Infinite Dimensional Topological-Holomorphic Symmetry in Three-Dimensions

TL;DR

This paper constructs a three-dimensional quantum field theory with an infinite-dimensional symmetry, extending 2D conformal methods to 3D via raviolo vertex algebras.

Contribution

It introduces a novel 3D quantum field theory with an affine graded Lie algebra symmetry, generalizing chiral symmetry from 2D models.

Findings

Constructed the Fock space via radial quantization.

Demonstrated the local operators form a raviolo vertex algebra.

Provides a framework for exact methods in 3D quantum field theory.

Abstract

We introduce a three-dimensional quantum field theory with an infinite-dimensional symmetry, realized explicitly through a centrally extended affine graded Lie algebra. This symmetry is a direct three-dimensional generalization of the chiral symmetry in the Wess-Zumino-Witten model. Upon performing radial quantization, we construct the Fock space of the theory and, via a three-dimensional analogue of the state-operator correspondence, we demonstrate that the algebra of local operators is endowed with the structure of a raviolo vertex algebra. Accordingly, this setup provides a framework for extending the methods of two-dimensional conformal field theory to three dimensions, and we expect it to lay the groundwork for exact methods in three-dimensional quantum field theory.