71 problems

Let $F^{(\ell)}_{L;m}(\varepsilon_1,\varepsilon_2,\ldots,\varepsilon_N)$ be the weighted sum over configurations with fixed interior edge variables, let $Z^{(\ell)}_{L;m}$ be the c…

In ${\cal N}=4$ supersymmetric Yang--Mills theory, consider dressed mean fields, written in terms of dressing functions and kernels. Gauge-independence conjecture. All gauge depend…

Let $R_k(x_1,\ldots,x_k)$ be the classical random-matrix correlation functions, and let … be the corresponding correlation functions including the real parts of the Green functions…

Consider the plurimer correlation in the random tiling model, where the results established in the paper assume triangular plurimers, with all but one monomer having side-length…

Let $\pone$ denote the projective line, let $F$ be a function on $\pone$, and let $\omega$ be a two-form on $\pone$. Assume that $F$ and $\omega$ are analytic in the real-analytic…

High-dimensional realizability conjecture. The known standard non-negativity conditions on $g_2$ should be sufficient to ensure the existence of point processes in all space dimens…

Same-loop connectivity conjectures. For $n=2,3,4$, respectively,

The spin-$\frac{1}{2}$ XXZ chain has correlation functions represented by multiple integrals, including one-dimensional integrals indexed by an integer $j$. The authors' results fo…

For the even-site ground state $|\Psi^{(2n)}\rangle$, define … and set $\alpha_i=(1+\sigma_i^z)/2$. Korepin–Izergin–Baxter ferromagnetic-string formula. The probabilities of format…

For the even-site ground state $|\Psi^{(2n)}\rangle$, define the normalized expectation of an operator $O$ by … and define $\alpha_i=(1+\sigma_i^z)/2$. The twisted XXZ next-nearest…

Let $N=2n+1$ be odd, and let $a_1,\ldots,a_k$ denote the local observables defining the ferromagnetic-string correlations. Ferromagnetic-string probability conjecture. The correlat…

Let $N=2n+1$ be odd, and let $a_j$ denote the local observable used for the spin-chain correlations. Correlation formula conjecture. The next-nearest-neighbor correlation is … The…

Let $N=2n+1$ be odd, let ${\mathcal N}_n$ be the norm of the ground state, and let the state vector have components in the spin-configuration basis. Sum-of-components conjecture. T…

Let $\langle\sigma_m^z\sigma_1^z\rangle$ denote the zero-temperature two-point correlation function of the $XXZ$ spin chain at separation $m$. In the critical regime, the distance…

Plane-wave/cylindrical-harmonic correlation-function conjecture. The correlation functions in the two bases are related by

Let $\Delta_\text{gap}$ denote the adjoint gap and let $\mu$ be the parameter approaching zero. We thus conjecture that the adjoint gap $\Delta_\text{gap}$ has a non-analytic behav…

Smooth multipoint representation conjecture.

Let $a_N$ be typically an affine function of $N$, let ${\mathcal{F}}_N({\boldsymbol{\lambda}}\mid{\boldsymbol{x}})$ be an explicit integrand, and let $\Gamma$ be a simple smooth cu…

In the $\mu=3$ quantum-group-symmetric XXZ$_q$ conformal field theory, let $\Lambda$ be the operator whose two- and three-point functions vanish in the calculations described, and…

Two-point correlation scaling conjecture. For arbitrary $\rho$ and $p$, the two-point correlation function satisfies

Let $S_i^z$ and $S_j^z$ denote the spin-$z$ operators at sites $i$ and $j$ of a Motzkin spin chain, and let $n$ be the chain length. The colorless-to-colorful correlation-scaling c…

Let $J_{(s_i,q_i)}$, $J'_{(s_i,q_i)}$, and $J”_{(s_i,q_i)}$ denote conserved currents with spin and label data $(s_i,q_i)$, and let $\bar{J}'_{(s_2,q_2)}$ denote the barred current…

Consider a root-$T\overline{T}$-deformed $\mathrm{CFT}_2$ and the corresponding perturbative correlation functions. The $T\overline{T}$-deformed planar stress-tensor two-point corr…

Let $\mathcal{O}_i$ be operators inserted at points $x_i$, and let $\langle\prod_i\mathcal{O}_i(x_i)\rangle^{(\lambda)}$ denote their correlation function in a $T\overline{T}$-defo…

Universal ACL conjecture. Triangular lattice models satisfying those three conditions can exhibit a unique ACL identical to that of the Ising model.