Classical Solutions for Poisson Sigma Models on a Riemann surface

TL;DR

This paper characterizes the classical solution space of Poisson Sigma Models on Riemann surfaces with specific Poisson structures, linking the moduli space dimension to topological and algebraic properties of the surface and Poisson tensor.

Contribution

It explicitly determines the moduli space of classical solutions for Poisson Sigma Models with vanishing Poisson form class on arbitrary Riemann surfaces, including its dimension and representatives.

Findings

Moduli space dimension depends on genus and Poisson tensor corank.

Constructs representatives of solutions using presymplectic forms.

Discusses potential generalizations beyond presymplectic cases.

Abstract

We determine the moduli space of classical solutions to the field equations of Poisson Sigma Models on arbitrary Riemann surfaces for Poisson structures with vanishing Poisson form class. This condition ensures the existence of a presymplectic form on the target Poisson manifold which agrees with the induced symplectic forms of the Poisson tensor upon pullback to the leaves. The dimension of the classical moduli space as a function of the genus of the worldsheet Sigma and the corank k of the Poisson tensor is determined as k(rank(H^1(Sigma))+1). Representatives of the classical solutions are provided using the above mentioned presymplectic 2-forms, and possible generalizations to cases where such a form does not exist are discussed. The results are compared to the known moduli space of classical solutions for two-dimensional BF and Yang-Mills theories.