A physicist-friendly reformulation of the Atiyah-Patodi-Singer index and its mathematical justification

Fukaya, Hidenori; Furuta, Mikio; Matsuo, Shinichiroh; Onogi, Tetsuya; Yamaguchi, Satoshi; Yamashita, Mayuko

doi:10.22323/1.363.0061

Cited by 5 publications

(6 citation statements)

References 43 publications

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“…where F is the field strength for the superconnection, and ch(F) is the corresponding Chern character. 3 Â is the Â-genus responsible for the usual gravitational contribution, and M 0 is the overall scale of the (renormalized) mass matrix. We also show some examples where the resulting parameterized invertible theory matches global anomalies in the parameter space in one lower dimensions.…”

Section: Summary Of Resultsmentioning

confidence: 99%

Higher Berry Phase of Fermions and Index Theorem

Choi,

Ohmori

2022

Preprint

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When a quantum field theory is trivially gapped, its infrared fixed point is an invertible field theory. The partition function of the invertible field theory records the response to various background fields in the long-distance limit. The set of background fields can include spacetimedependent coupling constants, in which case we call the corresponding invertible theory a parameterized invertible field theory. We study such parameterized invertible field theories arising from free Dirac fermions with spacetime-dependent mass parameters using the Atiyah-Patodi-Singer index theorem for superconnections. In particular, the response to an infinitesimal modulation of the mass is encoded into a higher analog of the Berry curvature, for which we provide a general formula. When the Berry curvature vanishes, the invertible theory can still be nontrivial if there is a remaining torsional Berry phase, for which we list some computable examples.

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Section: Summary Of Resultsmentioning

confidence: 99%

Higher Berry Phase of Fermions and Index Theorem

Choi,

Ohmori

2022

Preprint

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“…5 That is, we will realize our d-dimensional fermion system of interest on an interface of an auxiliary (d + 1)-dimensional fermion system by varing the (d + 1)dimensional mass parameters appropriately. A similar construction for massive fermions was considered in [14,15,18,49].…”

Section: Jhep09(2022)022mentioning

confidence: 99%

“…We can further turn on the background gauge field A for the U(N f ) global symmetry, if we allow the mass matrix to transform in the adjoint representation. 14 We can combine the gauge field A and the mass matrix M into the following superconnection:…”

Section: Superconnectionmentioning

confidence: 99%

“…(B. 14), where α is given by α = α 0 θ(w 0 − w) inside the collar neighborhood of the boundary and remains to be equal to constant α 0 everywhere inside the bulk. w 0 ∈ [0, − ) is a fixed Euclidean time inside the collar neighborhood, and θ(w 0 − w) is the Heaviside step function which vanishes for w > w 0 , i.e., near the boundary.…”

Section: Jhep09(2022)022mentioning

confidence: 99%

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Higher Berry phase of fermions and index theorem

Choi

Ohmori

2022

J. High Energ. Phys.

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When a quantum field theory is trivially gapped, its infrared fixed point is an invertible field theory. The partition function of the invertible field theory records the response to various background fields in the long-distance limit. The set of background fields can include spacetime-dependent coupling constants, in which case we call the corresponding invertible theory a parameterized invertible field theory. We study such parameterized invertible field theories arising from free Dirac fermions with spacetime-dependent mass parameters using the Atiyah-Patodi-Singer index theorem for superconnections. In particular, the response to an infinitesimal modulation of the mass is encoded into a higher analog of the Berry curvature, for which we provide a general formula. When the Berry curvature vanishes, the invertible theory can still be nontrivial if there is a remaining torsional Berry phase, for which we list some computable examples.

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“…Owing to an image of the projection operator, the vector space splits into two subspaces such that: (20). Operator ĥ acts on v ± as ĥ(v ± ) = 1 2 ĥ ± ĥ• ĥ…”

Section: Primary Superspacementioning

confidence: 99%

Topological indices of general relativity and Yang-Mills theory in four-dimensional space-time

Kurihara¹

2022

Preprint

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This report investigates general relativity and the Yang-Mills theory in four-dimensional space-time using the common mathematical framework, the Chern-Weil theory for principal bundles. The whole theory is described owing to the fibre bundle with the GL(4) symmetry, a model of the space-time, and twisting several principal bundles with the gauge symmetry that are a model of force and matter fields.We introduce the Hodge-dual connection in addition to the principal connection into the Lagrangian to make gauge fields have dynamics independent from the Bianchi identity. The duplex superstructure appears in the bundle when a Z2-grading operator exists in a total space of the bundle in general. The Dirac operator appears in the secondary superspace using the one-dimensional Clifford algebra. Thus, it provides topological indices owing to the Atiyah-Singer index theorem. We introduce a novel method, namely the theta-metric space, to discuss topological indices in the hyperbolic space-time manifold. The theta-metric treats the Euclidean and Minkowski spaces simultaneously and defines the topological index in the Minkowski space-time.In this framework, we show several examples of topological indices appearing in the field theory and clarify the physical significance of these indices in the Minkowski space-time.

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A physicist-friendly reformulation of the Atiyah-Patodi-Singer index and its mathematical justification

Cited by 5 publications

References 43 publications

Higher Berry Phase of Fermions and Index Theorem

Higher Berry Phase of Fermions and Index Theorem

Higher Berry phase of fermions and index theorem

Topological indices of general relativity and Yang-Mills theory in four-dimensional space-time

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