Heat Invariant E_2 for Nonminimal Operator on Manifolds with Torsion

TL;DR

The paper tackles the problem of extracting the heat kernel invariant $E_2$ for a nonminimal elliptic operator on manifolds with torsion and gauge fields. It employs the covariant pseudodifferential calculus in the Widom framework, implemented in C to produce $E_2$ for general dimension $n$ and specifically for $n=2$, including a Lorentz trace. The main contributions are the explicit full expression for $E_2$ with dimension- and parameter-dependent coefficients $C_i(a,n)$ and the corresponding $n=2$ forms, along with the trace formula. This work extends heat kernel methods to torsionful geometries and nonminimal operators, enabling applications in quantum field theory and spectral geometry, while highlighting substantial computational challenges and the need for automated simplification of large tensor expressions.

Abstract

Computer algebra methods are applied to investigation of spectral asymptotics of elliptic differential operators on curved manifolds with torsion and in the presence of a gauge field. In this paper we present complete expressions for the second coefficient (E_2) in the heat kernel expansion for nonminimal operator on manifolds with nonzero torsion. The expressions were computed for general case of manifolds of arbitrary dimension n and also for the most important for E_2 case n=2. The calculations have been carried out on PC with the help of a program written in C.