15 problems

Let $(X,H)$ be a smooth polarized surface with $H_1(X,\mathbb{Z})=0$ and $h^{2,0}(X)>0$. Let $r$ be prime and let $c_1\in H^2(X,\mathbb{Z})$ be algebraic. Let…

Let $[r-1]=\\{1,\ldots,r-1\\}$, and for $1\leq i<j\leq r-1$ define … and … Let $(X,H)$ be a smooth polarized surface with $H_1(X,\mathbb{Z})=0$ and $h^{2,0}(X)>0$. Let $r>1$ and…

Let $r>1$. The notation $\overline{\Delta}(q)$ and $\epsilon_r$ is as defined in the introduction. For a smooth polarized surface $(X,H)$ with $b_1(X)=0$ and $h^{2,0}(X)>0$, let…

Tanaka–Thomas's conjecture. There exist rational numbers $\mathsf{VW}(v_i)$ such that, for all $n\gg0$,

Let $\hat{\mathsf{VW}}^{\mathsf m,\mathsf a}_{{\mathsf d},0}$ denote the zero-degree Vafa–Witten invariants and set $\omega=e^{2\pi i/{\mathsf r}}$. S-duality without $\mu$-inserti…

Let $\hat{\mathsf{VW}}^{\mathsf m,\mathsf a}_{\mathsf d}(B,q)_t$ denote the Vafa–Witten generating series, with indices recording degree, monopole data, and flux. Geometric S-duali…

Let $q=e^{2\pi i\tau}$ and let $\tilde{\hat{\mathsf{VW}}}^{\mathsf m,\mathsf a}_{\mathsf d}(B,q)_t$ be the modified Vafa–Witten generating series with $\mu$-insertions, indexed by…

Let a 4d $\mathcal{N}=2$ theory be reachable from 4d $\mathcal{N}=4$ super-Yang–Mills theory by an RG flow, and let $M_3$ be a 3-manifold. Spectral-sequence conjecture. The RG flow…

Let $M_3$ be a 3-manifold and let $\mathcal{H}_{\mathrm{VW}}(M_3)$ denote the Vafa–Witten homology associated with it. No-instanton conjecture. In Vafa–Witten theory on 3-manifolds…

Let $M_3$ be a closed 3-manifold, and let $\mathcal{H}_{\mathrm{VW}}(M_3)$ denote its Vafa–Witten homology. Infinite-dimensionality conjecture. The space…

Let $X$ be a four-manifold equipped with the geometric data defining the adjoint Seiberg–Witten equations and the Vafa–Witten equations. The deformed Vafa–Witten equations are obta…

Tanaka--Thomas conjecture. The invariant $\mathrm{VW}_S^H(r,c_1,n)$ is independent of the choice of $\nu\gg0$.

VafaWitten S-duality conjecture. The two partition functions are Fourier expansions of these meromorphic functions, which satisfy

Tensoring invariance conjecture. For any such $(S,H)$, $c_1$, and $b3$, one has

Let $S$ be a smooth projective surface, let $Q(\alpha)=\int_S\alpha^2$, and let $a_1,\ldots,a_s$ be its Seiberg–Witten basic classes. Define the generating series…