Heat kernel for flat generalized Laplacians with anisotropic scaling

TL;DR

This paper derives an explicit heat kernel solution for anisotropic flat Laplacians, including UV and UV/IR interpolations, expressed via Fox-Wright functions, aiding future curved Hořava-Lifshitz geometry studies.

Contribution

It provides the first analytic closed-form heat kernel for anisotropic Laplacians with UV and UV/IR interpolation, expressed in terms of Fox-Wright functions.

Findings

Reproduces spectral dimension flow analytically.

Provides a foundation for heat kernel construction in curved Hořava-Lifshitz geometries.

Validates results through consistency checks.

Abstract

We calculate the closed analytic form of the solution of heat kernel equation for the anisotropic generalizations of flat Laplacian. We consider a UV as well as UV/IR interpolating generalizations. In all cases, the result can be expressed in terms of Fox-Wright psi-functions. We perform different consistency checks, analytically reproducing some of the previous numerical or qualitative results, such as spectral dimension flow. Our study should be considered as a first step towards the construction of a heat kernel for curved Ho\v{r}ava-Lifshitz geometries, which is an essential ingredient in the spectral action approach to the construction of the Ho\v{r}ava-Lifshitz gravity.