# One-loop Partition Functions in Hyperbolic Space $\mathbb{H}^n(\mathbb{R})$

## TL;DR

This paper develops a new method to compute one-loop partition functions for free quantum fields on hyperbolic spaces of arbitrary dimension, extending previous work limited to three dimensions.

## Contribution

Introduces a novel approach for calculating heat kernel solutions and asymptotics on hyperbolic spaces, enabling analysis in any dimension and including gauge fields.

## Key findings

- Derived the regular part and UV divergences of the effective action.
- Extended heat kernel coefficient calculations to all dimensions.
- Validated the method with DeWitt's approach.

## Abstract

We first study the problem of the one-loop partition function for a free massive quantum field theory living on a fixed background hyperbolic space on the field of real numbers, $\mathbb{H}^n(\mathbb{R}), \,\, n\geq 2$. Earlier attempts were limited to $n=3$ dimensions due to the computational complexity. We have developed a new method to determine the fundamental solution of the heat equation and techniques to specify its asymptotics in the small time limit. These enable us to determine the regular part and the ultra violet divergences of the one-loop effective action in the scalar case. The contribution of the Abelian gauge excitations to the one-loop partition function were treated separately using Fourier analysis and bi-tensor techniques on $\mathbb{H}^n(\mathbb{R})$. Finally, by employing the DeWitt's method, we confirmed the correctness and extended our results to any dimension regarding the first three heat kernel coefficients.