Holomorphic Poisson Field Theories

TL;DR

This paper introduces a new class of quantum field theories based on holomorphic Poisson structures, exploring their deformations, anomalies, and topological properties, with potential links to supergravity theories.

Contribution

It constructs holomorphic Poisson field theories, characterizes their deformations and anomalies via Gelfand-Fuchs cohomology, and discusses their topological nature and algebraic structures.

Findings

Theories depend on holomorphic Poisson structures.

Deformations and anomalies characterized by Gelfand-Fuchs cohomology.

Non-degenerate cases are 'de Rham topological' theories.

Abstract

We construct a class of quantum field theories depending on the data of a holomorphic Poisson structure on a piece of the underlying spacetime. The main technical tool relies on a characterization of deformations and anomalies of such theories in terms of the Gelfand-Fuchs cohomology of formal Hamiltonian vector fields. In the case that the Poisson structure is non-degenerate such theories are topological in a certain weak sense, which we refer to as "de Rham topological". While the Lie algebra of translations acts in a homotopically trivial way, we will show that the space of observables of such a theory does not define an E_n-algebra. Additionally, we will highlight a conjectural relationship to theories of supergravity in four and five dimensions.