Two-dimensional conformal field theory, full vertex algebra and current-current deformation

TL;DR

This paper constructs a mathematical framework for non-perturbative deformations of two-dimensional conformal field theories using full vertex algebras, revealing new structures and classifications of vertex operator algebras.

Contribution

It introduces the notion of full vertex algebra and develops a deformation theory for conformal field theories, including a formula for counting isomorphism classes of resulting vertex operator algebras.

Findings

Constructed a deformation family parameterized by a double coset space.

Provided a counting formula for isomorphism classes of deformed vertex operator algebras.

Applied the theory to holomorphic vertex operator algebras of central charge 24.

Abstract

The main purpose of this paper is a mathematical construction of a non-perturbative deformation of a two-dimensional conformal field theory. We introduce a notion of a full vertex algebra which formulates a compact two-dimensional conformal field theory. Then, we construct a deformation family of a full vertex algebra which serves as a current-current deformation of conformal field theory in physics. The parameter space of the deformation is expressed as a double coset of an orthogonal group, a quotient of an orthogonal Grassmannian. As an application, we consider a deformation of chiral conformal field theories, vertex operator algebras. A current-current deformation of a "vertex operator algebra" may produce new vertex operator algebras. We give a formula for counting the number of the isomorphic classes of vertex operator algebras obtained in this way. We demonstrate it for some holomorphic vertex operator algebra of central charge $24$.