Mathematical proof for “physicist-friendly” reformulation of Atiyah-Patodi-Singer index Hidenori Fukaya (Osaka U.) HF,. T Onogi, S. Yamaguchi PRD96(2017) no. 12, 125004 [arXiv:1710.03379] M. Furuta (U. Tokyo), S. Matsuo (Nagoya U.),T. Onogi (Osaka U.), S. Yamaguchi (Osaka U.), M. Yamashita (U.Tokyo) [arXiv: 19xx.xxxxx]
My talk today In 2017 we proposed “A physicist-friendly reformulation of the Atiyah-Patodi-Singer(APS) index theorem.” F,. Onogi, Yamaguchi PRD96(2017) no.12, 125004 [arXiv: 1710.033379] Recently, we invited 3 mathematicians and succeeded in a mathematical proof. F, Furuta, Matuso, Onogi, Yamaguchi, Yamashita, in progress
My talk today In 2017 we proposed “A physicist-friendly reformulation of the Atiyah-Patodi-Singer(APS) index theorem.” F,. Onogi, Yamaguchi PRD96(2017) no.12, 125004 [arXiv: 1710.033379] Recently, we invited 3 mathematicians and succeeded in a mathematical proof. F, Furuta, Matuso, Onogi, Yamaguchi, Yamashita, in progress
index on a manifold with boundary, Atiyah-Patodi-Singer index theorem [Atiyah-Patodi-Singer 1975] integer non-integer non-integer
Witten 2015 : APS index is a key to understand path integrals fermion APS index in topological insulator bulk-edge correspondence in symmetry protected <latexit sha1_base64="dE7EQpoW2Ob212jgjBz3Uq6UB8=">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</latexit> <latexit sha1_base64="dE7EQpoW2Ob212jgjBz3Uq6UB8=">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</latexit> <latexit sha1_base64="dE7EQpoW2Ob212jgjBz3Uq6UB8=">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</latexit> <latexit sha1_base64="dE7EQpoW2Ob212jgjBz3Uq6UB8=">ACi3ichVFNSxtBGH5cbWx1mgvBS9iUPRgeLcVlKAgiqA3v6KC0bC7meiS/WJ3EkyX/QPSew+eFKQUb1715sU/4MGfIB4VvHjwzWZBVFrfYWaeZ93nlmRvcsM5BE1y1Ka9uHj+0dn1Kdn7u+dKd7etcCt+obIm+4lutv6FogLNMReWlKS2x4vtBs3RLremW2sb9eE35gus6qrHtiy9Z2HLNsGpkqpgemhoeVUe2w4IUezIuF/qiFIWjBVuTu2Vfq4QLURQV0xnKUhz9b4GagAySWHTf1BACS4MVGFDwIFkbEFDwG0TKgec1sImfMZmfG+QIQUa6ucJThDY7bC4w6vNhPW4XWjZhCrDT7F4u6zsh+DdEV/6Y4u6YRu6PGftcK4RsNLnWe9qRVesXv/28rDuyqbZ4ndZ9V/PUuUMRF7Ndm7FzONWxhNfe3n7uV3PJgOERHdMv+D+maLvgGTu3eOF4SywdI8Qeor5/7LVj7nlUpqy6NZaZnkq/oQB8GMzvPY5pzGMReT73F05xhnOlS/mh5JTJZqrSkmi+4kUoc09emJjp</latexit> topological insulator: T-anomalous T-anomalous T is protected ! = ( − 1) − I [Related works: Metlitski 15, Seiberg-Witten 16, Tachikawa-Yonekura 16&18, Freed-Hopkins 16, Witten 16, Yonekura 16&19 … ]
What puzzled us
What puzzled us 1. APS boundary condition is non-local, while that of topological matter is local.
What puzzled us 1. APS boundary condition is non-local, while that of topological matter is local. 2. APS is for massless fermion but bulk fermion of topological insulator is massive (gapped).
What puzzled us 1. APS boundary condition is non-local, while that of topological matter is local. 2. APS is for massless fermion but bulk fermion of topological insulator is massive (gapped). 3. No edge-localized modes allowed.
What puzzled us 1. APS boundary condition is non-local, while that of topological matter is local. 2. APS is for massless fermion but bulk fermion of topological insulator is massive (gapped). 3. No edge-localized modes allowed. 4. No “physicist-friendly” description in the literature [except for Alvarez-Gaume et al. 1985 but boundary condition is obscure.]
What puzzled us 1. APS boundary condition is non-local, while that of topological matter is local. 2. APS is for massless fermion but bulk fermion of topological insulator is massive (gapped). 3. No edge-localized modes allowed. 4. No “physicist-friendly” description in the literature [except for Alvarez-Gaume et al. 1985 but boundary condition is obscure.] → We launched a study group reading original APS paper and it took 3 months to translate it into “physics language”, and we reached an alternative expression.
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.
Locality >> chirality for physicists Locality (=causality) is essential. We cannot accept APS condition even if it is beautiful. non-local boundary hit!
Locality >> chirality for physicists Locality (=causality) is essential. We cannot accept APS condition even if it is beautiful. non-local boundary hit! information
Locality >> chirality for physicists Locality (=causality) is essential. We cannot accept APS condition even if it is beautiful. non-local boundary hit! information information propagates faster than speed of light.
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