30 problems

Let $B>0$ and let $\ell$ be a positive integer. For $N$ particles, let … be the Laughlin lowest-Landau-level ground-state space, where $F$ is a correlation factor and…

The Chalker–Coddington model is a random unitary operator on $l^{2}({\mathbb Z}^{2})$ with real tunneling amplitudes $r$ and $t$ satisfying $r^{2}+t^{2}=1$. The parameter $|t|$ is…

Let $H_\circ$ be a Hamiltonian whose ground state $G_{\circ,{\bf l}}$ has log-extensive degeneracy, and let $G_{\epsilon,{\bf l}}$ be the strictly less degenerate ground state obta…

Wigner crystal conjecture. At higher fields and lower densities of electrons, the same system is conjectured to be in a Wigner crystal state.

Let the basis polynomials be indexed by link patterns, with spectral parameters $z_i$, and let the loop fugacity be the parameter of the loop model. Kasatani–Pasquier's positivity…

Let $\Gamma$ be a cocompact Fuchsian group and let $\sigma$ be a multiplier on $\Gamma$. Let $\theta$ denote the flux associated with $\sigma$, and let $H_{\sigma,V}$ be the corres…

Let ${\mathbf K}$ be a rational $n\times n$ matrix, and let ${\mathbf Q}$ and ${\mathbf u}$ be $n$-vectors. Define the universal chiral partition function … Here the sum is over no…

Projective flatness conjecture. The vector bundles of completely filled quantum Hall states over geometric parameter spaces are projectively flat for an appropriate connection.

Statistics-transmutation conjecture. In this ansatz, the energy expectation satisfies

Fractional-statistics emergence conjecture. As $N\to+\infty$, the identity

Let the local case be $N_f=N_c=\lambda$, and let quantum vortices be considered in the corresponding background flux. Fermi-vortex duality conjecture. Quantum vortices are fermions…

Let $\Delta$ be an irrational droplet ratio, and let $\mathcal K_{\Delta}(\alpha,\beta)$ denote the rescaled edge correlator as a function of angular coordinates $\alpha$ and…

The anomalous Hall effect is a Hall-like response observed in ferromagnetic materials without an external magnetic field. It is conjectured that this effect arises from internal ma…

Conformal-block conjecture. The edge contribution to the entanglement entropy is a particular linear combination of these conformal blocks, very much like correlation functions in…

Let IQHE and FQHE denote the integer and fractional quantum Hall effects, respectively. Suppose that correlations of all local observables decay faster than any exponential. Correl…

The discussion concerns gyrotropic photonic crystals, whose bulk phases are classified in class A by integer-valued Chern numbers, including a first Chern number in two dimensions.…

Holomorphic adiabatic-state conjecture. The property that the operators representing singularities transform as conformal primaries is a general property of any holomorphic adiabat…

Geodesic point-splitting conjecture. The coincident-point correlator is correctly regularized by

Let $c_H$ be the geometric central charge associated with a quantum Hall wave function, let $c$ be the chiral central charge, and let $\nu^{s}$ denote the orbital spin filling frac…

Let $d=3$ and let $chat{P}_+$ and $chat{P}_-$ be the surface Fermi projections associated with a chiral system described by a Fermi unitary $U$, under the assumption that the surfa…

Let ${\mathcal F}$ be the Hilbert space of states, and let $\psi^{N,M}_{Q,\mathbf n}$ denote the states defined in the paper for $N,M\in\mathbb{N}_0$, $Q\in\mathbb{Z}$, and…

Let $\hat h=(h,\tilde h)\in M_{2N}(\mathbb C)\times\widehat{\mathpzc A}_d$ with $d=3$, suppose BGH and CH apply, and assume ${\rm Ch}_3(u_F)\neq 0$. Then the boundary spectrum is d…

Let $\mathcal{F}$ be the expression defined in Eq., and let $\log \det'\Delta_{\rm L}$ denote the logarithm of the regularized spectral determinant of the Laplacian. Consider their…

Thermal Hall conductance conjecture. The thermal Hall conductance is given by

Let $\mathcal{Z}[g]$ be the partition function of the Coulomb plasma on a surface of constant curvature, let $N$ be the number of particles, and let $\chi$ be the Euler characteris…