Quantum heat traces

TL;DR

This paper introduces new quantum invariants called relativistic and quantum heat traces for elliptic operators on manifolds, deriving their formulas and asymptotics, linking them to classical heat trace coefficients and global invariants.

Contribution

It defines and analyzes new quantum heat trace invariants, providing reduction formulas and asymptotic expansions that connect to classical heat trace coefficients.

Findings

Derived reduction formulas expressing quantum heat traces in terms of classical heat trace transforms.

Computed asymptotic expansions of the new invariants.

Identified coefficients involving classical heat trace coefficients and new global invariants.

Abstract

We study new invariants of elliptic partial differential operators acting on sections of a vector bundle over a closed Riemannian manifold that we call the relativistic heat trace and the quantum heat traces. We obtain some reduction formulas expressing these new invariants in terms of some integral transforms of the usual classical heat trace and compute the asymptotics of these invariants. The coefficients of these asymptotic expansion are determined by the usual heat trace coefficients (which are locally computable) as well as by some new global invariants.