We use the conformal bootstrap equations to study the non-perturbative gravitational scattering between infalling and outgoing particles in the vicinity of a black hole horizon in AdS. We focus on irrational 2D CFTs with large c and only Virasoro symmetry. The scattering process is described by the matrix element of two light operators (particles) between two heavy states (BTZ black holes). We find that the operator algebra in this regime is (i) universal and identical to that of Liouville CFT, and (ii) takes the form of an exchange algebra, specified by an R-matrix that exactly matches with the scattering amplitude of 2+1 gravity. The R-matrix is given by a quantum 6j-symbol and the scattering phase by the volume of a hyperbolic tetrahedron. We comment on the relevance of our results to scrambling and the holographic reconstruction of the bulk physics near black hole horizons.

In this article, we show that eigenenergies and eigenstates of a system consisting of four onedimensional hard-core particles with masses 6m, 2m, m, and 3m in a hard-wall box can be found exactly using Bethe Ansatz. The Ansatz is based on the exceptional affine reflection group F 4 associated with the symmetries and tiling properties of an octacube-a Platonic solid unique to four-dimensions, with no three-dimensional analogues. We also uncover the Liouville integrability structure of our problem: the four integrals of motion in involution are identified as invariant polynomials of the finite reflection group F 4 , taken as functions of the components of momenta.

We show that the dynamics of particles in a one-dimensional harmonic trap with hard-core interactions can be solvable for certain arrangements of unequal masses. For any number of particles, there exist two families of unequal mass particles that have integrable dynamics, and there are additional exceptional cases for three, four and five particles. The integrable mass families are classified by Coxeter reflection groups and the corresponding solutions are Bethe ansatz-like superpositions of hyperspherical harmonics in the relative hyperangular coordinates that are then restricted to sectors of fixed particle order. We also provide evidence for superintegrability of these Coxeter mass families and conjecture maximal superintegrability.

We study the central hyperplane arrangement whose hyperplanes are the vanishing loci of the weights of the first and the second fundamental representations of gl n restricted to the dual fundamental Weyl chamber. We obtain generating functions that count flats and faces of a given dimension. This counting is interpreted in physics as the enumeration of the phases of the Coulomb and mixed Coulomb-Higgs branches of a five dimensional gauge theory with 8 supercharges in presence of hypermultiplets transforming in the fundamental and antisymmetric representation of a U (n) gauge group as described by the Intriligator-Morrison-Seiberg superpotential.2010 Mathematics Subject Classification. 05E10, 52C35, 05A15, 17B10, 17B81.

The Ratios Conjecture of Conrey, Farmer and Zirnbauer predicts the answers to numerous questions in number theory, ranging from n-level densities and correlations to mollifiers to moments and vanishing at the central point. The conjecture gives a recipe to generate these answers, which are believed to be correct up to square-root cancelation. These predictions have been verified, for suitably restricted test functions, for the 1-level density of orthogonal and symplectic families of L-functions. In this paper we verify the conjecture's predictions for the unitary family of all Dirichlet $L$-functions with prime conductor; we show square-root agreement between prediction and number theory if the support of the Fourier transform of the test function is in (-1,1), and for support up to (-2,2) we show agreement up to a power savings in the family's cardinality.Comment: Version 2: 24 pages, provided additional details, fixed some small mistakes and expanded the exposition in place