Finiteness of quantum gravity coupled with matter in three spacetime dimensions

TL;DR

This paper demonstrates an algorithmic approach to make three-dimensional quantum gravity coupled with matter finite by solving beta function equations, with explicit examples including Chern-Simons gauge theory with fermions.

Contribution

It introduces a method to achieve finiteness in 3D quantum gravity coupled with matter by solving beta functions and reabsorbing irrelevant terms, under specific conditions.

Findings

Finiteness equations are solvable due to beta function structures.

Irrelevant Riemann tensor terms can be reabsorbed via field redefinitions.

Explicit two-loop solution for gravity coupled with Chern-Simons gauge theory.

Abstract

As it stands, quantum gravity coupled with matter in three spacetime dimensions is not finite. In this paper I show that an algorithmic procedure that makes it finite exists, under certain conditions. To achieve this result, gravity is coupled with an interacting conformal field theory C. The Newton constant and the marginal parameters of C are taken as independent couplings. The values of the other irrelevant couplings are determined iteratively in the loop- and energy-expansions, imposing that their beta functions vanish. The finiteness equations are solvable thanks to the following properties: the beta functions of the irrelevant couplings have a simple structure; the irrelevant terms made with the Riemann tensor can be reabsorbed by means of field redefinitions; the other irrelevant terms have, generically, non-vanishing anomalous dimensions. The perturbative expansion is governed by an effective Planck mass that takes care of the interactions in the matter sector. As an example, I study gravity coupled with Chern-Simons U(1) gauge theory with massless fermions, solve the finiteness equations and determine the four-fermion couplings to two-loop order. The construction of this paper does not immediately apply to four-dimensional quantum gravity.