The heat kernel of the compactified D=11 supermembrane with non-trivial winding

TL;DR

This paper investigates the heat kernel of the compactified D=11 supermembrane with non-trivial winding, establishing its properties through perturbation theory and path integral formulations, with implications for quantum membrane models.

Contribution

It constructs a convergent Dyson series for the heat kernel of the supermembrane Hamiltonian, linking it to quantum models and path integral representations.

Findings

Heat kernel is of finite trace due to perturbation analysis.

Dyson series converges in von Neumann-Schatten topology.

Matrix Feynman-Kac formula derived for supermembrane heat kernel.

Abstract

We study the quantization of the regularized hamiltonian, $H$, of the compactified D=11 supermembrane with non-trivial winding. By showing that $H$ is a relatively small perturbation of the bosonic hamiltonian, we construct a Dyson series for the heat kernel of $H$ and prove its convergence in the topology of the von Neumann-Schatten classes so that $e^{-Ht}$ is ensured to be of finite trace. The results provided have a natural interpretation in terms of the quantum mechanical model associated to regularizations of compactified supermembranes. In this direction, we discuss the validity of the Feynman path integral description of the heat kernel for D=11 supermembranes and obtain a matrix Feynman-Kac formula.