Coupling a vertex algebra to a large center

TL;DR

The paper introduces a uniform construction of a vertex algebra with a large central center, $\tilde{\mathcal{V}}=(\mathcal{O}(G[[z]])\otimes \mathcal{V})^G$, extending a given $G$-acted vertex algebra $\mathcal{V}$ by functionals on regular $\mathfrak{g}$-connections. It shows that each fiber over a regular connection $\mathrm{d}+A$ is canonically isomorphic to $\mathcal{V}$ via a delta embedding, and that pulling back fiberwise yields a $(\mathrm{d}+A)$-twisted $\mathcal{V}$-module with an explicit twisted commutator formula; irregular cases are discussed with the role of Stokes data. The work develops the necessary commutative vertex algebras $\mathcal{O}(\mathfrak{g}[[z]])$ and $\mathcal{O}(G[[z]])$, their Poisson-vertex structures, and gauge automorphisms, and provides concrete SL$_2$ and doublet-vertex-algebra examples illustrating how central terms alter OPEs and recover known theories like $(\hat{\sl}_2)_1$ and symplectic fermions. Overall, the construction aims to realize twisted modules and potential higher-categorical Langlands-type structures in a geometric, gauge-theoretic framework, with broad implications for large-center limits and quantum groups.

Abstract

Suppose a Lie group $G$ acts on a vertex algebra $V$. In this article we construct a vertex algebra $\tilde{V}$, which is an extension of $V$ by a big central vertex subalgebra identified with the algebra of functionals on the space of regular $\mathfrak{g}$-connections $(d+A)$. The category of representations of $\tilde{V}$ fibres over the set of connections, and the fibres should be viewed as $(d+A)$-twisted modules of $V$, generalizing the familiar notion of $g$-twisted modules. In fact, another application of our result is that it proposes an explicit definition of $(d+A)$-twisted modules of $V$ in terms of a twisted commutator formula, and we feel that this subject should be pursued further. Vertex algebras with big centers appear in practice as critical level or large level limits of vertex algebras. I particular we have in mind limits of the generalized quantum Langlands kernel, in which case $G$ is the Langland dual and $V$ is conjecturally the Feigin-Tipunin vertex algebra and the extension $\tilde{V}$ is conjecturally related to the Kac-DeConcini-Procesi quantum group with big center. With the current article, we can give a uniform and independent construction of these limits.

Theorems & Definitions (25)

• Theorem 1: \ref{['thm_twistedRep']}
• Definition 2.1
• Lemma 2.2
• Lemma 2.3
• Corollary 2.4
• Lemma 2.5
• proof
• Lemma 2.6
• proof
• Definition 2.7