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

Yuto Moriwaki

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 $24$.

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

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 .

Paper Structure

This paper contains 36 sections, 71 theorems, 182 equations, 1 figure, 1 table.

Table of Contents

  1. From chiral vertex algebra to full vertex algebra
  2. The spaces of chiral / full formal power series
  3. Chiral / full vertex operators and expansions
  4. Compositions of chiral vertex operator and $\mathbb{Z}$-graded vertex algebra
  5. Non-chiral correlation functions and formal calculus
  6. Convergence
  7. Conformal singularity
  8. Generalized two-point correlation function
  9. Full vertex algebra
  10. Definition of full vertex algebra
  11. Chiral and anti-chiral vectors
  12. Tensor product of full vertex algebras
  13. Full conformal vertex algebra
  14. Generalized full vertex algebras
  15. Definition of generalized vertex algebra
  16. ...and 21 more sections

Key Result

Lemma 1.2

If $f({\underline{z}}) \in W((z,{\bar{z}},|z|^\mathbb{R}))$ satisfies $\frac{d}{d{\bar{z}}} f({\underline{z}})=0$, then $f({\underline{z}}) \in W((z))$.

Figures (1)

  • Figure 1: CFT moduli space of $(c,\bar{c})=(1,1)$

Theorems & Definitions (119)

  • Remark 1.1
  • Lemma 1.2
  • Lemma 1.3
  • Remark 1.4
  • Proposition 1.5
  • Lemma 1.6
  • proof
  • Proposition 1.7
  • proof
  • Remark 1.8
  • ...and 109 more