18 problems

Let $\mathfrak{g}\in\{\mathfrak{a}_0,\mathfrak{a}_1,\mathfrak{a}_2\}$, let $M_{\mathfrak{g}}(q)$ denote the quantum monodromy, and let $h^\vee$ be the dual Coxeter number. For inte…

Generalized Schur limit matching conjecture. The displayed relations are necessary for the two partition functions to be related in this way. The claim is attributed in the source…

Let $\hat{\mathcal{Z}}(q,\alpha)$ denote the generalized Schur limit, depending on a parameter $\alpha$. A modular differential equation conjecture. Based on several examples, the…

Gaussian-subzone dominance conjecture. Only the Gaussian subzone of the intermediate zones enters the competition with the near zones and the outer patch.

Near- and Gaussian-zone asymptotics conjecture. In this situation, the near zone or the Gaussian zone correctly captures the power-law and $O(\beta^0)$ asymptotics.

Puncture-zone dominance conjecture. A puncture $p_j$ contributes negligibly to Cardy-like asymptotics of the index, except through its near zone and the Gaussian zone.

Consider a four-dimensional $\mathcal{N}=1$ superconformal quiver theory described by a brane tiling, and let its superconformal index be the generalization of the Witten index. Le…

For a four-dimensional superconformal theory, let ${\cal I}_{HL}(t)$ denote the Hall–Littlewood limit of the superconformal index, and let…

Let $J$ be a simple Lie algebra with dual Coxeter number $h$, and let $J^h[k]$ be a general irregular puncture. Let $\psi_{\vec\lambda}^{I_J}(q)$ denote the pure Yang–Mills irregul…

Let $J$ be a simple Lie algebra with dual Coxeter number $h$, and let $J^h[k]$ be an irreducible irregular puncture with $b=h$ and no flavor symmetry. Let…

Let $I_{N,k}\equiv A_{N-1}^N[k]$ be the irregular puncture, and let $\psi_{\vec\lambda}^{I_{SU(N)}}(q)$ denote the wave function of the corresponding $SU(N)$ irregular puncture. So…

The 4d ${\cal N}=2$ superconformal field theory/vertex operator algebra correspondence associates a meromorphic chiral algebra to a 4d theory, whose operators arise in the cohomolo…

Let $p$, $q$, and $t$ be superconformal-index fugacities, let $a_1$ and $a_2$ be fugacities for the $SU(3)$ flavor symmetry, and let $z$ be the gauge-integration variable. Write…

Index-factorization conjecture. The index satisfies

Let a large-$N$ SU($N$) toric quiver SCFT be dual to a Sasaki–Einstein five-manifold $SE$, let $r_i$ be the associated zigzag-path charges, and let $\operatorname{vol}(SE)$ and…

Let $\mathfrak{h}_{qu}$ be the locus of minima of the Rains function of a four-dimensional Lagrangian SCFT, and let the unlifted Coulomb branch on $R^3\times S^1$ be the Coulomb br…

Consider the $\mathbb{Z}_2$ orbifold of $\mathcal{N}=4$ super-Yang–Mills theory with gauge group $\mathrm{SU}(N)\times\mathrm{SU}(N)$, and let $\mathfrak{h}_{qu}$ denote the quantu…

Let $Z_{S^3}$ denote the three-sphere partition function of the dimensionally reduced daughter theory, and let $\mathcal{I}(\beta)$ be the four-dimensional superconformal index at…