Universal structures in $\mathbb C$-linear enumerative invariant theories

Abstract

An enumerative invariant theory in Algebraic Geometry, Differential Geometry, or Representation Theory, is the study of invariants which 'count' $\tau$-(semi)stable objects $E$ with fixed topological invariants $[E]=\alpha$ in some geometric problem, using a virtual class $[{\cal M}_\alpha^{\rm ss}(\tau)]_{\rm virt}$ in some homology theory for the moduli spaces ${\cal M}_\alpha^{\rm st}(\tau)\subseteq{\cal M}_\alpha^{\rm ss}(\tau)$ of $\tau$-(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. Such theories have two moduli spaces ${\cal M},{\cal M}^{\rm pl}$, where the second author gives $H_*({\cal M})$ the structure of a graded vertex algebra, and $H_*({\cal M}^{\rm pl})$ a graded Lie algebra, closely related to $H_*({\cal M})$. The virtual classes $[{\cal M}_\alpha^{\rm ss}(\tau)]_{\rm virt}$ take values in $H_*({\cal M}^{\rm pl})$. Defining $[{\cal M}_\alpha^{\rm ss}(\tau)]_{\rm virt}$ when ${\cal M}_\alpha^{\rm st}(\tau)\ne{\cal M}_\alpha^{\rm ss}(\tau)$ (in gauge theory, when the moduli space contains reducibles) is a difficult problem. We conjecture that there is a natural way to define $[{\cal M}_\alpha^{\rm ss}(\tau)]_{\rm virt}$ in homology over $\mathbb Q$, and that the resulting classes satisfy a universal wall-crossing formula under change of stability condition $\tau$, written using the Lie bracket on $H_*({\cal M}^{\rm pl})$. We prove our conjectures for moduli spaces of representations of quivers without oriented cycles. Our conjectures in Algebraic Geometry using Behrend-Fantechi virtual classes are proved in the sequel arXiv:2111.04694.