Heat Kernel Coefficients for Chern-Simons Boundary Conditions in QED

TL;DR

This paper calculates heat kernel coefficients for Maxwell theory with Chern-Simons boundary conditions on a four-sphere, revealing that the one-loop counterterm coefficient is independent of the Chern-Simons coupling.

Contribution

It provides explicit formulas for heat kernel coefficients under Chern-Simons boundary conditions, highlighting their dependence or independence on the coupling constant.

Findings

The a_2 coefficient is independent of the Chern-Simons coupling.

The heat kernel becomes singular at a critical coupling value.

Explicit heat kernel coefficients are computed for the four-sphere case.

Abstract

We consider the four dimensional Euclidean Maxwell theory with a Chern-Simons term on the boundary. The corresponding gauge invariant boundary conditions become dependent on tangential derivatives. Taking the four-sphere as a particular example, we calculate explicitly a number of the first heat kernel coefficients and obtain the general formulas that yields any desired coefficient. A remarkable observation is that the coefficient $a_2$, which defines the one-loop counterterm and the conformal anomaly, does not depend on the Chern-Simons coupling constant, while the heat kernel itself becomes singular at a certain (critical) value of the coupling. This could be a reflection of a general property of Chern-Simons theories.