52 problems

Mathieu moonshine conjecture. Each $\mathcal{D}_i$ is, in some natural but unexplained way, a representation of the finite simple group $M_{24}$. The conjecture concerns the appear…

Let $\mathcal V^{(N)}$ be the vertex algebra associated with the $N$th power of the monodromy operator of a four-dimensional ${\mathcal N}=2$ superconformal field theory. Let $r$ b…

Let the $A_{N-1}$ $(2,0)$ superconformal field theory be the six-dimensional theory labelled by the simply laced Lie algebra $A_{N-1}$, and let $N$ denote the number of units of 4-…

Consider an $(A_{k-1}, A_{N-1})$ theory and its associated vertex operator algebra. Zhu's $C_2$-algebra is a truncation of the vertex operator algebra, and a Jacobi algebra is the…

Off-shell free-energy extremization conjecture. The entropy or central charge of a general class of $d=5,6$ SCFTs compactified on $\mathbb M_4$, with twists parameterized by…

Gepner's Fermat-quintic conjecture. The Gepner-model $N=2$ superconformal field theory should be isomorphic to the quantum nonlinear $\sigma$-model with the Fermat quintic threefol…

Let $\mathcal{T}/\simeq$ be the set of toric diagrams up to the equivalence relation used in the paper, and let … be the modulus defined by the maximum value of the polynomial…

A Sasaki–Einstein five-manifold $X_5$ is an Einstein manifold whose metric cone is Ricci-flat and Kähler, hence a Calabi–Yau threefold. The product $AdS_5\times X_5$, equipped with…

Gepner-model orbifold conjecture. The Gepner model $(2)^4$ agrees with the $\mathbb{Z}_4$ orbifold of $\mathcal{T}_0$.

Fubini–Study conjecture. The two-plane $\Omega^\ast$ agrees with the two-plane in equation $$ . Equivalently, $\mathcal{C}_{a,b,c}$ has a geometric interpretation on $X_{a,b,c}…

Natural hyperkähler structure conjecture. The theory $\mathcal{C}_X$ has a geometric interpretation whose full hyperkähler structure is the natural one induced on $X$ by its embedd…

Let $\Gamma$ and $\Gamma'$ be lattices with quadratic forms $q$ and $q'$, let $g:(\Gamma\oplus\Gamma^*,q)\to(\Gamma'\oplus{\Gamma'}^*,q')$ be a lattice isomorphism, and let…

Let an $N=1$ superconformal field theory have a maximal supersymmetry subalgebra, meaning the largest possible subalgebra in the relevant series indexed by $N$. Uniqueness conjectu…

Non-rationality conjecture for Higgsless Lagrangian SCFTs. Such a cancellation never occurs; consequently, all Higgsless Lagrangian SCFTs give rise to strongly finite non-rational…

Consider bifiltered affine schemes associated with four-dimensional $\mathcal{N}=2$ superconformal field theories having non-trivial Higgs branches, and let the jet scheme of such…

For $N>7$, consider the $W(3,3+N)$ vertex operator algebra associated with the relevant $(A_2,A_{N-1})$ theories, and let the ideals in Table encode the corresponding polynomial re…

Let a four-dimensional $\mathcal{N}=2$ superconformal field theory be given, and consider decoupling relations in its operator product expansion. A bifiltered affine scheme is an a…

Let $\mathcal{T}$ be a four-dimensional $\mathcal{N}=2$ superconformal field theory with associated vertex operator algebra $V_{\mathcal{T}}$. Define Zhu's $C_2$-algebra by…

Let $X$ be an affine scheme satisfying currently unknown algebro-geometric conditions. Let $\mathcal{T}_X$ denote a four-dimensional $\mathcal{N}=2$ superconformal field theory, co…

Let $\mathcal{T}$ be a four-dimensional $\mathcal{N}=2$ superconformal field theory, and let its Schur and Macdonald indices be particular limits of its supersymmetric partition fu…

Let ${\cal V}$ be the strongly finitely generated vertex operator algebra associated with a four-dimensional ${\cal N}=2$ superconformal field theory, and let … be its refined char…

Let a $d$-dimensional superconformal field theory, with $d$ even, be placed on a space with topology … Let $I_{d+2}$ denote its anomaly polynomial, and let the integral below be th…

Let a four-dimensional ${\mathcal N}=2$ superconformal field theory have Schur index given by a formal power series in $q$. A modular linear differential equation (MLDE) is a linea…

Nonexistence conjecture. For dimensions $d=4,5,6$, there are no interacting SCFTs with a trivial Coulomb branch.

Let an $\mathcal{N}=2$ superconformal field theory (SCFT) have rank equal to the complex dimension of its Coulomb branch. A rank-zero SCFT is therefore one without a Coulomb branch…