The Poisson sigma model on closed surfaces

TL;DR

This paper investigates the Poisson sigma model on closed surfaces using formal geometry, revealing conditions under which quantum corrections vanish and relating the partition function to topological invariants.

Contribution

It applies formal geometry methods to analyze the Poisson sigma model, demonstrating cases with no quantum corrections and linking the partition function to the Euler characteristic.

Findings

No quantum corrections on tori or regular unimodular Poisson structures

Partition function on the torus equals the Euler characteristic in certain cases

Formal geometry methods may extend to other AKSZ-type sigma models

Abstract

Using methods of formal geometry, the Poisson sigma model on a closed surface is studied in perturbation theory. The effective action, as a function on vacua, is shown to have no quantum corrections if the surface is a torus or if the Poisson structure is regular and unimodular (e.g., symplectic). In the case of a Kahler structure or of a trivial Poisson structure, the partition function on the torus is shown to be the Euler characteristic of the target; some evidence is given for this to happen more generally. The methods of formal geometry introduced in this paper might be applicable to other sigma models, at least of the AKSZ type.