The Virasoro conjecture for moduli of pairs

Let $X$ be a smooth projective variety, let $V$ be a fixed sheaf on $X$, and let $P$ be a moduli space of pairs consisting of a sheaf $F$ and a map $V\to F$, with universal pair $q^*V\to\mathbb{F}$ on $P\times X$. Let $\mathbb{D}^X$ be the descendent algebra and let $\mathsf{L}_k^V$ be the Virasoro operators associated with the pair theory. Virasoro conjecture for pairs. For every $k\geq0$ and $D\in\mathbb{D}^X$,

$$\int_{[P]^{\mathrm{vir}}}\xi_{\mathbb{F}}\bigl(\mathsf{L}_k^V(D)\bigr)=0.$$

The conjecture proposes universal Virasoro constraints for descendent invariants of pair moduli; the paper proves them in the cases covered by its main results, while the general formulation remains open.