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Higgs Branches in the Omega-background via the Category of Line Operators

Thomas Karabela, Wenjun Niu

TL;DR

The paper provides a categorical realization of the Omega-background quantization for the Higgs branch $M_H$ of a 3d ${\mathcal N}=4$ gauge theory by studying line operators in the B-twist. It shows that turning on the omega-background corresponds to passing to the $S^1$-invariant category of line defects, where the ribbon twist encodes the $S^1$-action and End of the unit object yields a deformation quantization of $\mathbb{C}[M_H]$, identified with quantum Hamiltonian reduction at a central character $J$. Using quadratic Koszul duality in a DG framework, the paper constructs an explicit model $\mathcal{CE}(\mathfrak{d}_{\mathfrak g,V},J)$ whose algebra structure gives the deformation quantization of $\mathbb{C}[\widehat{M}_H]$ and reproduces the quantum Hamiltonian reduction. This approach provides a rigorous, category-theoretic bridge between Omega-background quantization and Hamiltonian reduction, with potential extensions to quantizations of Poisson vertex algebras and Lagrangian subvarieties via line operators.

Abstract

The vacuum manifold $\mathcal{M}$ of a topological twist of a 3d $\mathcal{N}=4$ gauge theory is a hyper-Kähler variety; deformations and quantizations of $\mathcal{M}$ can be constructed in the framework of 3 dimensional topological quantum field theories. In particular, based on physics arguments, turning on an omega-background results in the quantization of $\mathbb{C}[\mathcal{M}]$ as a Poisson algebra. In this paper, we implement this idea mathematically, in the context of the B-twist of 3d $\mathcal{N}=4$ gauge theories, namely for Higgs branches. Our strategy is to implement omega-background in the category of line operators via the choice of a ribbon twist, and obtain the quantum Hamiltonian reduction as derived endomorphisms in the equivariant category, the category where the ribbon twist acts trivially. We apply quadratic Koszul duality to perform this computation.

Higgs Branches in the Omega-background via the Category of Line Operators

TL;DR

The paper provides a categorical realization of the Omega-background quantization for the Higgs branch of a 3d gauge theory by studying line operators in the B-twist. It shows that turning on the omega-background corresponds to passing to the -invariant category of line defects, where the ribbon twist encodes the -action and End of the unit object yields a deformation quantization of , identified with quantum Hamiltonian reduction at a central character . Using quadratic Koszul duality in a DG framework, the paper constructs an explicit model whose algebra structure gives the deformation quantization of and reproduces the quantum Hamiltonian reduction. This approach provides a rigorous, category-theoretic bridge between Omega-background quantization and Hamiltonian reduction, with potential extensions to quantizations of Poisson vertex algebras and Lagrangian subvarieties via line operators.

Abstract

The vacuum manifold of a topological twist of a 3d gauge theory is a hyper-Kähler variety; deformations and quantizations of can be constructed in the framework of 3 dimensional topological quantum field theories. In particular, based on physics arguments, turning on an omega-background results in the quantization of as a Poisson algebra. In this paper, we implement this idea mathematically, in the context of the B-twist of 3d gauge theories, namely for Higgs branches. Our strategy is to implement omega-background in the category of line operators via the choice of a ribbon twist, and obtain the quantum Hamiltonian reduction as derived endomorphisms in the equivariant category, the category where the ribbon twist acts trivially. We apply quadratic Koszul duality to perform this computation.

Paper Structure

This paper contains 6 sections, 1 theorem, 10 equations.

Table of Contents

  1. Introduction
  2. Quantization and omega-background
  3. Line defects in the B-twist and Lie superalgebras
  4. Mathematical statements and strategy
  5. Symplectic reductions and braided tensor categories
  6. Symplectic reductions and quantizations

Key Result

Theorem 1.1

Let $\mathbb C$ be the trivial module of $U(\mathfrak{d}_{\mathfrak g, V})[\psi]$, then is a deformation quantization of the Poisson algebra $\mathbb C[\widehat{{\mathcal{M}}}_H]$. It is identified with the usual quantum Hamiltonian reduction with respect to the same central character $J$. Here, $\mathrm{REnd}_{U(\mathfrak{d}_{\mathfrak g, V})[\psi]\mathrm{-Mod}}^{\mathbb C^\times}$ de

Theorems & Definitions (2)

  • Theorem 1.1
  • Remark 1.2