12 problems

Dolbeault S-duality matching conjecture. Over the $\diamondsuit$-locus, the following matching data hold under $\mathbf{S}_{\mathrm{Dol}}$: the dual of $A_{\mathrm{Dol}}(M_\rho')$…

Let $V_i$ and $V_j$ be finite-dimensional vector spaces, and let $\operatorname{GL}(V_i)$ and $\operatorname{GL}(V_j)$ act on the Hamiltonian spaces $\mathbf M_{\medcirc}(V_i,V_j)$…

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 ${\mathsf r}\geq 2$, and let $\check{\mathsf{QM}}^{\mathsf a}_{{\mathsf d},0}$ and $\hat{\mathsf{QM}}^{\mathsf a}_{{\mathsf d},0}$ denote the zero-degree quasimap invariants in…

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

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

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 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 $G$ be a group with Langlands dual group $G^{\vee}$, let $\calT$ be a theory with a $G$-action, and let $\calT^{\vee}$ be its $S$-dual theory with a $G^{\vee}$-action. Let…

Let $M_k^0$ be the hyperkähler moduli space of certain ${\rm SU}(2)$ magnetic monopoles on $\mathbb R^3$, and let $\widetilde{M}_k^0$ be its universal cover. Write…