Higgs and Coulomb branches from vertex operator algebras

TL;DR

The paper proposes a framework linking line defects in twisted 3d N=4 theories to derived categories of boundary VOAs and their Ext algebras, tested across basic and richer Abelian theories, non-Lagrangian examples, and mirror symmetric setups. It develops a BRST and D-module toolkit, connects to the Braverman-Finkelberg-Nakajima construction of Coulomb branches via affine Grassmannians, and argues for a perturbative capture of Higgs/Coulomb data by boundary VOAs. A central theme is that Ext algebras of boundary VOAs encode Higgs and Coulomb branch algebras, with simple current extensions providing a controlled method to compute these structures in both C and H twists. The results support a program in which line defects in 3d N=4 theories are governed, in a derived sense, by boundary vertex algebras, offering computational handles for both Lagrangian and non-Lagrangian theories and guiding future geometric representation theoretic explorations.

Abstract

We formulate a conjectural relation between the category of line defects in topologically twisted 3d ${\cal N} = 4$ supersymmetric quantum field theories and categories of modules for Vertex Operator Algebras of boundary local operators for the theories. We test the conjecture in several examples and provide some partial proofs for standard classes of gauge theories.

Figures (16)

• Figure 1: The Loewy diagram of the modules $\mathcal{E}_\pm$ and of the projective cover $\mathcal{P}$ of the Fc-vacuum $\mathcal{V}$.
• Figure 2: Loewy diagram of the projective cover $R_{s, r}$ of the simple triplet module $W_{s, r}$ for $1\leq s\leq p-1$.
• Figure 3: The Loewy diagram of the Zig-Zag $Z^n_{s,k}$.
• Figure 4: The Loewy diagram of the projective cover $P_{s,k}$ of the simple module $M_{s,k}$.
• Figure 5: New modules extending the simple module $M_{s,k}$ in $\mathcal{C}_{\text{log}}$. The dashed-line indicates the nilpotent action of $H$.

Theorems & Definitions (1)

• Conjecture