23 problems

Bounded-degree and quasi-modularity conjecture. For any genus, the tropical refined invariants have bounded degree, and the coefficients of these polynomials are given by quasi-mod…

Let $S$ be a del Pezzo surface, let $E$ be a suitable anticanonical divisor, and let $M_{\beta}$ be the moduli space of stable one-dimensional sheaves of class $\beta$ and Euler ch…

Square-dependence conjecture. If $2\nmid\beta$, then $\mathsf{GV}_{\beta}$ should depend only on the square $\beta^2$.

Enriques perverse Hodge generating-series conjecture. The perverse Hodge numbers satisfy the displayed generating-series identity involving the theta functions, eta functions, and…

Refined Gopakumar–Vafa conjecture. One has

Kool–Vafa–Witten independence conjecture. The quotient

Let refined BPS counting denote the refined enumeration of BPS states encoded by the refined partition functions, and let the refined topological string denote the corresponding re…

Refined Donaldson–Thomas product conjecture. One has

Let $\Phi^{\mathrm{ref}}(\tau,z,\nu,\sigma)$ be the refined multiplicative lift constructed from the Fourier coefficients of a Hodge–Elliptic genus. Motivic Donaldson–Thomas conjec…

Let $\mathtt{Q}=\prod_{i\in J}Q_i$ be the charge monomial associated with a wall-crossing combination, and let $t_1$ and $t_2$ be the refined variables. In the refined BPS partitio…

Let $\gamma_1,\ldots,\gamma_n$ be charges, let $\gamma_{kl}$ denote their pairing data entering the refined Moyal factor, let $\Delta$ be the deformation parameter, and define … Le…

Let $d_B$ and $d_E$ denote the base and fiber degrees, respectively, and let $\mathcal{F}^{(n,g)}$ be the base-degree-zero amplitude of genus data satisfying $n+g\geq 2$. Let $q$ b…

Let $X$ be a Calabi–Yau threefold, and let $F$ be a stable coherent sheaf on $X$ of pure dimension $1$, with Chern character $\operatorname{ch}(F)=\beta+\chi$, where $\beta$ is the…

Let $S$ be a smooth projective surface with $p_g(S)>0$ and $b_1(S)=0$, and let $L\in\operatorname{Pic}(S)$. Choose $H,c_1,c_2$ so that there are no rank-two strictly Gieseker $H$-s…

Let $g^{\rm (ref)}_n(\{\check\gamma_i,c_i\},y)$ be defined by the refined generating-function construction and the reference solution $F^{\rm (ref)}_n$, with coefficients $b_n$ fix…

Let $\alpha\in\operatorname{Pic}(S)$ be a positive primitive class on a K3 surface with $\langle\alpha,\alpha\rangle=2h-2$, and define $h[k]$ by $h[k]=k^2(h-1)+1$. Assuming Conject…

Let $\mathcal{P}^{CS}_{n,m,\lambda}(q,t)$ be the refined Chern–Simons invariant of a colored $(m,n)$ torus knot, defined using levels $k$ and $N$. Aganagic–Shakirov stabilization c…

Let the refined Chern–Simons invariants of an $(m,n)$ torus knot be given by the construction of Aganagic and Shakirov. Aganagic–Shakirov conjecture. These invariants match the Poi…

Dolbeault polynomial formula conjecture. The Dolbeault polynomial $\widetilde{Q}(\gamma;y,t)$ is given by the formula obtained from the corresponding formula for the refined index…

Let $q=e^{\hbar}$ tend to $1$ while $t$ and $x\equiv e^u=q^n$ remain fixed. Let $P_n$ denote the homological knot invariant and $Z_{\text{BPS}}^{\text{open}}(u,q,t)$ the refined BP…

Degree-independence conjecture. For fixed $r$, both $\overline H(r,e)(y)$ and $\overline H(r,e)(u,v)$ are independent of $e$; equivalently, they have the same value for all pairs…

Doubly refined wallcrossing and multicover conjecture. Under the same conditions as the preceding refined wallcrossing conjecture, there exist doubly refined invariants in…

Refined multicover conjecture. One has