27 problems

Virasoro conjecture. For any smooth projective variety,

Virasoro constraints conjecture. For all $k\geq -1$ and $D\in\mathbb D_{\mathrm{PT}}^X$,

Eguchi–Hori–Xiong conjecture. The generating function satisfies the Virasoro conditions

Let $X$ be a nonsingular projective 3-fold with only $(p,p)$-cohomology, let $\beta\in H_2(X,\mathbb{Z})$, and let $\mathbb{D}^X_{\mathrm{PT}}$ be the stable pairs descendent algeb…

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

Let $X$ be a smooth projective variety of dimension $r$, let $\gamma_a$ be a homogeneous basis of $H^\bullet(X,\mathbb{C})$ with $\gamma_a\in H^{p_a,q_a}(X)$, and set…

Matrix-integral conjecture. The generating function of Gromov–Witten invariants should have a matrix integral expression.

Let $X$ be a target with equivariant quantum cohomology, and suppose its non-equivariant counterpart is semisimple. An asymptotic solution is a formal solution of the quantum diffe…

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

Let $\mathcal{D}({\bf t};\hbar)$ be the homogeneous total descendent potential of a calibrated homogeneous CohFT with vacuum, and let the generalized Virasoro constraints denote th…

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…

Let $X$ be a smooth projective variety. Let $F_g$ be the genus-$g$ generating function, and write $\langle\langle\cdots;\lambda_g\rangle\rangle_g$ for the corresponding Hodge-integ…

Let $X$ be a target space, and let $Z_X$ be the Gromov--Witten partition function obtained from the moduli spaces of stable maps to $X$. Let the Virasoro conditions mean the corres…

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

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, let $M=M_\alpha$ be a moduli space of sheaves of topological type $\alpha$, and let $\mathbb{D}^X$ be the descendent algebra. Choose an alge…

Let $\overline{\mathcal M}_{g,n}$ be the moduli space of stable curves of genus $g$ with $n$ marked points, and let descendents be the natural cohomology classes on this space. Wit…

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

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 $F^o$ be a solution of the WDVV equations satisfying the unit and homogeneity properties, together with its calibration. Under possibly additional assumptions, let…

Let $\mathcal{F}^{\mathrm{pt},o}_g$ be the genus-$g$ open generating series arising from the intersection theory on moduli spaces of stable pointed disks, with…

Let $\tau_1({\bf T};{\bf S}_0)$ be the generating function for open intersection numbers with $Q=1$ and ${\bf S}_0=\{S_0,0,0,\dots\}$. PST's open KdV and Virasoro conjecture. The g…

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

Let $F^c$ and $F^o$ be the closed and open generating series of descendent integrals, respectively, and set … Here $u$, $s$, and $t_d$ are the formal variables appearing in the gen…

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