Modules and generalizations of Joyce vertex algebras

TL;DR

This paper extends Joyce vertex algebras to non-linear enumerative geometry settings, introducing twisted modules and variants applicable to diverse invariants.

Contribution

It generalizes Joyce vertex algebras to non-linear problems, including twisted modules and variants for different enumerative invariants.

Findings

Constructed twisted modules for Joyce vertex algebras in orthosymplectic geometry.

Developed variants applicable to Joyce's homological, DT4, and K-theoretic invariants.

Proposed a framework for wall-crossing formulas in non-linear moduli stacks.

Abstract

Joyce vertex algebras are vertex algebra structures defined on the homology of certain $\mathbb{C}$-linear moduli stacks, and are used to express wall-crossing formulae for Joyce's homological enumerative invariants. This paper studies the generalization of this construction to settings that come from non-linear enumerative problems. In the special case of orthosymplectic enumerative geometry, we obtain twisted modules for Joyce vertex algebras. We expect that our construction will be useful for formulating wall-crossing formulae for enumerative invariants for non-linear moduli stacks. We include several variants of our construction that apply to different flavours of enumerative invariants, including Joyce's homological invariants, DT4 invariants, and a version of $K$-theoretic enumerative invariants.