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 $[{\mathcal{M}}_{\alpha }^{\mathrm{ss}}(\tau ){]}_{\mathrm{virt}}$ in some homology theory for the moduli spaces ${\mathcal{M}}_{\alpha }^{\mathrm{st}}(\tau )\subseteq {\mathcal{M}}_{\alpha }^{\mathrm{ss}}(\tau )$ 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 $\mathcal{M},{\mathcal{M}}^{\mathrm{pl}}$, where the second author gives $H_{\ast }(\mathcal{M})$ the structure of a graded vertex algebra, and $H_{\ast }({\mathcal{M}}^{\mathrm{pl}})$ a graded Lie algebra, closely related to $H_{\ast }(\mathcal{M})$. The virtual classes $[{\mathcal{M}}_{\alpha }^{\mathrm{ss}}(\tau ){]}_{\mathrm{virt}}$ take values in $H_{\ast }({\mathcal{M}}^{\mathrm{pl}})$. Defining $[{\mathcal{M}}_{\alpha }^{\mathrm{ss}}(\tau ){]}_{\mathrm{virt}}$ when ${\mathcal{M}}_{\alpha }^{\mathrm{st}}(\tau )\ne {\mathcal{M}}_{\alpha }^{\mathrm{ss}}(\tau )$ (in gauge theory, when the moduli space contains reducibles) is a difficult problem. We conjecture that there is a natural way to define $[{\mathcal{M}}_{\alpha }^{\mathrm{ss}}(\tau ){]}_{\mathrm{virt}}$ 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 $H_{\ast }({\mathcal{M}}^{\mathrm{pl}})$. 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.

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• Theta-stratifications, Theta-reductive stacks, and applications
• Geometry from Donaldson-Thomas invariants
• Homological Lie brackets on moduli spaces and pushforward operations in twisted K-theory

Cited by in corpus (1)

• Wall-crossing for zero-dimensional sheaves and Hilbert schemes of points on Calabi-Yau 4-folds