ+TXxSRlFicmpOZlxpaklmSkxpRUJSamJuUkxqelO0Mkg8vSy0qzszPCympLEiNzU1Mz8tMy0xOLAEKxQtoxeQmlmSkFSVmV3vWKtgqxKSWJGrEpCfm5ibGm2q4aPtqasZVF6Wm1+obxQsoG+oZgIGCgZ6pgaGlmSGQARWBSSkzQEFAvsB2hhiGFIZ8hmSGUoZchlSGPIYSIDuHIZGhGAijGQwZDBgKgGKxDNVAsSIgKxMsn8pQy8AF1FsKVJUKVJEIFM0GkulAXjRUNA/IB5lZDNadDLQlB4iLgDoVGFQNrhqsNPhscMJgtcFLgz84zaoGmwFySyWQToLoTS2I5++SCP5OUFcukC5hyEDowqMjCagav59KGNIYLMB+yQT6rQAsAvJlMsT8sqrpn4OtglSr1QwWGbwG+m+hwU2Dw0Af5pV9SV4amBo0m4ELOYJwM8KM9AwN9AwDTZQdnKBRxcEgzaDEoAGMD3MGBwYPhgCGUKC9Exi2MexnOMBkyBTOFMeUAFHKxAjVI8yAApgyAQ9in50=</latexit>

Page 35: Domain-wall fermion and index theorem › ... › Fukaya-APSDW.pdf · 1.Introduction 2.Physicistʼs view of index theorems 3.Atiyah-Singer index with massive fermion operator 4.Index

More physical set-up?In physics, 1. Any boundary has “outside”: manifold + boundary → domain-wall.

outside

Page 36: Domain-wall fermion and index theorem › ... › Fukaya-APSDW.pdf · 1.Introduction 2.Physicistʼs view of index theorems 3.Atiyah-Singer index with massive fermion operator 4.Index

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)

Page 37: Domain-wall fermion and index theorem › ... › Fukaya-APSDW.pdf · 1.Introduction 2.Physicistʼs view of index theorems 3.Atiyah-Singer index with massive fermion operator 4.Index

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.

Page 38: Domain-wall fermion and index theorem › ... › Fukaya-APSDW.pdf · 1.Introduction 2.Physicistʼs view of index theorems 3.Atiyah-Singer index with massive fermion operator 4.Index

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.

Page 39: Domain-wall fermion and index theorem › ... › Fukaya-APSDW.pdf · 1.Introduction 2.Physicistʼs view of index theorems 3.Atiyah-Singer index with massive fermion operator 4.Index

Domain-wall Dirac operatorLet us consider

on 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 ]

Page 40: Domain-wall fermion and index theorem › ... › Fukaya-APSDW.pdf · 1.Introduction 2.Physicistʼs view of index theorems 3.Atiyah-Singer index with massive fermion operator 4.Index

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

Fujikawa method

coincides with APS index !

Page 41: Domain-wall fermion and index theorem › ... › Fukaya-APSDW.pdf · 1.Introduction 2.Physicistʼs view of index theorems 3.Atiyah-Singer index with massive fermion operator 4.Index

PV part = Atiyah-Singer index

Fujikawa-method does not contribute.

Page 42: Domain-wall fermion and index theorem › ... › Fukaya-APSDW.pdf · 1.Introduction 2.Physicistʼs view of index theorems 3.Atiyah-Singer index with massive fermion operator 4.Index

Domain-wall fermion partNow let’s compute

In the free fermion case,

→ eigenvalue problem = Schrodinger equation with δ-function-like potential.

Page 43: Domain-wall fermion and index theorem › ... › Fukaya-APSDW.pdf · 1.Introduction 2.Physicistʼs view of index theorems 3.Atiyah-Singer index with massive fermion operator 4.Index

Complete set in the free caseSolutions to are

where and 3D direction = conventional plane waves.

Edge mode appears !

Page 44: Domain-wall fermion and index theorem › ... › Fukaya-APSDW.pdf · 1.Introduction 2.Physicistʼs view of index theorems 3.Atiyah-Singer index with massive fermion operator 4.Index

“Automatic” boundary condition

We didn’t put any boundary condition by hand. But

is automatically satisfied due to the δ-function-like potential. This condition is LOCAL and PRESERVES angular-momentum in x4 direction but DOES NOT keep chirality.

Page 45: Domain-wall fermion and index theorem › ... › Fukaya-APSDW.pdf · 1.Introduction 2.Physicistʼs view of index theorems 3.Atiyah-Singer index with massive fermion operator 4.Index

Fujikawa-method

We insert our complete set

Perturbative expansion

( See our paper for the details. ) 100% edge-mode effect

=

Page 46: Domain-wall fermion and index theorem › ... › Fukaya-APSDW.pdf · 1.Introduction 2.Physicistʼs view of index theorems 3.Atiyah-Singer index with massive fermion operator 4.Index

Total index

Page 47: Domain-wall fermion and index theorem › ... › Fukaya-APSDW.pdf · 1.Introduction 2.Physicistʼs view of index theorems 3.Atiyah-Singer index with massive fermion operator 4.Index

Contents✔ 1. Introduction

2. Physicist’s view of index theorems AS index is physicist-friendly but APS is not. 3. Atiyah-Singer index with massive fermion operator coincides with the AS index. 4. Index with domain-wall fermion Dirac operator [F, Onogi, Yamaguchi

2017] coincides with the APS index.

5. Mathematical proof [this work] 6. Discussion 7. Summary

I = ⌘(�5(D +M))reg/2<latexit sha1_base64="9Hjmz73sFWiddmPGmw/3Dli7i7c=">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</latexit><latexit sha1_base64="9Hjmz73sFWiddmPGmw/3Dli7i7c=">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</latexit><latexit sha1_base64="9Hjmz73sFWiddmPGmw/3Dli7i7c=">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</latexit><latexit sha1_base64="9Hjmz73sFWiddmPGmw/3Dli7i7c=">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</latexit>

I = ⌘(�5(D +M✏(x4)))reg

/2<latexit sha1_base64="BND6Kli/5C5jJD6YujX9xRn1e0c=">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</latexit><latexit sha1_base64="BND6Kli/5C5jJD6YujX9xRn1e0c=">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</latexit><latexit sha1_base64="BND6Kli/5C5jJD6YujX9xRn1e0c=">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</latexit><latexit sha1_base64="BND6Kli/5C5jJD6YujX9xRn1e0c=">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</latexit>

Page 48: Domain-wall fermion and index theorem › ... › Fukaya-APSDW.pdf · 1.Introduction 2.Physicistʼs view of index theorems 3.Atiyah-Singer index with massive fermion operator 4.Index

This talk = A mathematical proof for

APS 1. massless Dirac

(even in bulk) 2. non-local boundary cond.

(depending on gauge fields) 3. SO(2) rotational sym. on

boundary is lost. 4. no edge mode appears. 5. manifold + boundary

Domain-wall fermion

1. massive Dirac in bulk (massless mode at edge)

2. local boundary cond. 3. SO(2) rotational sym. on boundary is

kept. 4. Edge mode describes eta-invariant. 5. closed manifold + domain-wall

Ind(DAPS) =1

2⌘(Hreg

DW )<latexit sha1_base64="EoRPPh9uCUnfA3cWijURhtihfY8=">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</latexit>

on general even-dimensional manifold.

Page 49: Domain-wall fermion and index theorem › ... › Fukaya-APSDW.pdf · 1.Introduction 2.Physicistʼs view of index theorems 3.Atiyah-Singer index with massive fermion operator 4.Index

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.

Page 50: Domain-wall fermion and index theorem › ... › Fukaya-APSDW.pdf · 1.Introduction 2.Physicistʼs view of index theorems 3.Atiyah-Singer index with massive fermion operator 4.Index

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 surfaceInd(�s(D

d + @s + i�sM(s))) =

Ind(Dd)⇥ Ind(�s@s +M(s))<latexit sha1_base64="U/+LkacMWAg/NQsEiODHibzT5gg=">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</latexit>

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

exists only in even-dim.

ZF ^ F ^ · · ·

<latexit sha1_base64="NHQVaUQhvfgUOhxRh5ItUsS+/NQ=">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</latexit>

Ind(Dodd�dim

APS

) =1

2

⇥⌘(Dboundary1)� ⌘(Dboundary2)

⇤<latexit sha1_base64="DYKpN+fKUFAel6kYrwNbGodjl+Y=">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</latexit>

Page 52: Domain-wall fermion and index theorem › ... › Fukaya-APSDW.pdf · 1.Introduction 2.Physicistʼs view of index theorems 3.Atiyah-Singer index with massive fermion operator 4.Index

5-dimensional Dirac operatorwe consider

where and is independent of s,

D

5D =

✓0 @s + �5(D4D +m(s, x))

�@s + �5(D4D +m(s, x)) 0

<latexit sha1_base64="agwpV6hLJnUZ61UpGJbAqlTC4qs=">AAADIHiclVHLahRBFL3dvmL7yKgbwU3hkDhDdLitCYogBM3CZR5OEkjFobpS0ynSL6prBsdmfsAfcJGVARfiB/gBbrITFy7iH4jLCG5ceKenQTQY8TbddevUOaf6cIMs0rlFPHDcEydPnT4zcdY7d/7CxcnapcuredozUrVlGqVmPRC5inSi2lbbSK1nRok4iNRasPNodL7WVybXafLEDjK1GYsw0V0thSWoU9tlC08LbmI2tzBkDxiPVNc2eKBCnRTCGDEYFlIOPWTTjGfCWC2iTs5mGA9FHIvOHGtU+lnSz7C4kd981mxy7t36DzqZo8dVslVdyY0Ot22zU6tjC8tiRxu/aupQ1WJaewcctiAFCT2IQUEClvoIBOT0bIAPCBlhm1AQZqjT5bmCIXik7RFLEUMQukPfkHYbFZrQfuSZl2pJt0T0GlIymMJP+AYPcR/f4hf88VevovQY/cuA1mCsVVln8sXVle//VMW0Wtj+pTpGERD7+EwWunCvzKIpW1Yio5Ry7N9//vJw5f7yVDGNe/iV8r3CA3xPCZP+N/l6SS3vgkcD8v8cx9Fm9XbLv9PCpdn6/MNqVBNwDa5Dg+ZxF+bhMSxCG6TjODccdHx3z913P7gfx1TXqTRX4LdyP/8EVn7DzA==</latexit>

m(x, s) =

8<

:

M for x4 > 0 & s > 0

0 for x4 = 0 & s = 0

�M2 otherwise

<latexit sha1_base64="VQSdUpxPqquH7/t8shljSh5D4Sk=">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</latexit>Aµ<latexit sha1_base64="8/onSvm0+gH97kwklOoGVPtlQzA=">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</latexit>

Page 53: Domain-wall fermion and index theorem › ... › Fukaya-APSDW.pdf · 1.Introduction 2.Physicistʼs view of index theorems 3.Atiyah-Singer index with massive fermion operator 4.Index

on X4D x R,s = x5

<latexit sha1_base64="ngVsWB+3MLhm/sJGqiJv0+dydOs=">AAACiHichVG7SgNBFD1ZXzG+ojaCTTAoVuHGBz5ACLGxzMOYgErYXUcd3Owuu5tgDP6AjaWKlYKF+AF+gI0/YOEniKWCjYU3mwVRUe8wM2fO3HNnDlezDel6RI8hpa29o7Mr3B3p6e3rH4gODq25VtXRRUG3DMspaaorDGmKgic9Q5RsR6gVzRBFbW+5eV+sCceVlrnq1W2xWVF3TLktddVjquAu7Zdny9E4JciP2E+QDEAcQWSs6C02sAULOqqoQMCEx9iACpfHOpIg2MxtosGcw0j69wKHiLC2ylmCM1Rm93jd4dN6wJp8btZ0fbXOrxg8HVbGME4PdE0vdE839ETvv9Zq+DWaf6nzrrW0wi4PHI3k3/5VVXj3sPup+kOhcfbfnjxsY973Itmb7TNNl3qrfu3g5CW/mBtvTNAlPbO/C3qkO3Zo1l71q6zInSPCDUp+b8dPsDaVSE4nKDsTT6WDVoUxijFMcj/mkMIKMijwuxLHOMWZElFImVMWWqlKKNAM40so6Q8O/pFd</latexit>

x4<latexit sha1_base64="UAsxXe+CpcoAsu0hQ34K726xFXg=">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</latexit>

and compute

in two different ways: 1. localization 2. eta-inv. at

Ind(D5D)<latexit sha1_base64="844Fk8waRowz9eblEafDas1cCY4=">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</latexit>

X4D<latexit sha1_base64="1Mr410h+DmLTbfElzgwmRbpGPiw=">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</latexit>

s = ±1.<latexit sha1_base64="yRC16mJDR3vvEjkejPSAa+LfTmA=">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</latexit>

Page 54: Domain-wall fermion and index theorem › ... › Fukaya-APSDW.pdf · 1.Introduction 2.Physicistʼs view of index theorems 3.Atiyah-Singer index with massive fermion operator 4.Index

LocalizationTheorem 2 tells us

and on the massless surface

theorem 1 indicates

s = x5<latexit sha1_base64="ngVsWB+3MLhm/sJGqiJv0+dydOs=">AAACiHichVG7SgNBFD1ZXzG+ojaCTTAoVuHGBz5ACLGxzMOYgErYXUcd3Owuu5tgDP6AjaWKlYKF+AF+gI0/YOEniKWCjYU3mwVRUe8wM2fO3HNnDlezDel6RI8hpa29o7Mr3B3p6e3rH4gODq25VtXRRUG3DMspaaorDGmKgic9Q5RsR6gVzRBFbW+5eV+sCceVlrnq1W2xWVF3TLktddVjquAu7Zdny9E4JciP2E+QDEAcQWSs6C02sAULOqqoQMCEx9iACpfHOpIg2MxtosGcw0j69wKHiLC2ylmCM1Rm93jd4dN6wJp8btZ0fbXOrxg8HVbGME4PdE0vdE839ETvv9Zq+DWaf6nzrrW0wi4PHI3k3/5VVXj3sPup+kOhcfbfnjxsY973Itmb7TNNl3qrfu3g5CW/mBtvTNAlPbO/C3qkO3Zo1l71q6zInSPCDUp+b8dPsDaVSE4nKDsTT6WDVoUxijFMcj/mkMIKMijwuxLHOMWZElFImVMWWqlKKNAM40so6Q8O/pFd</latexit>

x4<latexit sha1_base64="UAsxXe+CpcoAsu0hQ34K726xFXg=">AAACiHichVG7SgNBFD1ZXzE+ErURbIJBsQo3KviogjaWJjFRUAm766hDNrvL7iYYQ37AxlLFSsFC/AA/wMYfsMgniGUEGwtvNguiot5hZs6cuefOHK5mG9L1iJohpau7p7cv3B8ZGBwajsZGRguuVXF0kdctw3K2NNUVhjRF3pOeIbZsR6hlzRCbWmm1fb9ZFY4rLXPDq9lit6wemHJf6qrHVP6oWJ9vFGMJSpIf8Z8gFYAEgli3YvfYwR4s6KigDAETHmMDKlwe20iBYDO3izpzDiPp3ws0EGFthbMEZ6jMlng94NN2wJp8btd0fbXOrxg8HVbGMUVPdEsteqQ7eqb3X2vV/Rrtv9R41zpaYRejJ+O5t39VZd49HH6q/lBonP23Jw/7WPS9SPZm+0zbpd6pXz0+a+WWs1P1abqmF/Z3RU16YIdm9VW/yYjsJSLcoNT3dvwEhdlkai5JmflEeiVoVRgTmMQM92MBaaxhHXl+V+IU57hQIgopC8pSJ1UJBZoxfAll5QOmM5Gk</latexit>

Ind(D5D)|M,M2!1 = Ind(D4D

m=0surface)⇥ IndD1Dnormal| {z }

=1<latexit sha1_base64="CPpC3MFPdbiXtLOlDnkmlfVf2Dw=">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</latexit>

=

Ind(D4Dm=0surface) = Ind(D

X4Dx4>0

APS )<latexit sha1_base64="H8XN0fg5m56T5J04Cjn0MB0EFqc=">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</latexit>

X4Dx4>0

<latexit sha1_base64="YeTD7S5aX3Hj2nq5Bu3wmTCt870=">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</latexit>

X4Dx4>0

<latexit sha1_base64="YeTD7S5aX3Hj2nq5Bu3wmTCt870=">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</latexit>

Page 55: Domain-wall fermion and index theorem › ... › Fukaya-APSDW.pdf · 1.Introduction 2.Physicistʼs view of index theorems 3.Atiyah-Singer index with massive fermion operator 4.Index

Boundary eta invariantsTheorem 1 tells us

and from theorem 3, we obtain

therefore,

s = x5<latexit sha1_base64="ngVsWB+3MLhm/sJGqiJv0+dydOs=">AAACiHichVG7SgNBFD1ZXzG+ojaCTTAoVuHGBz5ACLGxzMOYgErYXUcd3Owuu5tgDP6AjaWKlYKF+AF+gI0/YOEniKWCjYU3mwVRUe8wM2fO3HNnDlezDel6RI8hpa29o7Mr3B3p6e3rH4gODq25VtXRRUG3DMspaaorDGmKgic9Q5RsR6gVzRBFbW+5eV+sCceVlrnq1W2xWVF3TLktddVjquAu7Zdny9E4JciP2E+QDEAcQWSs6C02sAULOqqoQMCEx9iACpfHOpIg2MxtosGcw0j69wKHiLC2ylmCM1Rm93jd4dN6wJp8btZ0fbXOrxg8HVbGME4PdE0vdE839ETvv9Zq+DWaf6nzrrW0wi4PHI3k3/5VVXj3sPup+kOhcfbfnjxsY973Itmb7TNNl3qrfu3g5CW/mBtvTNAlPbO/C3qkO3Zo1l71q6zInSPCDUp+b8dPsDaVSE4nKDsTT6WDVoUxijFMcj/mkMIKMijwuxLHOMWZElFImVMWWqlKKNAM40so6Q8O/pFd</latexit>

x4<latexit sha1_base64="UAsxXe+CpcoAsu0hQ34K726xFXg=">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</latexit>

Ind(D5D) = Ind(D5DAPS b.c.ats=±1

)<latexit sha1_base64="JinTDFsGQj7ORvapMSmte/S4K1A=">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</latexit>

Ind(D5DAPS b.c.ats=±1

) =1

2

⇥⌘(D4D

s=1)� ⌘(D4Ds=�1)

⇤<latexit sha1_base64="/OgwWfYHMvysCczgdV9cGZXhidE=">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</latexit>

=1

2

⇥⌘(�5(D

4D +M✏(x4))� ⌘(�5(D4D �M2)

⇤=

1

2⌘

PV reg.(�5(D4D +M✏(x4))

<latexit sha1_base64="dWyeE2RBlFwAHSTWXG9Od84LCZo=">AAADLXicjVFNaxNBGH53/apR26gXwctgqCRIw2yMKEJK0R68FNLWpIVsusyOk+3Q/WJ2EmyX/QP9Az14UhAUDx79AV78Az0UxKuIxwpePPhms+BHafVdZuedZ57nfefhdWNfJprSA8M8dfrM2XNT50sXLl6anilfvtJNoqHiosMjP1LrLkuEL0PR0VL7Yj1WggWuL9bcrYfj+7WRUImMwsd6Oxb9gHmhHEjONEJO+VXLHijGUytLG5nti4Hu2UKzqu2xIGDOHVJd3EhtFZDmYkZukSVbxIn0o7D61GnWamTuePLcktOo2Up6m7pfapHf26BmI213iRJePfvPVk65Qus0D3I0sYqkAkW0o/I7sOEJRMBhCAEICEFj7gODBL8eWEAhRqwPKWIKM5nfC8ighNohsgQyGKJb+Pfw1CvQEM/jmkmu5tjFx6VQSWCW7tPX9JB+oG/oF/rj2FppXmP8lm3c3YlWxM7M7rXV7/9UBbhr2PylOkHhIvtkTxoGcC/3ItFbnCNjl3xSf7Szd7h6f2U2vUlf0K/o7zk9oO/RYTj6xl8ui5VnUMIBWX+P42jSbdSt23W63KwsPChGNQXX4QZUcR53YQEeQRs6wI1po2m0jHnzrblvfjQ/TaimUWiuwh9hfv4JDurKKA==</latexit>

Ind(DAPS) =1

2⌘(Hreg

DW )<latexit sha1_base64="EoRPPh9uCUnfA3cWijURhtihfY8=">AAACtXichVHLShxBFD12YtTJw1E3gWwaB8O4GW6bgCIIalyY3ehkHMGZNN09NWNjv6iuGTBN/4AfoAtXCWQR8gHZZBcS8gMu/ISQpYKbLHKnp0ESiblFVZ06dc+tOlw78txYEZ2PaHfujt4bG58o3H/w8NFkcWp6Jw570hF1J/RCuWtbsfDcQNSVqzyxG0lh+bYnGvbBi8F9oy9k7IbBK3UYiZZvdQO34zqWYsoskv4yaOvlDTNpSl9fq9bS+ZVmR1pOYqTJQtoUyipvmslGI32dSNFN581iiSqUhX4TGDkoIY9qWPyEJtoI4aAHHwIBFGMPFmIeezBAiJhrIWFOMnKze4EUBdb2OEtwhsXsAa9dPu3lbMDnQc04Uzv8isdTslLHHJ3RB7qg7/SRftCvf9ZKshqDvxzybg+1IjInjx7Xrv6r8nlX2L9W3aKwOft2TwodLGVeXPYWZczApTOs339zclFb3p5LntI7+sn+3tI5fWGHQf/Seb8ltk9R4AYZf7fjJthZqBjPKrT1vLS6nrdqHE8wizL3YxGr2EQVdX73GJ/xFd+0Ra2ltbXOMFUbyTUz+CO08DdUzaJD</latexit> Q.E.D.

Page 56: Domain-wall fermion and index theorem › ... › Fukaya-APSDW.pdf · 1.Introduction 2.Physicistʼs view of index theorems 3.Atiyah-Singer index with massive fermion operator 4.Index

Contents✔ 1. Introduction 2. Physicist’s view of index theorems AS index is physicist-friendly but APS is not. 3. Atiyah-Singer index with massive fermion operator coincides with the AS index. 4. Index with domain-wall fermion Dirac operator [F, Onogi,

Yamaguchi 2017] coincides with the APS index.

5. Mathematical proof [this work] are different expressions of the same 5D Dirac index. 6. Discussion 7. Summary

I = ⌘(�5(D +M))reg/2<latexit sha1_base64="9Hjmz73sFWiddmPGmw/3Dli7i7c=">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</latexit><latexit sha1_base64="9Hjmz73sFWiddmPGmw/3Dli7i7c=">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</latexit><latexit sha1_base64="9Hjmz73sFWiddmPGmw/3Dli7i7c=">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</latexit><latexit sha1_base64="9Hjmz73sFWiddmPGmw/3Dli7i7c=">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</latexit>

I = ⌘(�5(D +M✏(x4)))reg

/2<latexit sha1_base64="BND6Kli/5C5jJD6YujX9xRn1e0c=">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</latexit><latexit sha1_base64="BND6Kli/5C5jJD6YujX9xRn1e0c=">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</latexit><latexit sha1_base64="BND6Kli/5C5jJD6YujX9xRn1e0c=">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</latexit><latexit sha1_base64="BND6Kli/5C5jJD6YujX9xRn1e0c=">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</latexit>

Ind(DAPS) and ⌘(�5(D +M✏(x4)))reg

/2<latexit sha1_base64="EnzAJ1KF4HTMKtkNcFNMh9czzQw=">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</latexit>

Page 57: Domain-wall fermion and index theorem › ... › Fukaya-APSDW.pdf · 1.Introduction 2.Physicistʼs view of index theorems 3.Atiyah-Singer index with massive fermion operator 4.Index

My main subject = lattice gauge theory.

Oakforest-PACS at U. of Tsukuba

Page 58: Domain-wall fermion and index theorem › ... › Fukaya-APSDW.pdf · 1.Introduction 2.Physicistʼs view of index theorems 3.Atiyah-Singer index with massive fermion operator 4.Index

On lattice, Dirac equation is a difference equation.

@

@x

(x) ! (x+ a)� (x)

a

<latexit sha1_base64="gr+hSkIR9vFrE9C179aikUWOAF0=">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</latexit>

For , we do not know how to realize the APS boundary condition. But

is easy and guaranteed to be an integer! [F, Kawai, Matsuki, Onogi, Yamaguchi, in progress.]

1

2⌘(�5(D

lat +M✏(x4))))<latexit sha1_base64="KXq60P8Y2PS8uXuW5ZIgtWcZwzs=">AAACvHichVHLShxBFD22mugkxtFshGwGB8OEwFA9URQXQRIXbgRfo4Jtmuq2Ziym+kF3zRDTzA/kB1xIFgZchHyAH+DGhS5d+AnBpYFsssidngZRid6iqk6duudWHa4TKhlrxi57jN6+/idPBwZzz54PvRjOj4yux0EzckXVDVQQbTo8Fkr6oqqlVmIzjAT3HCU2nMbHzv1GS0SxDPw1vReKbY/XfVmTLtdE2flpqxZxNzHbSaVtCc1LVp17HrenSvOfEivyCorr9ttFS4SxVIFf+mxPvqEo2PkiK7M0CveBmYEislgK8sewsIMALprwIOBDE1bgiGlswQRDSNw2EuIiQjK9F2gjR9omZQnK4MQ2aK3TaStjfTp3asap2qVXFM2IlAVMsAv2g12zU/aT/WJ//1srSWt0/rJHu9PVitAe/jq2+udRlUe7xu6N6gGFQ9kPe9KoYSb1IslbmDIdl263fuvL/vXq7MpE8pp9Z1fk75BdshNy6Ld+u0fLYuUAOWqQebcd98F6pWy+K1eWJ4tzH7JWDeAVxlGifkxjDgtYQpXe/YYTnOHceG/sGA3D66YaPZnmJW6F0foHEtOk5w==</latexit>

Ind(DAPS)<latexit sha1_base64="OwUMoHkUjEOLyCemqdI2i/cIoNI=">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</latexit>

Page 59: Domain-wall fermion and index theorem › ... › Fukaya-APSDW.pdf · 1.Introduction 2.Physicistʼs view of index theorems 3.Atiyah-Singer index with massive fermion operator 4.Index

How about 5D bulk and 4D boundary?It was known that 5D lattice Dirac operator ends up (after removing bulk effect) with a 4D lattice Dirac operator (overlap Dirac operator)

of which index is Atiyah-Singer index on 4D manifold + O(a^2) errors.

Dov

=1

a

�1 + �5sgn�5(D

lat � 1/a)�

<latexit sha1_base64="mSxUgHW6kTWAYwCza+jhexa7bpM=">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</latexit>

Page 60: Domain-wall fermion and index theorem › ... › Fukaya-APSDW.pdf · 1.Introduction 2.Physicistʼs view of index theorems 3.Atiyah-Singer index with massive fermion operator 4.Index

Lattice AS index = index in K-theory by Karoubi?

Dcontinuum

<latexit sha1_base64="da/ER83MHM68IGHCh26CzCFFXcM=">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</latexit> Dov

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2/a<latexit sha1_base64="h/sV4CXyCQTwfmsWVePJG8NxzcA=">AAAChnichVG7TgJBFD2sL8QHqI2JDZFgrHAWNRgroo0lD3kkSMjuOuKGZXezu5Ag8QdMbKWw0sTC+AF+gI0/YMEnGEtMbCy8LJsYJeKdzMyZM/fcmZMrm5pqO4x1fcLY+MTklH86MDM7Nx8MLSzmbaNhKTynGJphFWXJ5pqq85yjOhovmhaX6rLGC3Jtv39faHLLVg390GmZvFyXqrp6oiqSQ1Q2viFVQhEWY26Eh4HogQi8SBmhRxzhGAYUNFAHhw6HsAYJNo0SRDCYxJXRJs4ipLr3HOcIkLZBWZwyJGJrtFbpVPJYnc79mrarVugVjaZFyjCi7IXdsx57Zg/slX3+Wavt1uj/pUW7PNBysxK8WM5+/Kuq0+7g9Fs1QiFT9mhPDk6w43pRyZvpMn2XyqB+86zTy+5mou01dsveyN8N67Incqg335W7NM9cI0ANEn+3Yxjk4zFxMxZPb0WSe16r/FjBKtapHwkkcYAUcvRuFZe4QkfwCzFhW0gMUgWfp1nCjxCSX63skFE=</latexit>

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Page 61: Domain-wall fermion and index theorem › ... › Fukaya-APSDW.pdf · 1.Introduction 2.Physicistʼs view of index theorems 3.Atiyah-Singer index with massive fermion operator 4.Index

Summary1. APS index describes bulk-edge correspondence of topological insulators. 2. APS (as well as AS) index can be reformulated by massive domain-wall Dirac operator. 3. We have given a mathematical proof for general cases. 4. math<->phys communication is important.

Page 62: Domain-wall fermion and index theorem › ... › Fukaya-APSDW.pdf · 1.Introduction 2.Physicistʼs view of index theorems 3.Atiyah-Singer index with massive fermion operator 4.Index

Backup slides

Page 63: Domain-wall fermion and index theorem › ... › Fukaya-APSDW.pdf · 1.Introduction 2.Physicistʼs view of index theorems 3.Atiyah-Singer index with massive fermion operator 4.Index

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

APS index tells

consistent with Z8 classification of Kitaev’s interacting Majorana chain.

Z / exp

⇣2⇡i

n

8

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Page 64: Domain-wall fermion and index theorem › ... › Fukaya-APSDW.pdf · 1.Introduction 2.Physicistʼs view of index theorems 3.Atiyah-Singer index with massive fermion operator 4.Index

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

⌘(iD3D)

2=

CS

2⇡+ integer

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= surface term.