FQHE on curved backgrounds, free fields and large N

TL;DR

This paper analyzes the free energy of the Laughlin quantum Hall state on curved surfaces using free field methods, revealing geometric functionals that govern large magnetic field behavior and deriving a path integral for higher-order terms.

Contribution

It introduces a novel approach to compute the free energy expansion of the Laughlin state on curved backgrounds using gravitational effective actions and geometric functionals.

Findings

Leading order given by the Aubin-Yau functional

Subleading order given by the Mabuchi functional

Next-to-next-to-leading order involves Liouville action

Abstract

We study the free energy of the Laughlin state on curved backgrounds, starting from the free field representation. A simple argument, based on the computation of the gravitational effective action from the transformation properties of Green functions under the change of the metric, allows to compute the first three terms of the expansion in large magnetic field. The leading and subleading contributions are given by the Aubin-Yau and Mabuchi functionals respectively, whereas the Liouville action appears at next-to-next-to-leading order. We also derive a path integral representation for the remainder terms. They correspond to a large mass expansion for a related interacting scalar field theory and are thus given by local polynomials in curvature invariants.