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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
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
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
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>
What puzzled us
What puzzled us1. APS boundary condition is non-local, while that of
topological matter is local.
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).
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.)]
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].
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=">AAACwXichVFNSxtRFD2OWm1sm6ibQjeDwaKLyo0IiiiIblz60ajg2DAzeYkP58uZl4AO+QP+AReuFFxIf4A/wI1bBRf+hOJSwY0LbyYDYm3jHd6888695753uFbgyEgR3XZonV3dH3p6P2b6Pn3+ks31D6xFfi20RdH2HT/csMxIONITRSWVIzaCUJiu5Yh1a2ehmV+vizCSvvdT7QViyzWrnqxI21RMlXIzhlDmyOKoPqsbUc39FYei2ijFhsMtyqZRFbs6NX78IzXDfCmXpzFKQn8LCinII40lP3cOA2X4sFGDCwEPirEDExF/myiAEDC3hZi5kJFM8gINZFhb4yrBFSazO/yv8mkzZT0+N3tGidrmWxxeISt1DNMNndE9XdJv+kNP/+0VJz2ab9nj3WppRVDKHnxdfXxX5fKusP2iaqOwuLq9J4UKphIvkr0FCdN0abf61/cP71enV4bj73RCd+zvmG7pgh169Qf7dFmsHCHDAyr8PY63YG18rMB4eSI/N5+OqhffMIQRnsck5rCIJRT53hNc4grX2oImtUALW6VaR6oZxKvQ4mdHXaf2</latexit><latexit sha1_base64="/Wo4FR0mfFjpoHBvuUTvFHngC1U=">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</latexit><latexit sha1_base64="/Wo4FR0mfFjpoHBvuUTvFHngC1U=">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</latexit>
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>
Overview
with physicist-unfriendly boundary condition [APS 1975]
Ind(DAPS)<latexit sha1_base64="OwUMoHkUjEOLyCemqdI2i/cIoNI=">AAAClnichVG7SgNBFD1ZXzE+ErUJ2ASDok2YREGxkPhC7fIwKmgIu5tJXNwXu5uIBn/AH7AQCwUV8QP8ABt/wCKfIJYRbCy82SyIivEOM3PmzD135nAlU1Vsh7G6T+jo7Oru8fcG+voHBoOhoeEt26hYMs/JhmpYO5Joc1XRec5RHJXvmBYXNUnl29LBcvN+u8otWzH0TefI5HlNLOtKSZFFh6hCaGhDL0YmVwq1PUuLLKayJ1OFUJTFmBuR3yDugSi8SBmhB+yhCAMyKtDAocMhrEKETWMXcTCYxOVRI84ipLj3HCcIkLZCWZwyRGIPaC3TaddjdTo3a9quWqZXVJoWKSMYZ8/sjjXYE7tnL+zjz1o1t0bzL0e0Sy0tNwvB03D2/V+VRruD/S9VG4VE2e09OShhzvWikDfTZZou5Vb96vFZIzufGa9NsCv2Sv4uWZ09kkO9+iZfp3nmHAFqUPxnO36DrUQsPh1LpGeiySWvVX6MYgyT1I9ZJLGOFHL07iEucINbISwsCKvCWitV8HmaEXwLIfUJTpOVsg==</latexit>
= =with physicist-friendly
set-up (topological insulator) [FOY 2017]
1
2⌘(HDW )
<latexit sha1_base64="wgj8OT7RSel3V3JC/ODlv1aMMgs=">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</latexit>
in physicist’s sense
=
THIS WORK! [FMOYY 2019]
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
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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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@
µ +m] = 0.<latexit sha1_base64="GW9YQ529SGOoNa8xz4UUbHC734I=">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</latexit><latexit 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sha1_base64="GW9YQ529SGOoNa8xz4UUbHC734I=">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</latexit><latexit sha1_base64="GW9YQ529SGOoNa8xz4UUbHC734I=">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</latexit><latexit sha1_base64="GW9YQ529SGOoNa8xz4UUbHC734I=">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</latexit><latexit 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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=">AAACrXichVFNSxtRFD1O1aapH7HdFLoZDEpEsC9FUAoFvxYu1ZgPyMTwZnymj8wXMy8BG/oHunDroqsKLopbwZVu3PgHushPKC4tuHHhncmA2FC9w5t37nn33PcO1/RtGSrGugPai8Gh4ZepV+nXI6Nj45mJN6XQawWWKFqe7QUVk4fClq4oKqlsUfEDwR3TFmWzuRqdl9siCKXnbqt9X9Qc3nDlnrS4Iqqeya3pn3WjwR2H7xhOS88ZPg+U5HY9ymZ1uRyDGb2eybI5FofeD/IJyCKJDS9zBgO78GChBQcCLhRhGxwhfVXkweATV0OHuICQjM8FviFN2hZVCargxDbp36CsmrAu5VHPMFZbdItNKyCljin2m/1iN+yKnbA/7O6/vTpxj+gt+7SbPa3w6+Pf3xVun1U5tCt8eVA9oTCp+mlPCntYjL1I8ubHTOTS6vVvfz28KXzamupMsyN2Tf5+si67JIdu+691vCm2fiBNA8r/O45+UPo4lye8OZ9dWklGlcJ7TCJH81jAEtaxgSLde4BTnONC+6AVNUPb6ZVqA4nmLR6F1rgHvemeMw==</latexit><latexit sha1_base64="VHDPLeMhoyNawD/KYqdCNgxnyiw=">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</latexit>
i
@
@t
+1
2m
@
2
@x
2i
� = 0.
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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=">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</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=">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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>
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=">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</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
◆
<latexit sha1_base64="GkG9IRLLuNKHKAh0gm4qFpSM9GU=">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</latexit>
�i<latexit sha1_base64="ZoqmMi3HMoQ+OWkMh52J0Mtts5Y=">AAACi3ichVG7SgNBFD1Z3/EVtRFsgkGxCrNRUMRCFMEyD6MBlbC7jnFwX+xOAjH4A5Y2FrFRsBA/wA+w8Qcs8gliGcHGwrubBVFR77IzZ87cc3cPR3dN4UvGWjGlq7unt69/ID44NDwymhgb3/adqmfwouGYjlfSNZ+bwuZFKaTJS67HNUs3+Y5+vB7c79S45wvH3pJ1l+9bWsUWh8LQJFGlPV9ULK0syokUS7Owkj+BGoEUoso6iXvs4QAODFRhgcOGJGxCg0/PLlQwuMTto0GcR0iE9xyniJO2Sl2cOjRij2mt0Gk3Ym06BzP9UG3QV0x6PVImMcOe2C1rs0d2x57Z+6+zGuGM4F/qtOsdLXfLo2eThbd/VRbtEkefqj8UOnX/7UniEEuhF0He3JAJXBqd+bWTi3ZhOT/TmGXX7IX8XbEWeyCHdu3VuMnxfBNxCkj9HsdPsJ1Jq/PpTG4htboWRdWPKUxjjvJYxCo2kUUxzOEcTVwqw8q8sqysdFqVWKSZwJdSNj4AqQKS9g==</latexit>
Pauli matrices
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=">AAACj3ichVG7SgNBFD2u7/iK2gg2waDYGGZFUQQlaKOdGqOCCWF3ncTBfbE7CWjwB/wALSx8gIX4AX6AjT9g4SeIZQQbC+9uFkSD8Q4zc+bMPXfmcHXXFL5k7KVFaW1r7+js6o719Pb1D8QHh7Z9p+wZPGs4puPt6prPTWHzrBTS5LuuxzVLN/mOfrgS3O9UuOcLx96SRy7PW1rJFkVhaJKoXK6kWZZWmF2cUhOFeJKlWBiJRqBGIIko1p34A3LYhwMDZVjgsCEJm9Dg09iDCgaXuDyqxHmERHjPcYIYacuUxSlDI/aQ1hKd9iLWpnNQ0w/VBr1i0vRImcA4e2Z3rMae2D17ZZ9/1qqGNYK/HNGu17XcLQycjmQ+/lVZtEscfKuaKHTKbu5Jooj50Isgb27IBC6Nev3K8Xkts7A5Xp1gN+yN/F2zF/ZIDu3Ku3G7wTcvEKMGqb/b0Qi2p1Mq4Y2ZZHo5alUXRjGGSerHHNJYxTqy9K6LM1ziShlU5pQlJV1PVVoizTB+hLL2BUvuk5M=</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=">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</latexit>
=
0
BB@
1
2
3
4
1
CCA =
0
BB@
os
1
| "i| #i os
4
1
CCA
<latexit sha1_base64="kILF5h5z0TMptmkCEf9jwGiKqrU=">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</latexit><latexit sha1_base64="kILF5h5z0TMptmkCEf9jwGiKqrU=">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</latexit><latexit sha1_base64="kILF5h5z0TMptmkCEf9jwGiKqrU=">AAADTHichVHNahRBEK6ZaIzjT1a9CF4Wl0g8GHpiwBAIhHjxmGTdJJBel5ne3kmT3u6hp3fDOu4L5AVyyEnBg/gAPoAHPXnzkEcQjwl4UMGan0VMdK1hpr/+qr6q+agwliKxhBw77sSFi5OXpi57V65euz5duXFzM9E9w3iDaanNdhgkXArFG1ZYybdjw4NuKPlWuPc4y2/1uUmEVk/tIObNbhAp0REssEi1Kh9onIjqcpVK3rGzNOSRUGlgTDAYpmzoZdmWT2kB5kfg4QgsZICrdinxqBHRrr3vLY/t9yzVyRCVL2gvxpzepyZQkeQ51db76gxZjBqp/jbPa1VqZI7kUT0P/BLUoIw1XXkHFNqggUEPusBBgUUsIYAEnx3wgUCMXBNS5Awikec5DMFDbQ+rOFYEyO7hN8LbTskqvGc9k1zNcIrE16CyCjPkM3lDTshH8pZ8IT/+2SvNe2T/MsAzLLQ8bk0f3K5/+6+qi6eF3d+qMYoQq8d7stCBxdyLQG9xzmQuWdG///zwpL60MZPeI6/IV/T3khyT9+hQ9U/Z63W+cQTZgvyz6zgPNufnfMTrC7WV1XJVU3AH7sIs7uMRrMATWIMGMOeBU3eo03Q/uafud/dnUeo6peYW/BETk78Akfjb1Q==</latexit><latexit sha1_base64="kILF5h5z0TMptmkCEf9jwGiKqrU=">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</latexit>
* os=off-shell modes, non-classical, not satisfyingE = mc2
p1 + p2/m2c2
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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.
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=">AAACmHichVFLSgNBEH2O//jJqBvRTTAoLiRURFEEIaig7kw0KhgJM2MbB+fHzCQQQy7gBVy4EAXB4AE8gBsv4CJHEJcKblxYmQyIilpNd79+Xa+6H6U6hu75RPUWqbWtvaOzqzvS09vXH5UHBrc9u+hqIqvZhu3uqoonDN0SWV/3DbHruEIxVUPsqMfLjfudknA93ba2/LIj9k2lYOmHuqb4TOXloVxlZSpXUExTyc/mqrHFGCXycpwSFETsJ0iGII4wNmz5DjkcwIaGIkwIWPAZG1Dg8dhDEgSHuX1UmHMZ6cG9QBUR1hY5S3CGwuwxrwU+7YWsxedGTS9Qa/yKwdNlZQzj9Eg1eqEHuqUnev+1ViWo0fhLmXe1qRVOPno6vPn2r8rk3cfRp+oPhcrZf3vycYj5wIvO3pyAabjUmvVLJ2cvmwuZ8coEXdEz+7ukOt2zQ6v0ql2nReYcEW5Q8ns7foLt6USScXomnloKW9WFUYxhkvsxhxTWsIEsv1vGBW5Qk0aklLQqrTdTpZZQM4QvIWU+AMnvlko=</latexit><latexit sha1_base64="bMa++0fT2CXNiHgwTBoNs+GlVO8=">AAACmHichVFLSgNBEH2O//jJqBvRTTAoLiRURFEEIaig7kw0KhgJM2MbB+fHzCQQQy7gBVy4EAXB4AE8gBsv4CJHEJcKblxYmQyIilpNd79+Xa+6H6U6hu75RPUWqbWtvaOzqzvS09vXH5UHBrc9u+hqIqvZhu3uqoonDN0SWV/3DbHruEIxVUPsqMfLjfudknA93ba2/LIj9k2lYOmHuqb4TOXloVxlZSpXUExTyc/mqrHFGCXycpwSFETsJ0iGII4wNmz5DjkcwIaGIkwIWPAZG1Dg8dhDEgSHuX1UmHMZ6cG9QBUR1hY5S3CGwuwxrwU+7YWsxedGTS9Qa/yKwdNlZQzj9Eg1eqEHuqUnev+1ViWo0fhLmXe1qRVOPno6vPn2r8rk3cfRp+oPhcrZf3vycYj5wIvO3pyAabjUmvVLJ2cvmwuZ8coEXdEz+7ukOt2zQ6v0ql2nReYcEW5Q8ns7foLt6USScXomnloKW9WFUYxhkvsxhxTWsIEsv1vGBW5Qk0aklLQqrTdTpZZQM4QvIWU+AMnvlko=</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
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
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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�<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>
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>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=">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</latexit><latexit sha1_base64="I7GfGAPgiGlSFncQPFOz0BiA9qo=">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</latexit>
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)
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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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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
✔
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=">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>
H�0 = �5M�0 = ±M�0.<latexit sha1_base64="C0aKYE6gmA0VJvi6oqtiXI2qLeg=">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</latexit><latexit sha1_base64="C0aKYE6gmA0VJvi6oqtiXI2qLeg=">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</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>
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=">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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>
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>
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=">AAACmnichVFdKwRRGH6M7/Wxi5TiYrKtSG3viEgp4YKk2LVLoW1mHGsyX83MbrH5A/6AC1eUC/wAP8CNP+DCT5BLyo0L785OCeE9nXOe85z3ec95ejXXNPyA6LFOqm9obGpuaY21tXd0xhNd3XnfKXm6yOmO6XibmuoL07BFLjACU2y6nlAtzRQb2sF89X6jLDzfcOz14NAVO5ZatI09Q1cDpgqJ3kV5Rt4uqpalFibk4QV5VF6RRwqJJKUpDPknUCKQRBSrTuIW29iFAx0lWBCwETA2ocLnsQUFBJe5HVSY8xgZ4b3AMWKsLXGW4AyV2QNei3zailibz9WafqjW+RWTp8dKGSl6oCt6oXu6oSd6/7VWJaxR/csh71pNK9xC/KQv+/avyuI9wP6n6g+Fxtl/ewqwh6nQi8He3JCputRr9ctHpy/Z6UyqMkQX9Mz+zumR7tihXX7VL9dE5gwxbpDyvR0/QX4srVBaWRtPzs5FrWpBPwYxzP2YxCwWsYocv1vBOa5xIw1Ic9KStFxLleoiTQ++hLT+Aaz1lbU=</latexit><latexit sha1_base64="p5LMQXIq6ZnPd6BFmqhXjlSY4WQ=">AAACmnichVFdKwRRGH6M7/Wxi5TiYrKtSG3viEgp4YKk2LVLoW1mHGsyX83MbrH5A/6AC1eUC/wAP8CNP+DCT5BLyo0L785OCeE9nXOe85z3ec95ejXXNPyA6LFOqm9obGpuaY21tXd0xhNd3XnfKXm6yOmO6XibmuoL07BFLjACU2y6nlAtzRQb2sF89X6jLDzfcOz14NAVO5ZatI09Q1cDpgqJ3kV5Rt4uqpalFibk4QV5VF6RRwqJJKUpDPknUCKQRBSrTuIW29iFAx0lWBCwETA2ocLnsQUFBJe5HVSY8xgZ4b3AMWKsLXGW4AyV2QNei3zailibz9WafqjW+RWTp8dKGSl6oCt6oXu6oSd6/7VWJaxR/csh71pNK9xC/KQv+/avyuI9wP6n6g+Fxtl/ewqwh6nQi8He3JCputRr9ctHpy/Z6UyqMkQX9Mz+zumR7tihXX7VL9dE5gwxbpDyvR0/QX4srVBaWRtPzs5FrWpBPwYxzP2YxCwWsYocv1vBOa5xIw1Ic9KStFxLleoiTQ++hLT+Aaz1lbU=</latexit>
Index(D) =
1
2
⌘(H)
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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="hewny92FYCHpAVdPl03ZDQuDG1Y=">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</latexit><latexit sha1_base64="hewny92FYCHpAVdPl03ZDQuDG1Y=">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</latexit><latexit sha1_base64="hewny92FYCHpAVdPl03ZDQuDG1Y=">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</latexit><latexit sha1_base64="hewny92FYCHpAVdPl03ZDQuDG1Y=">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</latexit>
�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=">AAACcXichVHLSgMxFD0d3/XRqhvFTWlRBKHcEUERBNGN7my1VfDFzJi2g/NiJi1o8Qf8AQVXCiLiZ7jxB1z4CeKyghsX3k4HREW9IcnJyT03J4nuWWYgiZ5iSlt7R2dXd0+8t69/IJEcHCoGbtU3RMFwLdff0rVAWKYjCtKUltjyfKHZuiU29cPl5v5mTfiB6Tob8sgTu7ZWdsySaWiSqb0dW5MVQ7PqqyepBXU/maEshZH6CdQIZBDFmpu8wQ4O4MJAFTYEHEjGFjQE3LahguAxt4s6cz4jM9wXOEGctVXOEpyhMXvIY5lX2xHr8LpZMwjVBp9icfdZmcI4PdItNeiB7uiZ3n+tVQ9rNL0c8ay3tMLbT5yOrL/9q7J5lqh8qv70LFHCXOjVZO9eyDRvYbT0teOzxvp8frw+QVf0wv4v6Ynu+QZO7dW4zon8BeL8Aer35/4JitNZlbJqbiazuBR9RTfGkMYkv/csFrGCNRT4XB/nuMRVrKGMKikl3UpVYpFmGF9CmfoAG2WPBw==</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=">AAACanichVG7SgNBFD1Z3/EVtVFsglGJTbgrvqugjaWvGMEE2V0nurjZXXYnAQ3+gJ2VYCoFEfEzbPwBCz9BtFOwsfBms6IW0TvM3Dtn7rlz5o7uWqYviR4jSlNzS2tbe0e0s6u7pzfW17/pOyXPEBnDsRxvS9d8YZm2yEhTWmLL9YRW1C2R1Q+WaufZsvB807E35KEr8kVtzzYLpqFJhrI5IbXk8sROLEEpCiz+I5gmdX5GjashkkBoK07sGjnswoGBEooQsCE5tqDB57ENFQSXsTwqjHkcmcG5wDGizC1xluAMjdEDXvd4tx2iNu9rNf2AbfAtFk+PmXGM0QPd0Cvd0y090UfDWpWgRk3LIXu9zhXuTu/J4Pr7v6wie4n9b9afmiUKmAu0mqzdDZDaK4w6v3x09rq+sDZWGadLemb9F/RId/wCu/xmXK2KtSqi/AFfXY43DjYnUyql1NWpRHox/Ip2DGMESe73LNJYxgoygbpTnKMaeVH6lSFluJ6qRELOAH6ZMvoJdleMAQ==</latexit><latexit sha1_base64="A89dKLbbcDrqFW1tQ1LXIFhsn3g=">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</latexit><latexit sha1_base64="A89dKLbbcDrqFW1tQ1LXIFhsn3g=">AAACanichVG7SgNBFD1Z3/EVtVFsglGJTbgrvqugjaWvGMEE2V0nurjZXXYnAQ3+gJ2VYCoFEfEzbPwBCz9BtFOwsfBms6IW0TvM3Dtn7rlz5o7uWqYviR4jSlNzS2tbe0e0s6u7pzfW17/pOyXPEBnDsRxvS9d8YZm2yEhTWmLL9YRW1C2R1Q+WaufZsvB807E35KEr8kVtzzYLpqFJhrI5IbXk8sROLEEpCiz+I5gmdX5GjashkkBoK07sGjnswoGBEooQsCE5tqDB57ENFQSXsTwqjHkcmcG5wDGizC1xluAMjdEDXvd4tx2iNu9rNf2AbfAtFk+PmXGM0QPd0Cvd0y090UfDWpWgRk3LIXu9zhXuTu/J4Pr7v6wie4n9b9afmiUKmAu0mqzdDZDaK4w6v3x09rq+sDZWGadLemb9F/RId/wCu/xmXK2KtSqi/AFfXY43DjYnUyql1NWpRHox/Ip2DGMESe73LNJYxgoygbpTnKMaeVH6lSFluJ6qRELOAH6ZMvoJdleMAQ==</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=">AAACm3ichVHLShxBFD12NOpE4yTZBCQwZFDGzXBbghEhIJqFCVn4yKhgy9Ddc8c09ovummG06R/ID7hINhFCCP5FsgnoNgs/IWRpIBsX3ulpkETUW1TVqVP33DpVZYWuEyui0z7tTv/A3cGh4cK9kdH7Y8UHD9fjoBXZXLMDN4g2LTNm1/G5phzl8mYYselZLm9Yu4vd/Y02R7ET+G/VXsjbnrnjO03HNpVQ9eKM4VlBJ3nlN7iTVl5OlV6UDMUdlVVOIm6kidGMTDvR02Q6TQ1WZmVpqlovlqlKWZSuAj0HZeSxHBS/wEADAWy04IHhQwl2YSKWtgUdhFC4bSTCRYKcbJ+RoiDalmSxZJjC7sq4I6utnPVl3a0ZZ2pbTnGlR6IsYYJ+0lc6ox90RL/o/NpaSVaj62VPZqun5bA+9v7x2t9bVZ7MCu8uVTd6VmhiNvPqiPcwY7q3sHv69v7B2drc6kQySYf0W/x/olP6Ljfw23/szyu8+gEF+QD9/+e+CtanqzpV9ZVn5fmF/CuGMI6nqMh7P8c8lrCMmpz7Ed9wjBPtibaovdbe9FK1vlzzCP+EVrsAoXae6w==</latexit><latexit sha1_base64="CGEzvhiJuRoQzxJu69hjKjylcSU=">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</latexit>
unpairedpaired
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 )] .
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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=">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</latexit>
✔
More physical set-up?In physics, 1. Any boundary has “outside”: manifold + boundary → domain-wall.
outside
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 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 ]
“new” APS index [F-Onogi-Yamaguchi 2017]
Fujikawa method
coincides with APS index !
PV part = Atiyah-Singer index
Fujikawa-method does not contribute.
Domain-wall fermion partNow let’s compute
In the free fermion case,
→ eigenvalue problem = Schrodinger equation with δ-function-like potential.
Complete set in the free caseSolutions to are
where and 3D direction = conventional plane waves.
Edge mode appears !
“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.
Fujikawa-method
We insert our complete set
Perturbative expansion
( See our paper for the details. ) 100% edge-mode effect
=
Total 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
✔
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✔
I = ⌘(�5(D +M✏(x4)))reg
/2<latexit sha1_base64="BND6Kli/5C5jJD6YujX9xRn1e0c=">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</latexit><latexit sha1_base64="BND6Kli/5C5jJD6YujX9xRn1e0c=">AAACvHichVHNThRBEP4YBXHlZ9WLiZeNG8xuSJYaAqImGqIe9GDC3wIJA5OeoXfp7PxlpncjTvYFfAEPxoMkHIwP4ANw8aBHDzyC8YiJFw/Wzg4BDkB1uqv6q/qq+0s5kacSTXQ4YFy5Ojh0bfh64cbI6Nh48eat1SRsx66su6EXxuuOSKSnAlnXSntyPYql8B1Prjmt5738WkfGiQqDFb0byU1fNAPVUK7QDNnFOcsXeqcRi1b6qlt6UrKkFhWrKXxf2LOVF5OvLRklyguDyht7plqtbqWxbHanpu1i2axRZiWqzZL56IHJQY4cp8rIbSEsfoWFbYRw0YYPiQCaYw8CCa8NmCBEjG0iZSzmSGV5iS4KzG1zleQKwWiLzybfNnI04HuvZ5KxXX7F4x0zs4QJ+kmf6Yi+0Rf6Rf/O7ZVmPXp/2WXv9Lkyssff3Vn+eynLZ6+xc8K6gOFw9cWaNBp4mGlRrC3KkJ5Kt9+/8/b90fLjpYn0Pu3Rb9b3iQ7pgBUGnT/u/qJc+oDC6QGdH6xO10yqmYsz5fln+aiGcRf3UOF5zGEeL7GAOr/7EQf4jh/GU2PbaBl+v9QYyDm3ccaMzn/VBKTR</latexit><latexit sha1_base64="BND6Kli/5C5jJD6YujX9xRn1e0c=">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</latexit><latexit sha1_base64="BND6Kli/5C5jJD6YujX9xRn1e0c=">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</latexit>
✔
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.
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 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>
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>
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=">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</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>
on X4D x R,s = x5
<latexit sha1_base64="ngVsWB+3MLhm/sJGqiJv0+dydOs=">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</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>
LocalizationTheorem 2 tells us
and on the massless surface
theorem 1 indicates
s = x5<latexit sha1_base64="ngVsWB+3MLhm/sJGqiJv0+dydOs=">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</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>
Boundary eta invariantsTheorem 1 tells us
and from theorem 3, we obtain
therefore,
s = x5<latexit sha1_base64="ngVsWB+3MLhm/sJGqiJv0+dydOs=">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</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=">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</latexit>
Ind(DAPS) =1
2⌘(Hreg
DW )<latexit sha1_base64="EoRPPh9uCUnfA3cWijURhtihfY8=">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</latexit> Q.E.D.
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
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My main subject = lattice gauge theory.
Oakforest-PACS at U. of Tsukuba
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=">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</latexit>
Ind(DAPS)<latexit sha1_base64="OwUMoHkUjEOLyCemqdI2i/cIoNI=">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</latexit>
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
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<latexit sha1_base64="mSxUgHW6kTWAYwCza+jhexa7bpM=">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</latexit>
Lattice AS index = index in K-theory by Karoubi?
Dcontinuum
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2/a<latexit sha1_base64="h/sV4CXyCQTwfmsWVePJG8NxzcA=">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</latexit>
We need more communication between mathematicians and physicists!
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.
Backup slides
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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Eta invariant = Chern Simons term + integer (non-local effect)
⌘(iD3D)
2=
CS
2⇡+ integer
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= surface term.