Week 06.11.2023 – 12.11.2023

In these series of lectures we will explore initial value problem in general relativity and how it can be solved in a computer in practical situations. We will first cover the necessary mathematical foundations, including the concepts of well-posedness and strong hyperbolicity, and then explore the current formulations of Einstein’s theory of gravity that are implemented in modern numerical codes, namely generalised harmonic coordinates and the BSSN formulation. We shall see how the latter can be implemented in a toy code so as to get some hands on experience. Time permitting, we will also explore the initial boundary value problem in asymptotically anti-de Sitter spaces and how it can be solved in practice using the characteristic formulation of the Einstein equations in applications of holography.

Tambara-Yamagami (TY) 1-categories provide the mathematical framework to describe the algebra of extended operators of (1+1)-d theories that admit a duality defect. In this talk I will define what is the generalization of TY 1-categories for fusion 2-categories, and how to construct them from fusion 2-categories that are group-theoretical. I will also explain that group-theoretical fusion 2-categories are completely characterized by the property that the braided fusion 1-category of endomorphisms of the monoidal unit is Tannakian. Using this characterization, I will show when a fusion 2-category admits a fiber 2-functor.

**Send email to jeremy.mann@kcl.ac.uk for link to online seminar.** In general relativity, spacetime metrics satisfy the Einstein equations, which are wave equations in the harmonic gauge. The Cauchy problem for the vacuum Einstein equations has been well-understood since the work of Choquet-Bruhat. For an initial data set satisfying the vacuum constraint equations, there exists a solution to the vacuum Einstein equations and it is geometrically unique in the domain of dependence of the initial surface. On contrast, the initial boundary value problem (IBVP) has been much less understood. To solve for an vacuum metric in a region with time-like boundary, one needs to impose boundary conditions to guarantee geometric uniqueness of the solution. However, due to gauge issues occurring on the boundary, there has not been a satisfying choice of boundary conditions. In this talk I will discuss obstacles in establishing a well-defined IBVP for vacuum Einstein equations and the geometric uniqueness problem. Then I will talk about an existence and geometric uniqueness result in a joint work with Michael Anderson.

We study ensembles of half-BPS bound states of fundamental strings and NS-fivebranes (NS5-F1 states) in the decoupling limit. We revisit a solution corresponding to an ensemble average of these bound states, and find that the appropriate duality frame for describing the near-source structure is the T-dual NS5-P frame in which the fivebranes are generically well-separated; this property results in the applicability of perturbative string theory. The geometry sourced by the typical microstate is not close to that of the black hole that carries the same charges. When members of the ensemble spin with two fixed angular potentials about two orthogonal planes, we find that the ensemble average geometry has an ellipsoidal structure. This contrasts with ring structures obtained when fixing the angular momenta instead of the angular potentials; we trace this difference of ensembles to large fluctuations of the angular momentum in the ensemble of fixed angular potential.

We consider black hole linear perturbation theory in a four-dimensional Schwarzschild (anti-)de Sitter background. We describe two methods that provide the quantization condition for the quasinormal mode frequencies of the perturbation field. The first consists of using the Nekrasov-Shatashvili functions, or, equivalently, the classical Virasoro conformal blocks, to obtain the connection coefficients for the differential equation encoding the spectral problem. The second method is based on a perturbative expansion of the local solutions of the differential equation, that involves multiple polylogarithmic functions. We conclude by showing how the two methods shed light on the mathematical structure of the quasinormal mode frequencies, and discussing how they can be generalized to problems in different backgrounds, emphasising their effectiveness.