43 problems

Vafa–Witten's blowup conjecture. There is a universal function $Z_a$, independent of $X$ and $v$, such that

Seiberg–Witten duality conjecture. The group $\Gamma(2)$ acts by autoequivalences on the derived category of $\mathcal M_V$. The claim is the categorical implication of the Seiberg…

Let $G$ be a gauge group with Lie algebra ${\mathfrak g}$, and let ${{}^LG}$ be its Langlands dual group. Let $\tau$ and ${{}^L\tau}$ be the gauge couplings, and let…

Let a finite ${\mathcal N}=2$ super-Yang–Mills theory have electric and magnetic charges, Wilson and 't Hooft operators, and a gauge coupling. S-duality conjecture. Some or perhaps…

Let $G$ be a gauge group and let the hypermultiplet transform in the adjoint representation of $G$. Write ${{}^LG}$ for the Langlands dual group. Langlands-dual S-duality conjectur…

Let $G$ be the gauge group of a finite ${\mathcal N}=4$ super-Yang–Mills theory, with complexified gauge coupling $\tau$ in the upper half-plane. S-duality conjecture. The theory i…

Let $CY_3$ be a Calabi–Yau threefold, and consider the topological A and B models targeting this same $CY_3$. S-duality conjecture. There is an S-duality between the A and B models…

Green–Gutperle–Vanhove conjecture. The $D^4\mathcal R^4$ terms in the type IIB effective action are of the form

Let $\tau=\chi+i e^{-\phi}$ be the axion/dilaton scalar of Type IIB string theory, and let $E_{3/2}(\tau,\bar \tau)$ denote the non-holomorphic Eisenstein series. In the string fra…

Non-renormalization conjecture. The $Q$-cohomology is exact at tree level, so that the classical $Q$-cohomology gives the exact quantum $Q$-cohomology.

Choi's conjecture. The Coulomb branch cohomology exhausts the monotone cohomology. Consequently, the monotone cohomologies of the $\mathrm{SO}(2N+1)$ and $\mathrm{USp}(2N)$ theorie…

The four-dimensional and five-dimensional theories are obtained as different dimensional reductions of a certain six-dimensional theory. Six-dimensional origin conjecture. S-dualit…

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

A hyperspherical Hamiltonian action is a distinguished class of Hamiltonian boundary condition arising in the relative Langlands program. Relative Langlands conjecture. Hyperspheri…

Let $G$ be a complex semisimple group, let $e$ be a nilpotent element, and let $S_e$ be a slice for $e$. Generalized S-duality conjecture. The same result also holds for more gener…

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 $\mathbf G$ be a group and let $\mathbf G^\vee$ denote its Langlands dual. Regard $(\mathbf G^\vee\curvearrowright\mathbf G^\vee\times\mathcal S^\vee)$ and…

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 ${\mathsf r}\geq 2$, 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…

Let $V$ be a vertex algebra associated with a four-dimensional quantum field theory, and let quantum geometric Langlands kernel vertex operator algebras act by a convolution operat…

Let $X$ be the smooth projective threefold under consideration, let $H$ be an ample divisor, and let $b_2(X)$ denote its second Betti number. For fixed sufficiently large $n$, let…

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