2D quantum gravity at three loops: a counterterm investigation

TL;DR

This paper investigates three-loop divergences in 2D quantum gravity, demonstrating that all but the most non-local divergence can be renormalized with local counterterms, implying a measure action renormalization.

Contribution

It provides a detailed analysis of three-loop divergences and shows that local counterterms suffice for renormalization, clarifying the structure of divergences in 2D quantum gravity.

Findings

Non-vanishing leading divergence coefficient in three-loop partition function.

All divergences except the maximally non-local one can be canceled by local counterterms.

The structure of counterterms indicates a renormalization of the measure action.

Abstract

We analyse the divergences of the three-loop partition function at fixed area in 2D quantum gravity. Considering the Liouville action in the Kahler formalism, we extract the coefficient of the leading divergence in $\sim A\Lambda^2 (\ln A\Lambda^2)^2$. This coefficient is non-vanishing. We discuss the counterterms one can and must add and compute their precise contribution to the partition function. This allows us to conclude that every local and non-local divergence in the partition function can be balanced by local counterterms, with the only exception of the maximally non-local divergence $(\ln A\Lambda^2)^3$. Yet, this latter is computed and does cancel between the different three-loop diagrams. Thus, requiring locality of the counterterms is enough to renormalize the partition function. Finally, the structure of the new counterterms strongly suggests that they can be understood as a renormalization of the measure action.