Infinite Dimensional Topological-Holomorphic Symmetry in Three-Dimensions
Hank Chen, Joaquin Liniado
TL;DR
The paper develops a concrete three-dimensional quantum field theory with an infinite-dimensional symmetry by embedding it in a transverse holomorphic foliation framework and a Lie 2-algebra 𝔊. Through the raviolo formalism, they identify cohomology classes that encode the mode expansions of higher currents and symmetry parameters, perform radial quantisation, and derive a centrally extended affine graded Lie algebra that governs the symmetry. This leads to the construction of a raviolo vertex algebra, providing a 3d analogue to the well-known 2d vertex algebras and offering a path toward exact techniques in 3d QFT. The work thus generalizes chiral symmetry concepts to 3d, laying groundwork for systematic operator algebras and potential exact results in three dimensions.
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.