120 problems

Let $\mathfrak{M}$ be the Higgs branch of a three-dimensional ${\cal N}=4$ supersymmetric Yang–Mills theory, and let $\mathfrak{M}^{!}$ denote its Koszul dual when this dual exists…

Let the instanton partition functions of $\mathcal{S}$-class $\mathcal{N}=2$ gauge theories be defined in the $Omega$-background. AGT conjecture. These instanton partition function…

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…

Nekrasov–Rosly–Shatashvili conjecture. The superpotential $\mathcal{W}^{\mathrm{eff}}$ may be obtained as a generating function of opers in terms of these Darboux coordinates.

Framed no-exotics conjecture. Framed BPS states form the trivial representation under $SU(2)_R$; consequently, ${\overline{\underline\Omega}}(\mathbb{L},u,\gamma,q)$ is a character…

Gaiotto's conjecture. The Nekrasov partition function equals the norm of the corresponding Whittaker vector in the Virasoro Verma module.

The perturbative partition function of supersymmetric Yang–Mills theory on suitable toric Sasaki–Einstein manifolds admits a factorisation into contributions associated with the to…

Let $\mathfrak{g}$ be a simple finite-dimensional Lie algebra over $\mathbb{C}$, let $g$ be its dual Coxeter number, and let $A$ be the quotient of…

Geometric engineering conjecture. Partition functions in $4$-dimensional $\mathcal N=2$ supersymmetric gauge theories are equal to limits of generating functions of local Gromov–Wi…

Dijkgraaf–Vafa conjecture. The superpotential $W(S)$ is the sum of the zero-momentum planar diagrams of the $\mathcal{N}=1$ theory under consideration.

Consider four-dimensional minimal $N=1$ super Yang–Mills theory with gauge group $G$, whose classical $U(1)$ $R$ symmetry is reduced by instantons to ${\bf Z}_{2h}$, where $h$ is t…

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 $p,q$ be positive coprime integers, let $b^2=p/q$, and let $\boldsymbol{Y}$ range over tuples of partitions with corresponding terms $Z_{\vec{Y}}$. Let $\epsilon_1$ and…

Consider the four-dimensional superconformal Yang–Mills theory discussed in the source, including four-dimensional instantons and insertions of constituents of chiral-algebra compo…

Non-graviton absence conjecture. There are no non-graviton operators in the Macdonald (or Schur) sector of the $\mathcal{N}=4$ $SU(2)$ theory.

One-loop exactness conjecture. The BPS spectrum is one-loop exact: states that remain BPS at one-loop do not lift at higher perturbative or non-perturbative orders.

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…

Argyres–Douglas $H_0$ conjecture. The building block $\mathcal Z$ coincides with the partition function of the four-dimensional $\mathcal N=2$ supersymmetric Argyres–Douglas theory…

For algebra $\mathfrak{su}(N)$ on $\mathbb{R}^3\times S^1$, let $I^{\mathcal{N}=1^{\ast}}(N)$ denote the total number of vacua over all global variants. Let…

The discussion concerns the twisted potential for non-simply laced algebras, with Langlands duality sending $\tau$ to $-\frac{1}{s\tau}$; the self-dual cases $B_2$, $G_2$, and…

Let $N>1$ and consider the four-dimensional $Sp(N)$ gauge theory with four hypermultiplets in the fundamental representation and one hypermultiplet in the anti-symmetric representa…