Abstract
Recently, the Atiyah-Patodi-Singer(APS) index theorem attracts attention for understanding physics on the surface of materials in topological phases. Although it is widely applied to physics, the mathematical set-up in the original APS index theorem is too abstract and general (allowing non-trivial metric and so on) and also the connection between the APS boundary condition and the physical boundary condition on the surface of topological material is unclear. For this reason, in contrast to the Atiyah-Singer index theorem, derivation of the APS index theorem in physics language is still missing. In this talk, we attempt to reformulate the APS index in a "physicist-friendly" way, similar to the Fujikawa method on closed manifolds, for our familiar domain-wall fermion Dirac operator in a flat Euclidean space. We find that the APS index is naturally embedded in the determinant of domain-wall fermions, representing the so-called anomaly descent equations.
Highlights
Massive fermions in D-dimensional bulk spacetime with (D-1)-dimensional boundary attracts attention in various fields of physics
The common feature in these systems is the existence of a massless fermion excitation at the boundary in the case when the massive fermion at the bulk is in the symmetry protected topological (SPT) phase
While the former case is known as the Callan-Harvey mechanism [3], the latter is less familiar in particle physics but is known to be related to the mathematical theorem called Atiyah-Patodi-Singer (APS) index theorem [4,5,6], which is an extension of Atiyah-Singer index theorem [7, 8] for massless
Summary
Massive fermions in D-dimensional bulk spacetime with (D-1)-dimensional boundary attracts attention in various fields of physics. The common feature in these systems is the existence of a massless fermion excitation at the boundary in the case when the massive fermion at the bulk is in the symmetry protected topological (SPT) phase. For D=even, it is the cancellation of time reversal symmetry anomaly (T-anomaly) While the former case is known as the Callan-Harvey mechanism [3], the latter is less familiar in particle physics but is known to be related to the mathematical theorem called Atiyah-Patodi-Singer (APS) index theorem [4,5,6] , which is an extension of Atiyah-Singer index theorem [7, 8] for massless. Where D4d is the gauge covariant 4-dimensional Dirac operator, M is the mass of the fermion and Λ is the mass of the Pauli-Villars regulator field. Atiyah-Patodi-Singer [4] have proven that the index of D4d which is defined by the difference of the numbers of eigenmodes with positive and negative chirality can be given as
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
