18 problems

The refined Vafa–Witten multiple-cover conjecture. The refined Vafa–Witten invariant satisfies

Let $S$ be the surface under consideration, let $c_1(S)$ denote its first Chern class, and let $v_0(S)$ be the null vector specified in the source. Write…

Let $\Lambda_S$ be the lattice of a del Pezzo surface. Let $v_0,v'_0\in\Lambda_S$ be null vectors satisfying … Let $\{E_I\}_{I=1}^{b_2-2}$ be an orthonormal basis of the orthogonal…

Jiang's twisted multiple-cover conjecture. If $H^{0,1}(S)=H^{0,2}(S)=0$, there exist rational numbers $\operatorname{VW}^{\operatorname{tw}}_{\alpha_i}({\mathfrak S})$ such that

S-duality conjecture for Vafa–Witten invariants. The two partition functions satisfy the S-duality transformation formula

Let $S$ be a smooth projective surface, let ${\mathfrak S}\to S$ be a $\mu_r$-gerbe, and let $\alpha=(\operatorname{rk},L,c_2)$. Let…

Let $S$ be a smooth projective surface and let $r\in{\mathbb Z}_{>0}$. For $L\in\operatorname{Pic}(S)$, define the $\operatorname{SU}(r)$ Vafa–Witten partition function by … Define…

Let $S$ be a smooth projective surface, let $q=e^{2\pi i\tau}$, and let $\eta(q)=q^{1/24}\prod_{k\geq 1}(1-q^k)$ be the Dedekind eta function. Define the normalized partition funct…

Let $S$ and $c_1$ satisfy the hypotheses of Theorem 1 in the source, and let $r$ be prime. Write $\mathrm{VW}_{r,c_1,c_2}^{\mathrm{monopole}}(S,y)$ for the monopole contribution to…

Let $S$ be a K3 surface, let $r$ be a positive integer, and let $\mathsf{VW}_{r,n}(t)$ denote the refined Vafa–Witten invariant for charge $(r,0,n)$. Let…

Let $S$ be a K3 surface, let $\alpha$ be a charge, and let $\chi_S(\alpha,\alpha)$ denote its Mukai pairing. Write $[r]_t=(t^{r/2}-t^{-r/2})/(t^{1/2}-t^{-1/2})$, and let…

Let $(S,\mathcal O_S(1))$ be a polarised surface, let $X=K_S$, and let $\alpha=(r,c_1,c_2)\in H^{\operatorname{ev}}(S)$ be a charge. Let $P^\perp_\alpha(n,t)$ be the refined invari…

Let $S$ be a K3 surface with polarization $H$, first Chern class $c_1$, and prime rank $r$. Let $\mathsf Z_{K3,H,r,c_1}^{\mathrm{inst}}(q,y)$ and…

For integers $(\beta_1,\beta_2,\beta_3,\beta_4)$ and $v\geq0$, let $F_v(\beta_1,\beta_2,\beta_3,\beta_4,y)$ and $G_v(\beta_1,\beta_2,\beta_3,\beta_4,y)$ be the universal functions…

Let $S$ be a smooth projective surface satisfying $H_1(S,\mathbb Z)=0$ and $p_g>0$. Let $H$ be a polarization, let $r=1$ or a prime, and let $c_1\in H^2(S,\mathbb Z)$. Let…

Let $S$ be a smooth projective surface satisfying $H_1(S,\mathbb Z)=0$ and $p_g>0$. For any polarization $H$ and first Chern class $c_1$, let…

Thomas's virtual localisation conjecture. The invariants should furthermore satisfy $\mathsf{VW}_\alpha=\mathsf{vw}_\alpha$ whenever $\deg K_S\leq0$. The preceding formulas conject…

Let $S$ be a smooth projective surface, let $P^\perp_{\alpha}(n)$ be the Joyce–Song pair invariant for charge $\alpha=(r,c_1(L),c_2)$, and let $\chi(\alpha(n))$ denote the relevant…