Forty-Eighth Meeting of the North British Mathematical Physics Seminar

Abstract 1

The elliptic genus of K3 is a topological invariant on the moduli space of theories describing superstrings propagating on K3, and is known to exhibit a phenomenon coined 'Mathieu Moonshine'. It may be calculated using data from the underlying 2d N=4 superconformal field theory, which include the Witten index of irreducible, unitary representations of the corresponding N=4 superconformal algebra. In theories with a specific extension of this N=4 symmetry, called 'large N=4 symmetry', the Witten index of all unitary, irreducible representations is trivial, but a new index that generalises the Witten index was introduced a while ago by Gukov and collaborators. In this talk, I focus on representations of the large N=4 superconformal algebra with non trivial Gukov index and determine the nature of the states which contribute to this index.

The quantum Knizhnik-Zamolodchikov equations are difference equations of great significance in mathematical physics, representation theory and elsewhere. In their original form they have been studied since the late 1980s by Smirnov, Frenkel and Reshetikhin, Jimbo and Miwa, and many others. These equations can be formulated in terms of solutions of the Yang-Baxter equation. Cherednik developed affine root system generalizations of these equations in which the above equations correspond to type A. The type B, C and D cases in this framework correspond to the boundary quantum KZ equations, where one also needs solutions of the reflection equation. We will review some known results and open questions, including recent joint work with Reshetikhin and Stokman.

It is usually hard to compute entanglement entropy and mutual information for conformal field theories (CFT). Ryu-Takayanagi proposals allows us to find the same quantities using calculations in gravity. In this talk I will show how to find holographic entanglement entropy and scrambling time for BTZ black hole perturbed by a heavy (backreacting) particle. Holographic bulk description improves on the shock-wave approximation in 3d bulk dimensions. I will also discuss current attempts to generalize this calculation to the rotating BTZ black hole.

I will describe constraints on the three-point functions of operators with spin which follow from conformal symmetry, unitarity, and some extra assumptions, natural for CFTs with gravity duals.

The Temperley-Lieb algebra was introduced in relation to lattice models in statistical mechanics, and can be constructed as the centralizer of the quantum group Uq(sl2). Recent work in logarithmic conformal field theory has brought interest to a restricted version of this quantum group. We generalize the Temperley-Lieb construction to the restricted case, giving generators and relations, and conjecture a formula for projections onto indecomposable modules.