Virasoro constraints and representations for quiver moduli spaces

We study the Virasoro constraints for moduli spaces of representations of quiver with relations by Joyce's vertex algebras. Using the framed Virasoro constraints, we construct a representation of half of the Virasoro algebra on the cohomology of moduli stacks of quiver representations under smoothness assumption. By exploiting the non-commutative nature of the Virasoro operators, we apply our theory for quivers to del Pezzo surfaces using exceptional collections. In particular, the Virasoro constraints and representations are proven for moduli of sheaves on $\mathbb{P}^2$, $\mathbb{P}^1\times \mathbb{P}^1$ and $\text{Bl}_{\mathsf{pt}}(\mathbb{P}^2)$. Lastly, we unravel the Virasoro constraints for Grassmannians in terms of symmetric polynomials and Hecke operators.