2D Quantum Gravity on Compact Riemann Surfaces and Two-Loop Partition Function: Circumventing the c=1 Barrier?

TL;DR

This paper performs a detailed two-loop analysis of 2D quantum gravity on Riemann surfaces, revealing that different renormalization choices might allow bypassing the traditional c=1 barrier in quantum gravity.

Contribution

It provides a first-principles two-loop computation of the partition function including counterterms, suggesting alternative renormalization conditions could circumvent the c=1 barrier.

Findings

Finite coefficients depend on counterterm finite parts.

Standard KPZ scaling is just one possible renormalization choice.

Potential to bypass the c=1 barrier with alternative renormalizations.

Abstract

We study two-dimensional quantum gravity on arbitrary genus Riemann surfaces in the Kaehler formalism where the basic quantum field is the (Laplacian of the) Kaehler potential. We do a careful first-principles computation of the fixed-area partition function $Z[A]$ up to and including all two-loop contributions. This includes genuine two-loop diagrams as determined by the Liouville action, one-loop diagrams resulting from the non-trivial measure on the space of metrics, as well as one-loop diagrams involving various counterterm vertices. Contrary to what is often believed, several such counterterms, in addition to the usual cosmological constant, do and must occur. We consistently determine the relevant counterterms from a one-loop computation of the full two-point Green's function of the Kaehler field. Throughout this paper we use the general spectral cutoff regularization developed recently and which is well-suited for multi-loop computations on curved manifolds. At two loops, while all "unwanted" contributions to $\ln (Z[A]/Z[A_0])$ correctly cancel, it appears that the finite coefficient of $\ln (A/A_0)$ does depend on the finite parts of certain counterterm coefficients, i.e. on the finite renormalization conditions one has to impose. There exists a choice that reproduces the famous KPZ-scaling, but it seems to be only one consistent choice among others. Maybe, this hints at the possibility that other renormalization conditions could eventually provide a way to circumvent the famous $c=1$ barrier.