12 problems

Virasoro conjecture for Hodge integrals. For any smooth projective variety,

Let $X$ be a smooth projective variety, and let $\Psi_{g,n}$ denote the universal expression for Hodge integrals with one $\lambda_{g-1}$ or $\lambda_g$ insertion defined in the so…

Virasoro conjecture. For every $k\geq -1$ and every $D\in\mathbb D^S$,

Let $S$ be a K3 surface, let $\gamma_a$ be a homogeneous basis of $H^*(S)$ with $\gamma_a\in H^{p_a,q_a}(S)$, and set $b_a=p_a-1/2$. Let $[\alpha]^p_q$ denote the coefficient of…

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…

Let $X$ be a smooth projective variety and let $M$ be a moduli space of sheaves on $X$ with virtual fundamental class $[M]^{\mathrm{vir}}$. Let $\mathbb{D}^X$ be the descendent alg…

Let $(W,G)$ be an admissible Landau–Ginzburg pair, and let $\mathcal{A}^{\clubsuit}_{W,G}$ denote its total ancestor potential in FJRW, Polishchuk–Vaintrob, or Kiem–Li theory. Let…

Let $X$ be a simply-connected smooth projective 3-fold. Let $\mathbb D_{\textup{PT}}^{X}$ be the algebra generated by stable-pairs descendents $\textup{ch}_i(\gamma)$, and let…

Let $\mathbb D^+$ be the descendent operator space, let $\mathcal L_k$ be the operators defined in the source, let $\mathsf D\in\mathbb D^+$, and let $d\mathsf L$ range over curve…

Pandharipande–Solomon–Tessler's open Virasoro constraint conjecture. The partition function satisfies

Open Virasoro conjecture. The operators $\mathcal{L}_n$ annihilate the full partition function:

Let $\mathcal{X}$ be a compact orbifold, let $\mathcal{D}_\mathcal{X}$ denote its total descendant Gromov–Witten potential, and let $\mathcal{L}_m$ be the Virasoro operators define…