Universal Virasoro constraints for quivers with relations

Abstract: Following our reformulation of sheaf-theoretic Virasoro constraints with applications to curves and surfaces joint with Lim-Moreira, I describe in the present work the quiver analog. After phrasing a universal approach to Virasoro constraints for moduli of quiver-representations, I prove them for any finite quiver with relations, with frozen vertices, but without cycles. I use partial flag varieties which are special cases of moduli spaces of framed representations as a guiding example throughout. These results are applied to give an independent proof of Virasoro constraints for all Gieseker semistable sheaves on $\mathbb{P}^2$ and $\mathbb{P}^1 \times \mathbb{P}^1$ by using derived equivalences to quivers with relations. Combined with an existing universality argument for Virasoro constraints on Hilbert schemes of points on surfaces, this leads to the proof of this rank 1 case for any $S$ which is independent of the previous results in Gromov-Witten theory.