Universal Structures in ℂ-Linear Enumerative Invariant Theories

Abstract

An enumerative invariant theory in algebraic geometry, differential geometry, or representation theory, is the study of invariants which "count" 𝜏-(semi)stable objects 𝐸 with fixed topological invariants ⟦𝐸⟧ = α in some geometric problem, by means of a virtual class [ℳˢˢα(𝜏)]ᵥᵢᵣₜ in some homology theory for the moduli spaces ℳˢᵗα(𝜏) ⊆ ℳˢˢα(𝜏) of 𝜏-(semi)stable objects. Examples include Mochizuki's invariants counting coherent sheaves on surfaces, Donaldson-Thomas type invariants counting coherent sheaves on Calabi-Yau 3- and 4-folds and Fano 3-folds, and Donaldson invariants of 4-manifolds. We make conjectures on new universal structures common to many enumerative invariant theories. Any such theory has two moduli spaces ℳ, ℳᵖˡ, where the second author (see https://people.maths.ox.ac.uk/~joyce/hall.pdf) gives 𝐻∗(ℳ) the structure of a graded vertex algebra, and 𝐻∗(ℳᵖˡ) a graded Lie algebra, closely related to 𝐻∗(ℳ). The virtual classes [ℳˢˢα(𝜏)]ᵥᵢᵣₜ take values in 𝐻∗(ℳᵖˡ). In most such theories, defining [ℳˢˢα(𝜏)]ᵥᵢᵣₜ when ℳˢᵗα(𝜏) ≠ ℳssα(𝜏) (in gauge theory, when the moduli space contains reducibles) is a difficult problem. We conjecture that there is a natural way to define invariants [ℳˢˢα(𝜏)]ᵢₙᵥ in homology over Q, with [ℳˢˢα(𝜏)]ᵢₙᵥ = [ℳˢˢα(𝜏)]ᵥᵢᵣₜ when ℳˢᵗα(𝜏) = ℳˢˢα(𝜏), and that these invariants satisfy a universal wall-crossing formula under change of stability condition 𝜏, written using the Lie bracket on 𝐻∗(ℳᵖˡ). We prove our conjectures for moduli spaces of representations of quivers without oriented cycles. Versions of our conjectures in algebraic geometry using Behrend-Fantechi virtual classes are proved in the sequel [arXiv:2111.04694].