Universal Structures in ℂ-Linear Enumerative Invariant Theories

• Published in: Symmetry, Integrability and Geometry: Methods and Applications
• Date: 2022
• ISSN: 1815-0659
• Main Authors: Gross, Jacob, Joyce, Dominic, Tanaka, Yuuji
• Format: Article
• Language: English
• Published: Інститут математики НАН України 2022
• Online Access: https://nasplib.isofts.kiev.ua/handle/123456789/211720
• Journal Title: Digital Library of Periodicals of National Academy of Sciences of Ukraine
• Cite this: Universal Structures in ℂ-Linear Enumerative Invariant Theories. Jacob Gross, Dominic Joyce and Yuuji Tanaka. SIGMA 18 (2022), 068, 61 pages

An enumerative invariant theory in algebraic geometry, differential geometry, or representation theory, is the study of invariants which "count" -(semi)stable objects with fixed topological invariants ⟦⟧ = α in some geometric problem, by means of a virtual class [ℳˢˢα()]ᵥᵢᵣₜ in some homology theory for the moduli spaces ℳˢᵗα() ⊆ ℳˢˢα() of -(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 ∗(ℳ) the structure of a graded vertex algebra, and ∗(ℳᵖˡ) a graded Lie algebra, closely related to ∗(ℳ). The virtual classes [ℳˢˢα()]ᵥᵢᵣₜ take values in ∗(ℳᵖˡ). In most such theories, defining [ℳˢˢα()]ᵥᵢᵣₜ when ℳˢᵗα() ≠ ℳssα() (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()]ᵢₙᵥ in homology over Q, with [ℳˢˢα()]ᵢₙᵥ = [ℳˢˢα()]ᵥᵢᵣₜ when ℳˢᵗα() = ℳˢˢα(), and that these invariants satisfy a universal wall-crossing formula under change of stability condition , written using the Lie bracket on ∗(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].