Heat kernel coefficients on the sphere in any dimension

TL;DR

This paper derives explicit formulas for heat kernel coefficients on spheres of any dimension for various tensor types, providing tools useful in quantum gravity and related fields.

Contribution

It presents comprehensive, easily implementable formulas for heat kernel coefficients on symmetric spaces in any dimension, including new Green's functions and integral representations.

Findings

Explicit heat kernel coefficients for scalars, vectors, tensors derived

Green's functions for transverse traceless tensors obtained

New integral representations for heat kernels provided

Abstract

We derive all heat kernel coefficients for Laplacians acting on scalars, vectors, and tensors on fully symmetric spaces, in any dimension. Final expressions are easy to evaluate and implement, and confirmed independently using spectral sums and the Euler-Maclaurin formula. We also obtain the Green's function for Laplacians acting on transverse traceless tensors in any dimension, and new integral representations for heat kernels using known eigenvalue spectra of Laplacians. Applications to quantum gravity and the functional renormalisation group, and other, are indicated.