Heat kernel coefficients for compact fuzzy spaces

TL;DR

This paper introduces a method to extract effective geometric quantities from the heat kernel trace of compact fuzzy spaces by using an approximate power-law expansion suitable for intermediate t values, overcoming limitations of traditional asymptotic expansions.

Contribution

It proposes a novel approach to determine geometric features of fuzzy spaces through an intermediate t expansion, with an efficient method demonstrated on known examples.

Findings

The method accurately extracts geometric coefficients from fuzzy spaces.

It works effectively for intermediate t values, unlike traditional asymptotic methods.

Validation on known fuzzy spaces confirms the approach's validity.

Abstract

I discuss the trace of a heat kernel Tr[e^(-tA)] for compact fuzzy spaces. In continuum theory its asymptotic expansion for t -> +0 provides geometric quantities, and therefore may be used to extract effective geometric quantities for fuzzy spaces. For compact fuzzy spaces, however, an asymptotic expansion for t -> +0 is not appropriate because of their finiteness. It is shown that effective geometric quantities are found as coefficients of an approximate power-law expansion of the trace of a heat kernel valid for intermediate values of t. An efficient method to obtain these coefficients is presented and applied to some known fuzzy spaces to check its validity.