34 problems

For four-dimensional supersymmetric Yang–Mills theories with $\mathcal{N}=4$ supersymmetry and gauge groups $U(N)$ of arbitrary rank, the associated Vafa–Witten invariants are defi…

Vafa–Witten modularity conjecture. The series $Z_r(X,{\mathcal N})$ is a modular form of weight $-\frac{\chi(X)}{2}$.

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

Kool–Vafa–Witten independence conjecture. The quotient

Let $S$ be a complex surface, with geometric genus $p_g(S)$ and positive second Betti number $b_2^+(S)$. Vafa–Witten wall-crossing conjecture. The wall-crossing for Vafa–Witten inv…

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…

Universal holomorphic ambiguity conjecture. For all null vectors of del Pezzo surfaces satisfying $r_1>0$, this choice continues to hold and ensures that the normalized generating…

Consider the three-node quiver relevant for $\mathds{P}^2$, with dimension vector $\vec N=(N_1,N_2,N_3)$, and let $\Omega_{\rm S}(\vec N)$ denote its single-centered invariant. Sin…

Let $\alpha$ be a dimension vector of a quiver associated to a Fano surface, and let $\Omega_{\rm S}(\alpha,y)$ denote its single-centered index. A pure D0-brane is the correspondi…

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 $X$ be a surface and let ${\mathfrak X}\to X$ range over all $\mu_r$-gerbes. For each such gerbe, consider the twisted Vafa–Witten invariants defined using the moduli space of…

Let $S$ be an almost Fano surface with $b_1(S)=0$ and $b_2^+(S)=1$, and let $K_S$ be its canonical class. For rank $N$, let the generating function of refined Vafa–Witten invariant…

Let $S$ be an almost complex four-manifold, with Euler number $chi(S)$ and signature $sigma(S)$. For rank $N$, let $h^{\rm VW,ref}_{N,\mu}$ denote the generating function of refine…

Rank-two K3 S-duality conjecture. Following the S-duality conjecture, this partition function is the rank-two Langlands-dual partition function associated with the fixed data…

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 $S$ be a smooth projective surface and let ${\mathfrak S}\to S$ be a $\mu_r$-gerbe. The twisted Vafa–Witten invariants are defined using twisted stable sheaves and Higgs sheave…

Let $A^{(3)}(y)$, $B^{(3)}(y)$, $C_{12}^{(3)}(y)$, $C_{11}^{(3)}(y)$, and $C_{22}^{(3)}(y)$ denote the universal series appearing in Theorem 1 of the source, and let $\phi_{-2,1}$,…

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 smooth projective surface, and let the refined Vafa–Witten invariants be decomposed into contributions from the instanton and monopole branches. Gottsche–Kool's modula…

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…