Universal Structures in $\mathbb C$-Linear Enumerative Invariant Theories

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, by means of a virtual class $[\mathcal{M}_\alpha^{{\rm ss}}(\tau)]_{{\rm virt}}$ in some homology theory for the moduli spaces $\mathcal{M}_\alpha^{{\rm st}}(\tau)\subseteq\mathcal{M}_\alpha^{{\rm ss}}(\tau)$ 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. Any such theory has two moduli spaces $\mathcal{M}$, $\mathcal{M}^{{\rm pl}}$, where the second author (see https://people.maths.ox.ac.uk/~joyce/hall.pdf) gives $H_*(\mathcal{M})$ the structure of a graded vertex algebra, and $H_*\big(\mathcal{M}^{{\rm pl}}\big)$ a graded Lie algebra, closely related to $H_*(\mathcal{M})$. The virtual classes $[\mathcal{M}_\alpha^{{\rm ss}}(\tau)]_{{\rm virt}}$ take values in $H_*\big(\mathcal{M}^{{\rm pl}}\big)$. In most such theories, defining $[\mathcal{M}_\alpha^{{\rm ss}}(\tau)]_{{\rm virt}}$ when $\mathcal{M}_\alpha^{{\rm st}}(\tau)\ne\mathcal{M}_\alpha^{{\rm ss}}(\tau)$ (in gauge theory, when the moduli space contains reducibles) is a difficult problem. We conjecture that there is a natural way to define invariants $[\mathcal{M}_\alpha^{{\rm ss}}(\tau)]_{\mathrm{inv}}$ in homology over $\mathbb{Q}$, with $[\mathcal{M}_\alpha^{{\rm ss}}(\tau)]_{\mathrm{inv}}=[\mathcal{M}_\alpha^{{\rm ss}}(\tau)]_{{\rm virt}}$ when $\mathcal{M}_\alpha^{{\rm st}}(\tau)=\mathcal{M}_\alpha^{{\rm ss}}(\tau)$, and that these invariants satisfy a universal wall-crossing formula under change of stability condition $\tau$, written using the Lie bracket on $H_*\big(\mathcal{M}^{{\rm pl}}\big)$. 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].