Mathematical Tools for Calculation of the Effective Action in Quantum Gravity

TL;DR

This paper reviews covariant methods in quantum gravity, focusing on heat kernel techniques to compute the effective action, including new algebraic approaches for symmetric spaces and covariantly constant backgrounds.

Contribution

It introduces novel algebraic methods for calculating the heat kernel on symmetric spaces, advancing the computation of the effective action in quantum gravity.

Findings

Developed a covariant method for heat kernel asymptotic expansion.

Created algebraic techniques for heat kernel calculation on homogeneous bundles.

Enabled low-energy non-perturbative effective action computation.

Abstract

We review the status of covariant methods in quantum field theory and quantum gravity, in particular, some recent progress in the calculation of the effective action via the heat kernel method. We study the heat kernel associated with an elliptic second-order partial differential operator of Laplace type acting on smooth sections of a vector bundle over a Riemannian manifold without boundary. We develop a manifestly covariant method for computation of the heat kernel asymptotic expansion as well as new algebraic methods for calculation of the heat kernel for covariantly constant background, in particular, on homogeneous bundles over symmetric spaces, which enables one to compute the low-energy non-perturbative effective action.