21 problems

Let $\mathfrak{g}$ be a simple finite-dimensional Lie algebra over $\mathbb{C}$, let $g$ be its dual Coxeter number, and let $A=\wedge(\mathfrak{g}\oplus\mathfrak{g})/I$, where $I$…

Superfield realization conjecture. For every $N$ and every topology of engineerable Adinkras, there exists a corresponding set of superderivative constraints such that the set of s…

A GLSM is a gauged linear sigma model, and worldsheet $(0,1)$-supersymmetry is supersymmetry with this chirality structure. De Sitter realization conjecture. GLSMs with worldsheet…

Let $\mathcal{R}(n;r,\ell)$ be the multigraded $S_n$-module with commuting and anticommuting variable sets, let $\operatorname{ch}\mathcal{R}(n;r,\ell)$ be its…

Let $L=2n+1$ with $n\geqslant 0$, let $\eta=pi/3$, and let $V$ be the two-dimensional spin space. For inhomogeneity parameters $u1,\dots,u{2n+1}$, consider the eigenvalue problem ……

Macdonald superpolynomial positivity conjecture. For all superpartitions $\Lambda$ and $\Om$, the coefficients satisfy

Clustering-basis conjecture. The Jack superpolynomials $P_{\Lambda}^{(\alpha_{k,2})}(x_1,\ldots,x_N,\ta_1,\ldots,\ta_N)$, with $\Lambda$ $(k,2,N)$-admissible, form a basis of…

Integral-form conjecture. The coefficients $v_{{\Lambda}\Om}(\alpha)$ are polynomials in $\alpha$ with integer coefficients, so that $v_{{\Lambda}\Om}(\alpha)\in\mathbb Z[\alpha]$.…

The evaluation conjecture. The evaluation of the Jack superpolynomial is

The norm conjecture. The superpolynomial ${P}^{(\alpha)}_\Lambda$ has norm, with respect to the combinatorial scalar product, given by

Consider a single $M5$ brane on $\mathbb{R}^7\times(\mathbb{R}^4/\mathbb{Z}_N)$, located at the singular point of the $A_N$ singularity and spanning six of the seven directions in…

Consider type IIA string theory on $\mathbb{R}^{10}$ with one $NS5$ brane on $\mathbb{R}^6\subset\mathbb{R}^{10}$, where $x_6,\ldots,x_9=0$, together with $k$ semi-infinite $D6$ br…

BCOV theory is a closed-string field theory for the topological $B$-model, formulated here on complex affine space. Type IIB supergravity is the ten-dimensional supergravity theory…

Let $\mathcal{L}_1$ be a maximal finite-dimensional superalgebra of Lie point symmetries of a nonlinear system of partial differential equations, and let $\mathcal{L}_2$ be a maxim…

For odd $N=2n+1$, consider the cyclic spin-chain Hamiltonian with parameters $\lambda,\zeta$ and its three translation-invariant zero-energy singlets $\Psi_N^a$, indexed by…

In the trigonometric case $y=0$, choose the singlet components to be coprime polynomials in $x$ and normalise their leading terms as specified in the source. Let $N$ be written as…

For twist angle $\phi=\pi$, write the supersymmetry singlet as $|\Psi\rangle=\sum_\mu\psi_\mu|\mu\rangle$, with the components chosen as coprime polynomials in $x,y$. The norm-fact…

Let $H_N(x,y)$ be the Hamiltonian on $N$ sites, with twist angle $\phi$, and let $S_N^3$ denote the total third spin component. The representative singlet conjecture. For every $N$…

Let $H_N(x,y)$ be the spin-chain Hamiltonian with twist angle $\phi$, depending on couplings $x$ and $y$. The spectral exchange conjecture. For twist angle $\phi=\pi$, the spectrum…

Let $h'$ be the modified local Hamiltonian and let $h_{\mathrm{XXZ}_{\ell/2}}$ be the spin-$\ell/2$ higher-spin XXZ Hamiltonian density defined from the quadratic Casimir construct…

Homogeneity conjecture. All supergravity backgrounds with