2D sigma models and differential Poisson algebras

TL;DR

This paper introduces a two-dimensional topological sigma model with a Poisson algebra structure on differential forms, linking geometric algebraic properties to a covariant quantization framework.

Contribution

It constructs a novel topological sigma model with a Poisson algebra on differential forms, connecting gauge symmetries, supersymmetry, and quantization of star product algebras.

Findings

Model has a Cartan integrable system of equations of motion.

Symmetries correspond to Poisson compatibility with de Rham operator.

Perturbative quantization relates to Kontsevich's covariant star product.

Abstract

We construct a two-dimensional topological sigma model whose target space is endowed with a Poisson algebra for differential forms. The model consists of an equal number of bosonic and fermionic fields of worldsheet form degrees zero and one. The action is built using exterior products and derivatives, without any reference to any worldsheet metric, and is of the covariant Hamiltonian form. The equations of motion define a universally Cartan integrable system. In addition to gauge symmetries, the model has one rigid nilpotent supersymmetry corresponding to the target space de Rham operator. The rigid and local symmetries of the action, respectively, are equivalent to the Poisson bracket being compatible with the de Rham operator and obeying graded Jacobi identities. We propose that perturbative quantization of the model yields a covariantized differential star product algebra of Kontsevich type. We comment on the resemblance to the topological A model.