558 problems

Verlinde's conjecture. The modular $S$-matrix governs the fusion rules through

Let $b\in\mathbb{R}$, and define $a=a(b)>0$ by … Set $\kappa=2/a^2$ and let $\kappa'=\kappa(-b)$, so that $\kappa\kappa'=16$. Duplantier's SLE duality conjecture. When $\kappa>4$,…

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…

Let $\mathcal{J}^{(m,n)}_{\alpha}$ be the Coulomb gas integral associated with a $(n,m)$ chordal link pattern $\alpha$, where $m\leq (n+2)/2$. These functions satisfy the null vect…

Let $(\Sigma,[g])$ be a Riemann surface with marked points, let $\mathcal{M}_{\mathfrak{g},m}$ be its moduli space, and let $\mathcal{T}_{\mathfrak{g},m}$ be Teichmüller space, the…

Let $(\cF,\cD,U,\Omega)$ be a potentially non-unitary Wightman conformal field theory on $S^1$ with invariant nondegenerate Hermitian form. Let $\cQ(I)=\cP(I)”$ be the associated a…

Finite-set bijection conjecture. There is a bijection

Let $\kappa\in(0,8)$, and consider a multiple-SLE$_\kappa$ process growing $2N$ curves in the upper half-plane from points $x_1<x_2<\ldots<x_{2N}$, with SLE$_\kappa$ partition func…

A Calabi–Yau manifold is a compactification space for the extra six dimensions of the heterotic string, while an $\mathcal{N}{=}2$ superconformal field theory (SCFT) describes the…

Zamolodchikov's exponentiation conjecture. In this semiclassical limit, the conformal blocks exponentiate:

Let the diagonal unitary minimal model $(D_6,A_{10})$ be deformed by the relevant operator $\phi_{\frac9{11}}$, with no other relevant operators turned on, and suppose the flow is…

In a $(2+1)$D topological quantum field theory, the entanglement spectrum is the spectrum obtained from the reduced density matrix of a spatial subsystem, while a two-dimensional r…

Let $\mathcal{K}^{(m,n)}_{\alpha}$ denote the screening solution constructed by Coulomb gas integrals from a chordal link pattern $\alpha$, in the presence of a marked boundary poi…

Let $\mathcal{J}(\boldsymbol{x},u)$ be a screening solution obtained by integrating the Coulomb gas integrand over an admissible choice of screening contours. These functions satis…

Figueroa-O'Farrill's conjecture. The BRST cohomology of any string theory and, more generally, of any two-dimensional topological conformal field theory is isomorphic to the chiral…

A conformal field theory is deformed by relevant bulk operators, and its ground state is considered in relation to conformal boundary states. Cardy's conjecture. The ground state o…

Zamolodchikov's conjecture. In the semi-classical limit, conformal blocks on a four-point sphere have an exponential structure.

Malliavin–Kontsevich–Suhov (MKS) loop measures are natural measures on curves and loops on Riemann surfaces, characterized by conformal restriction covariance and formulated using…

Energy-boundedness conjecture. Every unitary simple vertex operator algebra is energy-bounded.

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

Let the ${\cal Z}_2$-invariant inhomogeneous six-vertex model have anisotropy parameter $q$ satisfying … Let the Euclidean black-hole nonlinear sigma model be viewed as a conformal…

Liouville spectrum conjecture. The spectrum of Liouville theory is

In a two-dimensional conformal field theory with Virasoro symmetry, consider its conformal blocks, and in four-dimensional gauge theory consider the instanton partition functions c…

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…

All-orders effective-action conjecture. The lambda-deformed action $S_{k,\lambda}(g)$ is the effective action for the non-Abelian Thirring model, valid to all orders in $\lambda$ a…