Quantum Anomaly in a Dirac Fermion System with Spacetime Dependent Mass

Quantum anomaly, a situation in which the symmetry of the classical action fails to be a symmetry of the full quantum theory, is a topic of great interest in quantum field theory. This is because it plays various important roles in physics and has beautiful mathematical structures.

In a new study, researchers from Japan explored perturbative anomalies in a system of N Dirac fermions with spacetime dependent mass (or the Higgs field that couples with the fermions through the Yukawa coupling) that included external gauge fields associated with U(N)₊ x U(N)_- chiral symmetry for even dimensions and U(N) flavor symmetry for odd dimensions. Using a technique called “Fujikawa’s method,” they went on to show that the anomalies associated with the chiral and flavor symmetry could be written in terms of a concept called “superconnection” introduced by the mathematician D. Quillen.

Interestingly, their results allowed for a natural string theory interpretation which independently suggests the emergence of superconnections in the anomaly formulas. Moreover, when the team applied these formulas to systems with interfaces and spacetime boundaries, they found that some of the known results for these systems were reproduced in a simple and unified manner.

The formalism developed in this study could be used to describe a broad range of topics in theoretical physics from quantum chromodynamics to topological matter with defects and boundaries, allowing us to explore deeper mathematical structures of physical theories that could lead us to realize new technologies.