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Poisson Vertex Algebra of Seiberg-Witten Theory

Ahsan Z. Khan

Abstract

The space of local operators in the $Q$-cohomology of the holomorphic-topological supercharge in a four-dimensional $\mathcal{N}=2$ theory carries the structure of a Poisson vertex algebra. This note studies the Poisson vertex algebra associated to the pure $\mathcal{N}=2$ gauge theory with gauge group $SU(2)$. We propose an explicit Poisson vertex algebra $A$, claimed to be isomorphic to the algebra of holomorphic-topological observables to all orders in perturbation theory. We compute the Hilbert-Poincaré series of $A$ and show that it refines the Schur index of the pure $SU(2)$ theory. We show that $A$ admits a further differential $Q_{\text{inst}}$ which we hypothesize captures non-perturbative corrections, and compute the cohomology of this differential. We thus present an explicit candidate for the space of non-perturbative holomorphic-topological observables of Seiberg-Witten theory.

Poisson Vertex Algebra of Seiberg-Witten Theory

Abstract

The space of local operators in the -cohomology of the holomorphic-topological supercharge in a four-dimensional theory carries the structure of a Poisson vertex algebra. This note studies the Poisson vertex algebra associated to the pure gauge theory with gauge group . We propose an explicit Poisson vertex algebra , claimed to be isomorphic to the algebra of holomorphic-topological observables to all orders in perturbation theory. We compute the Hilbert-Poincaré series of and show that it refines the Schur index of the pure theory. We show that admits a further differential which we hypothesize captures non-perturbative corrections, and compute the cohomology of this differential. We thus present an explicit candidate for the space of non-perturbative holomorphic-topological observables of Seiberg-Witten theory.

Paper Structure

This paper contains 15 sections, 197 equations, 1 table.

Table of Contents

  1. Introduction
  2. The Poisson Vertex Algebra $A$
  3. Background
  4. The Poisson Vertex Algebra $A$
  5. Hilbert-Poincaré Series of $A$
  6. Holomorphic-Topological Observables
  7. Holomorphic-Topological Twist
  8. Holomorphic-Topological BF Theory
  9. BRST Reduction of Classical $bc$ System from BF Theory
  10. Basic Invariants and $\text{Obs}_{\text{HT}}(\mathfrak{g})$.
  11. Recovering the Schur index from the Basic Complex
  12. The Case $\mathfrak{g}= \mathfrak{sl}_2$
  13. The Differential $Q_{\text{inst}}$ and its Cohomology
  14. Future Directions
  15. AI Assisted Exploration

Theorems & Definitions (5)

  • Remark 3.1
  • Remark 3.2
  • Remark 3.3
  • Remark 3.4
  • Conjecture 1