Non-perturbative Heat Kernel Asymptotics on Homogeneous Abelian Bundles

TL;DR

This paper develops a new short-time asymptotic expansion for the heat kernel on complex vector bundles with a U(1) connection having parallel curvature, effectively summing all orders of the curvature F in the expansion.

Contribution

It introduces a novel local asymptotic expansion for the heat kernel that sums all orders of the curvature F, including polynomial and non-polynomial dependencies, for bundles with a parallel U(1) curvature.

Findings

Derived explicit formulas for the first three heat kernel coefficients.

Established a universal structure for coefficients depending on curvature F.

Provided a framework to sum the heat kernel expansion to all orders in F.

Abstract

We study the heat kernel for a Laplace type partial differential operator acting on smooth sections of a complex vector bundle with the structure group $G\times U(1)$ over a Riemannian manifold $M$ without boundary. The total connection on the vector bundle naturally splits into a $G$-connection and a U(1)-connection, which is assumed to have a parallel curvature $F$. We find a new local short time asymptotic expansion of the off-diagonal heat kernel $U(t|x,x')$ close to the diagonal of $M\times M$ assuming the curvature $F$ to be of order $t^{-1}$. The coefficients of this expansion are polynomial functions in the Riemann curvature tensor (and the curvature of the $G$-connection) and its derivatives with universal coefficients depending in a non-polynomial but analytic way on the curvature $F$, more precisely, on $tF$. These functions generate all terms quadratic and linear in the Riemann curvature and of arbitrary order in $F$ in the usual heat kernel coefficients. In that sense, we effectively sum up the usual short time heat kernel asymptotic expansion to all orders of the curvature $F$. We compute the first three coefficients (both diagonal and off-diagonal) of this new asymptotic expansion.