Count invariant theory in algebraic geometry, differential geometry, or representation theory is the study of invariants that “count” fixed topological invariants $\tau$-(semi)stable objects $E$[E]=\alpha$ in some geometric problem, using virtual class $[{\cal M}_\alpha^{\rm
ss}(\tau)]_{\rm virt}$ ( \tau)$ of $\tau$-(semi)stable objects. There are Donaldson-Thomas type invariants for counting, Donaldson invariants for 4-manifolds, and so on.

We speculate about new universal structures common to many count-invariant theories. In such a theory he has two moduli spaces ${\cal M}, {\cal M}^{\rm pl}$, where his second author writes $H_*({\ cal M}) gives $ to the structure of stepwise vertex algebra. and $H_*({\cal M}^{\rm pl})$ $H_*({\cal M})$ closely related graded Lie algebras. virtual class $[{\cal M}_\alpha^{\rm
ss}(\tau)]_{\rm virt}$ takes the value of $H_*({\cal M}^{\rm pl})$.

Definition of $[{\cal M}_\alpha^{\rm ss}(\tau)]_{\rm virt}$ when ${\cal M}_\alpha^{\rm st}(\tau)\ne{\cal M}_\alpha^{\rm ss}(\tau)$ (at Gauge theory (when the moduli space contains divisors) is a hard problem, I guess there is a natural way to define $[{\cal M}_\alpha^{\rm
ss}(\tau)]_{\rm virt}$ is homologous to $\mathbb Q$ and the resulting class is $H_*({\ cal M}^{\rm pl})$. We prove the conjecture for the moduli space of quiver representations without directed cycles.

Conjectures in algebraic geometry using Behrend-Fantechi virtual classes are proved in the sequel arXiv:2111.04694.



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