Universal structures in C-linear enumerative invariant theories - Gross, J; Joyce, D; Tanaka, Y (National Academy of Science of Ukraine, 2022)
An enumerative invariant theory in algebraic geometry, differential geometry, or representation theory, is the study of invariants which `count' τ-(semi)stable objects E with fixed topological invariants [[E]]=α in some geometric problem, by means of a virtual class [Mssα(τ)]virt in some homology theory for the moduli spaces Mstα(τ)⊆Mssα(τ) of τ-(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 H∗(M) the structure of a graded vertex algebra, and H∗(Mpl) a graded Lie algebra, closely related to H∗(M). The virtual classes [Mssα(τ)]virt take values in H∗(Mpl). In most such theories, defining [Mssα(τ)]virt when Mstα(τ)≠Mssα(τ) (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(τ)]inv in homology over Q, with [Mssα(τ)]inv=[Mssα(τ)]virt when Mstα(τ)=Mssα(τ), and that these invariants satisfy a universal wall-crossing formula under change of stability condition τ, written using the Lie bracket on H∗(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].