Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 18 (2022), 068, 61 pages      arXiv:2005.05637      https://doi.org/10.3842/SIGMA.2022.068
Contribution to the Special Issue on Enumerative and Gauge-Theoretic Invariants in honor of Lothar Göttsche on the occasion of his 60th birthday

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

Jacob Gross a, Dominic Joyce a and Yuuji Tanaka b
a) The Mathematical Institute, Radcliffe Observatory Quarter, Woodstock Road, Oxford, OX2 6GG, UK
b) Department of Mathematics, Faculty of Science, Kyoto University, Kitashirakawa Oiwake-cho, Sakyo-ku, Kyoto 606-8502, Japan

Received November 22, 2021, in final form September 06, 2022; Published online September 23, 2022

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, 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 $[{\mathcal M}_\alpha^{{\rm ss}}(\tau)]_{{\rm inv}}$ in homology over $\mathbb Q$, with $[{\mathcal M}_\alpha^{{\rm ss}}(\tau)]_{{\rm inv}}=[{\mathcal M}_\alpha^{{\rm ss}}(\tau)]_{{\rm virt}}$ 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].

Key words: invariant; stability condition; vertex algebra; wall crossing formula; quiver.

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