Spectral asymmetry on the ball and asymptotics of the asymmetry kernel

TL;DR

This paper computes the spectral asymmetry of the Dirac operator on a ball with specific boundary conditions in 2 and 4 dimensions, and analyzes related heat trace asymptotics for Dirac-type operators.

Contribution

It provides explicit calculations of spectral asymmetry and heat trace asymptotics for Dirac operators with chiral bag boundary conditions in low dimensions.

Findings

Spectral asymmetry computed for D=2 and D=4.

Asymptotics of heat trace analyzed under boundary conditions.

Results contribute to understanding spectral properties of Dirac operators.

Abstract

Let $\ui\di$ be the Dirac operator on a $D=2d$ dimensional ball $\mcB$ with radius $R$. We calculate the spectral asymmetry $\eta(0,\ui\di)$ for D=2 and D=4, when local chiral bag boundary conditions are imposed. With these boundary conditions, we also analyze the small-$t$ asymptotics of the heat trace $\Tr (F P e^{-t P^2})$ where $P$ is an operator of Dirac type and $F$ is an auxiliary smooth smearing function.