Modules and generalizations of Joyce vertex algebras

Abstract Joyce vertex algebras are vertex algebra structures defined on the homology of certain double struck upper C $\mathbb {C}$ 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 nonlinear 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 nonlinear 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.