Why is domain-wall fermion mathematically interesting? Hidenori - PowerPoint PPT Presentation

Why is domain-wall fermion mathematically interesting? Hidenori Fukaya (Osaka U.) HF,. T Onogi, S. Yamaguchi PRD96(2017) no. 12, 125004 [arXiv:1710.03379] M. Furuta (U. Tokyo), S. Matsuo (Nagoya U.),T. Onogi (Osaka U.), S. Yamaguchi (Osaka U.), M. Yamashita (U.Tokyo) [arXiv: 19xx.xxxxx]

Menu : Atiyah-Patodi-Singer (APS) index theorem and domain-wall fermion My talk (appetizer): perturbative finding that an APS index coincides with the eta-invariant of domain-wall Dirac op. Furuta’s talk (main dish): mathematical proof that every APS index is equivalent to the eta-invariant of domain-wall Dirac op. [ F, Onogi, Yamaguchi2017] [ F, Furuta, Matsuo, Onogi, Yamaguchi, Yamashita in progress.]

Menu : Atiyah-Patodi-Singer (APS) index and domain-wall fermion My talk (appetizer): 4D domain-wall fermion w/ SU(N) gauge field in the flat continuum Euclidean space with Pauli-Villars regulator. Furuta’s talk (main dish): More general set-up including curved metric. APS index on a lattice? -> on going. Please wait for lattice 2019 conference [F, Kawai, Matsuki, Mori, Onogi, Yamaguchi in progress.].

4-dim. domain-wall fermion <latexit sha1_base64="9wq5BLe+h6/VGREVbKWgw+4YxH0=">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</latexit> <latexit sha1_base64="S3CsvewxXJeQKV5SEFjlxI7bfmQ=">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</latexit> Massless Dirac fermion localized at 3-dim edge. No gauge anomaly, but T(or parity) anomaly. good model for topological insulator. D DW = D + M ✏ ( x 4 ) x 4

index on a manifold with boundary, Atiyah-Patodi-Singer index theorem [Atiyah-Patodi-Singer 1975] integer non-integer non-integer

Witten 2015 : APS index is a key to understand fermion APS index in topological insulator bulk-edge correspondence in symmetry protected topological insulator: path integrals T-anomalous T-anomalous T is protected ! [Related works: Metlitski 15, Seiberg-Witten 16, Tachikawa-Yonekura 16, Freed- Hopkins 16, Witten 16, Yonekura 16 … ]

What puzzled us

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

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

What puzzled us 1. APS boundary condition is non-local, while that of topological matter is local. 2. APS is for massless fermion but bulk fermion of topological insulator is massive (gapped). 3. No “physicist-friendly” description in the literature [except for Alvarez-Gaume et al. 1985 (bulk part is limitted to an integer due to adiabatic approximation, and boundary condition is obscure.)]

What puzzled us 1. APS boundary condition is non-local, while that of topological matter is local. 2. APS is for massless fermion but bulk fermion of topological insulator is massive (gapped). 3. No “physicist-friendly” description in the literature [except for Alvarez-Gaume et al. 1985 (bulk part is limitted to an integer due to adiabatic approximation, and 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 propose an alternative expression.

Contents 2. What is APS index theorem? 
 3. Index from massive Dirac operator 4. New index from domain-wall operator 5. What’s good with eta-invariant 6. Summary ✔ 1. Introduction

Difficulty with boundary 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.

Atiyah-Patodi-Singer boundary condition <latexit sha1_base64="esrqkDvu2FjxU7ro8/y0gLZqzs=">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</latexit> <latexit sha1_base64="esrqkDvu2FjxU7ro8/y0gLZqzs=">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</latexit> <latexit sha1_base64="esrqkDvu2FjxU7ro8/y0gLZqzs=">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</latexit> <latexit sha1_base64="esrqkDvu2FjxU7ro8/y0gLZqzs=">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</latexit> Beautiful! But physicist- unfriendly. [Atiyah, Patodi, Singer 75] Gives up the locality and rotational symmetry but keeps the chirality. Eg. 4 dim gauge They impose a non-local boundary b.c. index = n + − n −

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.