Poisson Vertex Algebras and Three-Dimensional Gauge Theory

TL;DR

This paper constructs a 3D gauge theory based on Poisson vertex algebras, revealing deep links between algebraic structures and topological quantum field theories, with implications for 3D gravity and deformation quantization.

Contribution

It introduces a novel 3D gauge theory framework associated with Poisson vertex algebras, connecting algebraic identities to gauge invariance and symmetry enhancement.

Findings

Gauge invariance linked to Poisson vertex algebra Jacobi identity

Symmetry enhancement occurs with a Virasoro element in the PVA

Connections established between PVAs of W-type and 3D gravity

Abstract

We introduce a mixed holomorphic-topological gauge theory in three dimensions associated to a (freely generated) Poisson vertex algebra. The $\lambda$-bracket of the PVA plays the role of the structure constants of the gauge algebra and the gauge invariance of the theory holds if and only if the $\lambda$-bracket Jacobi identity is satisfied. We show that the holomorphic-topological symmetry of the theory enhances to full topological symmetry if the Poisson vertex algebra contains a Virasoro element. We outline examples associated to PVAs of $\mathcal{W}$-type and demonstrate their connections to various versions of $3d$ gravity. We expect the three-dimensional Poisson sigma model to play an important role in the deformation quantization of Poisson vertex algebras.