Generators of a Bosonic VOA and Connections to a Boundary VOA

TL;DR

The paper advances the construction of bosonic vertex operator algebras (VOAs) arising from 3D ${ m N}=4$ gauge theories by explicitly implementing BRST reduction on boundary systems with symplectic bosons, and by introducing covariant-derivative-like deformations that generate nontrivial, BRST-closed operators. For the abelian quivers associated with $T_{[n-1,1]}^{[1^n]}(SU(n))$, it identifies a complete set of generators—encompassing $U$, $\,\mathcal{X}$, $\mathcal{Y}$, the $T_i$, $W_i$, and the $\mathcal{X}_I$, $\mathcal{Y}_I$ families—constructed via covariant derivatives acting on Higgs-branch-like operators, and proves primary-status for the key operators while detailing the OPE structure (with many results relegated to appendices). The second theory, $T^{[2,1^{n-1}]}_{[n-1,1^2]}(SU(n+1))$, is treated analogously, yielding a parallel bosonic VOA with generators $U,H,E,F,X_I^{\pm},Y_I^{\pm},T_i,W_i$, and their composites, and demonstrating a consistent boundary-VOA perspective where the bosonic subalgebra sits inside a larger fermionic extension. A central theme is a Catalan-number–controlled counting of independent XI- and YI-type primaries, and a combinatorial framework (via maps $F$ and $G$) that supports independence and basis arguments. The work also strengthens the link between a bosonic VOA and a boundary VOA, showing that a boundary VOA can naturally contain and thus inherit properties from its bosonic subalgebra, and it sketches how fermionic extensions further enrich this interplay. Overall, the paper solidifies the generator-based construction of these VOAs and clarifies their OPEs and structure, setting the stage for broader applications and generalizations to nonabelian setups and related algebraic structures.

Abstract

We show how to identify generators of bosonic VOAs associated with $T_{[n-1,1]}^{[1^n]}(SU(n))$ and $T_{[n-1,1^2]}^{[2,1^{n-1}]}(SU(n+1))$, and conjecture that the algebraic structure of these VOAs can be constructed by these generators. We also find out that the boundary VOA associated with $T_{[n-1,1]}^{[1^n]}(SU(n))$ naturally includes the bosonic VOA.

Figures (8)

• Figure 1: The quiver diagram for $T_{[2,1]}^{[1^3]}(SU(3))$.
• Figure 2: The case of $D_i -D_j$ for $1 \leq i < j \leq n$.
• Figure 3: The case of $(D_i -D_j)(D_k -D_l)$ for $1 \leq i < j <k < l \leq n$.
• Figure 4: The case of $(D_i -D_j)(D_k -D_l)$ for $1 \leq i < k < j < l \leq n$.
• Figure 5: The case of $(D_i -D_j)(D_k -D_l)$ for $1 \leq i < k < l < j \leq n$.

Theorems & Definitions (37)

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• Lemma 7.1
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• Lemma 7.2
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• Lemma 7.3
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