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Poisson Vertex Algebras and Three-Dimensional Gauge Theory

Ahsan Z. Khan, Keyou Zeng

TL;DR

This work introduces a three-dimensional mixed holomorphic-topological gauge theory built from a freely generated Poisson vertex algebra, showing that gauge invariance is equivalent to the $\\lambda$-bracket Jacobi identity and that a Virasoro element enhances the theory to be fully topological. It develops the BV/BRST framework, analyzes the gauge algebra as the PV-algebra data, and demonstrates how PVAs of affine, Virasoro, super-Virasoro, and $W_3$ types yield HT models with rich links to $3d$ gravity and higher-spin theories. The paper then studies deformation quantization via boundary vertex algebras, predicting central-charge shifts and exploring Koszul duality between boundary theories, while outlining a program to realize a VOA/PSM correspondence and to explore higher-dimensional and holographic extensions. Overall, the HT Poisson sigma model provides a natural bridge from PVAs to quantum algebras and gravitational theories, with potential implications for conformal blocks, boundary algebras, and twisted holography.

Abstract

We introduce a mixed holomorphic-topological gauge theory in three dimensions associated to a (freely generated) Poisson vertex algebra. The $λ$-bracket of the PVA plays the role of the structure constants of the gauge algebra and the gauge invariance of the theory holds if and only if the $λ$-bracket Jacobi identity is satisfied. We show that the holomorphic-topological symmetry of the theory enhances to full topological symmetry if the Poisson vertex algebra contains a Virasoro element. We outline examples associated to PVAs of $\mathcal{W}$-type and demonstrate their connections to various versions of $3d$ gravity. We expect the three-dimensional Poisson sigma model to play an important role in the deformation quantization of Poisson vertex algebras.

Poisson Vertex Algebras and Three-Dimensional Gauge Theory

TL;DR

This work introduces a three-dimensional mixed holomorphic-topological gauge theory built from a freely generated Poisson vertex algebra, showing that gauge invariance is equivalent to the -bracket Jacobi identity and that a Virasoro element enhances the theory to be fully topological. It develops the BV/BRST framework, analyzes the gauge algebra as the PV-algebra data, and demonstrates how PVAs of affine, Virasoro, super-Virasoro, and types yield HT models with rich links to gravity and higher-spin theories. The paper then studies deformation quantization via boundary vertex algebras, predicting central-charge shifts and exploring Koszul duality between boundary theories, while outlining a program to realize a VOA/PSM correspondence and to explore higher-dimensional and holographic extensions. Overall, the HT Poisson sigma model provides a natural bridge from PVAs to quantum algebras and gravitational theories, with potential implications for conformal blocks, boundary algebras, and twisted holography.

Abstract

We introduce a mixed holomorphic-topological gauge theory in three dimensions associated to a (freely generated) Poisson vertex algebra. The -bracket of the PVA plays the role of the structure constants of the gauge algebra and the gauge invariance of the theory holds if and only if the -bracket Jacobi identity is satisfied. We show that the holomorphic-topological symmetry of the theory enhances to full topological symmetry if the Poisson vertex algebra contains a Virasoro element. We outline examples associated to PVAs of -type and demonstrate their connections to various versions of gravity. We expect the three-dimensional Poisson sigma model to play an important role in the deformation quantization of Poisson vertex algebras.

Paper Structure

This paper contains 32 sections, 193 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Holomorphic-Topological Poisson Sigma Model
  3. Poisson Vertex Algebras
  4. The Model
  5. Global definition
  6. Examples
  7. Affine PVA and Chern-Simons Theory
  8. Virasoro PVA and Gravity
  9. Phase space and Quantization
  10. Wilson-type Line Defects
  11. SuperVirasoro and Supergravity
  12. Classical phase space
  13. Wilson-type Line Defects
  14. $W_3$ PVA and Higher Spin Theory
  15. Further properties
  16. ...and 17 more sections

Figures (4)

  • Figure 1: A hierarchy of Poisson vertex algebras and their corresponding Poisson sigma models.
  • Figure 2: Deformation quantization of PVA and quantization of the Poisson sigma model (with boundary).
  • Figure 3: The tree level boundary Feynman diagram
  • Figure 4: The first loop digram

Theorems & Definitions (2)

  • Remark 2.1
  • Remark 4.1