Universal Structures in C-Linear Enumerative Invariant Theories

An enumerative invariant theory in Algebraic Geometry, Differential or Representation Theory, is the study of invariants which 'count' $\tau$-(semi)stable objects $E$ with fixed topological $[E]=\alpha$ some geometric problem, using a virtual class $[{\cal M}_\alpha^{\rm ss}(\tau)]_{\rm virt}$ homology for moduli spaces ${\cal st}(\tau)\subseteq{\cal ss}(\tau)$ objects. Examples include Mochizuki's counting coherent sheaves on surfaces, Donaldson-Thomas type Calabi-Yau 3- and 4-folds Fano 3-folds, Donaldson 4-manifolds. We make conjectures new universal structures common to many theories. Such theories have two M},{\cal M}^{\rm pl}$, where second author gives $H_*({\cal M})$ structure graded vertex algebra, pl})$ Lie closely related M})$. The classes take values pl})$. Defining when st}(\tau)\ne{\cal (in gauge theory, space contains reducibles) difficult problem. conjecture that there natural way define over $\mathbb Q$, resulting satisfy wall-crossing formula under change stability condition $\tau$, written bracket prove our representations quivers without oriented cycles. Our Geometry Behrend-Fantechi are proved sequel arXiv:2111.04694.