Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

$\hbar$-Vertex algebras and chiralization of star products

Simone Castellan

TL;DR

This paper develops the theory of ${\hbar}$-vertex algebras as deformations of vertex algebras with a modified translation covariance and builds a comprehensive structure theory paralleling the classical framework, including a generalized Borcherds identity, the ${\hbar}$-bracket, and ${\hbar}$-Zhu algebras. It then applies this formalism to the chiralization of classical star-products, proving that every star-product on the symmetric algebra of a Lie algebra (and its central extensions) admits a chiralization and providing explicit formulas for Moyal–Weyl and Gutt-type chiral star-products. The construction yields deformation results for Poisson vertex algebras in the ${\hbar}\to 0$ limit and connects to well-known vertex algebras via a change of variables that links ordinary vertex algebras to ${\hbar}$-vertex algebras. Overall, the work offers a unified, computationally tractable framework to lift classical deformations to the chiral (vertex) setting and to obtain explicit chiral analogues of key star-products.

Abstract

We develop the theory of $\hbar$-vertex algebras, algebraic structures closely related to vertex algebras but with a deformed translation covariance axiom. We establish their structure theory, including analogues of Goddard's Uniqueness Theorem, the Reconstruction Theorem, Borcherds Identity, and the OPE Expansion Formula, and introduce the associated notions of $\hbar$-Lie conformal and $\hbar$-Poisson vertex algebras. The formalism provides a natural and simplified construction of the Zhu algebra. The main application is to the chiralization of classical star-products: we show that every star-product on the symmetric algebra of a Lie algebra (or its central extensions) admits a chiralization, and we derive explicit formulae for these chiral star-products, including the Moyal-Weyl and Gutt star-products. Setting $\hbar=0$ recovers explicit deformation quantizations of a broad class of Poisson vertex algebras, including the classical limits of free-boson, $βγ$-system, affine, and Virasoro vertex algebras.

$\hbar$-Vertex algebras and chiralization of star products

TL;DR

This paper develops the theory of -vertex algebras as deformations of vertex algebras with a modified translation covariance and builds a comprehensive structure theory paralleling the classical framework, including a generalized Borcherds identity, the -bracket, and -Zhu algebras. It then applies this formalism to the chiralization of classical star-products, proving that every star-product on the symmetric algebra of a Lie algebra (and its central extensions) admits a chiralization and providing explicit formulas for Moyal–Weyl and Gutt-type chiral star-products. The construction yields deformation results for Poisson vertex algebras in the limit and connects to well-known vertex algebras via a change of variables that links ordinary vertex algebras to -vertex algebras. Overall, the work offers a unified, computationally tractable framework to lift classical deformations to the chiral (vertex) setting and to obtain explicit chiral analogues of key star-products.

Abstract

We develop the theory of -vertex algebras, algebraic structures closely related to vertex algebras but with a deformed translation covariance axiom. We establish their structure theory, including analogues of Goddard's Uniqueness Theorem, the Reconstruction Theorem, Borcherds Identity, and the OPE Expansion Formula, and introduce the associated notions of -Lie conformal and -Poisson vertex algebras. The formalism provides a natural and simplified construction of the Zhu algebra. The main application is to the chiralization of classical star-products: we show that every star-product on the symmetric algebra of a Lie algebra (or its central extensions) admits a chiralization, and we derive explicit formulae for these chiral star-products, including the Moyal-Weyl and Gutt star-products. Setting recovers explicit deformation quantizations of a broad class of Poisson vertex algebras, including the classical limits of free-boson, -system, affine, and Virasoro vertex algebras.
Paper Structure (30 sections, 51 theorems, 278 equations)

This paper contains 30 sections, 51 theorems, 278 equations.

Table of Contents

  1. Introduction
  2. Chiralization of star-products
  3. Organization of the paper.
  4. Acknowledgments.
  5. Notation.
  6. Preliminaries
  7. Structure theory of vertex algebras
  8. The conformal structure
  9. lambda-bracket formalism
  10. The integral lambda-bracket
  11. Classical limit
  12. The Zhu algebra
  13. h-Vertex algebras
  14. Structure theory of h-vertex algebras
  15. Notation.
  16. ...and 15 more sections

Key Result

Theorem 2.2

Let $V$ be a vertex algebra. The vertex operators $Y(a,z)$ satisfy the following identities:

Theorems & Definitions (183)

  • Definition 2.1
  • Theorem 2.2
  • Remark 2.3
  • Theorem 2.4: Goddard's uniqueness theorem
  • Theorem 2.5: Reconstruction Theorem
  • Definition 2.6
  • Definition 2.7
  • Remark 2.8
  • Definition 2.9
  • Theorem 2.10
  • ...and 173 more