Twenty-Second Meeting of the North British Mathematical Physics Seminar

In this talk I will review the recent developments in generating stationary axisymmetric solutions of five-dimensional Einstein's equations. I will explain in some detail how to obtain the black saturn and the doubly spinning ring using the inverse scattering method for gravity. As a further development, I will show how these techniques can be combined with the SL(3,R) transformations to generate a black ring in Taub-NUT.

Recently there has been interest in so-called non-Hermitian quantum systems and the question under what circumstances such systems allow for the explicit construction of an inner product which renders the quantum Hamiltonian Hermitian. We apply this discussion to the U_q(sl_2)-invariant anisotropic Heisenberg chain and show using techniques from representation theory that the Hamiltonian can be made Hermitian w.r.t. to a particular scalar product which we explicitly construct for special values of q on the unit circle. These values are of special interest as it has been argued in the literature that the QG invariant Heisenberg chain and its algebraic features are closely related to similar stuctures in logarithmic CFT (LCFT). We make contact with this discussion for q^2=-1 which ought to correspond to the simplest LCFT with c=-2.

Multi-matrix models are toy-models for gluon dynamics in quantum non-abelian gauge theories, such as Yang-Mills theory. They also appear in models for quantum gravity and various combinatorial problems. The 't Hooft limit as the matrices become large is a `classical' limit and expected to be a good starting point for an approximation method. However, determining the limiting correlations is itself difficult and requires further approximations. We present a variational principle for Euclidean large-N matrix models. It can be regarded as a classical action for a discrete model of closed string field theory. It involves the confluence of ideas on the entropy of operator-valued random variables and the cohomology of automorphism groups of tensor algebras.