Open AccessUnderstanding the index theorems with massive fermions
Author(s) -
Hidenori Fukaya
Publication year - 2021
Publication title -
international journal of modern physics a
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.581
H-Index - 108
eISSN - 1793-656X
pISSN - 0217-751X
DOI - 10.1142/s0217751x21300155
Subject(s) - physics , atiyah–singer index theorem , dirac operator , anomaly (physics) , theoretical physics , fermion , massless particle , chiral anomaly , operator (biology) , mixed anomaly , dirac (video compression format) , gauge theory , boundary value problem , dirac fermion , mathematical physics , gauge anomaly , supersymmetric gauge theory , quantum mechanics , neutrino , mathematics , pure mathematics , biochemistry , chemistry , repressor , transcription factor , gene
The index theorems relate the gauge field and metric on a manifold to the solution of the Dirac equation on it. In the standard approach, the Dirac operator must be massless to make the chirality operator well defined. In physics, however, the index theorem appears as a consequence of chiral anomaly, which is an explicit breaking of the symmetry. It is then natural to ask if we can understand the index theorems in a massive fermion system which does not have chiral symmetry. In this review, we discuss how to reformulate the chiral anomaly and index theorems with massive Dirac operators, where we find nontrivial mathematical relations between massless and massive fermions. A special focus is placed on the Atiyah–Patodi–Singer index, whose original formulation requires a physicist-unfriendly boundary condition, while the corresponding massive domain-wall fermion reformulation does not. The massive formulation provides a natural understanding of the anomaly inflow between the bulk and edge in particle and condensed matter physics.