64th North British Mathematical Physics Seminar

Abstract: In the search for a complete description of quantum mechanical and gravitational phenomena, we are inevitably led to consider observables on boundaries at infinity. This is the common mantra that there are no local observables in quantum gravity and gives rise to the tantalising possibility of a purely boundary--or holographic--description of physics in the interior. The AdS/CFT correspondence provides an important working example of these ideas, where the boundary description of quantum gravity in anti-de Sitter (AdS) space is an ordinary quantum mechanical system-- in particular, a Lorentzian Conformal Field Theory (CFT)--where the rules of the game are well understood. It would be desirable to have a similar level of understanding for the universe we actually live in. In this talk I will explain some recent efforts that aim to understand the rules of the game for observables on the future boundary of de Sitter (dS) space. Unlike in AdS, the boundaries of dS space are purely spatial with no standard notion of locality and time. This obscures how the boundary observables capture a consistent picture of unitary time evolution in the interior of dS space. I will explain how, despite this difference, the structural similarities between dS and AdS spaces allow to forge relations between boundary correlators in these two space-times. These can be used to import techniques, results and understanding from AdS to dS.

Abstract: Toroidally compactified string theory comes with a set of differential form fields and scalars, which should be described by a connection in a certain higher groupoid — but which one? We construct this groupoid and show that it comes with an E_n(Z) discrete U-duality action and the correct non-geometric moduli (Q- and R-fluxes). Using this, we can model non-geometric string compactifications (T-folds, R-spaces, U-folds) as higher groupoid bundles with connection. Based on ongoing joint work with C. Saemann and L. Borsten.

Abstract: As surprising as it may seem, there is no agreed upon notion of higher arity associativity, in particular, our understanding of the analogue of associativity for ternary operations is sorrowfully incomplete. In this talk I will present the motivation to study ternary (and higher) algebraic operations in the context of mathematical physics and differential geometry by reviewing some results about the integration problem for 3-Lie algebras. I will report on my recent progress in understanding existing notions of generalized ternary associativity in heaps, which are a kind of ternary generalization of groups, with the aid of the language of hypergraph rewriting systems.

Abstract: I will discuss two approaches to mass in General Relativity. One quasi—local, and applicable to closed surfaces in space times (like that of the Kerr horizon), and one global, based on causal properties of space-times near space-like infinity.

Abstract: We establish a framework for computing graviton scattering amplitudes on curved self-dual radiative space-times; these are chiral, source-free, and asymptotically flat spaces, determined by free characteristic data at null infinity. Such space-times admit an elegant twistor description via Penrose's non-linear graviton, which manifests their underlying integrability. The tree-level S-matrix is written in terms of an integral over the moduli space of holomorphic maps from the Riemann sphere to twistor space, with the degree of the map corresponding to the helicity configuration of the external gravitons. For the MHV sector, we derive the amplitudes directly from general relativity, while other helicity configurations arise from a natural family of generating functionals and pass several consistency checks. The amplitudes exhibit many novel features which are absent in Minkowski space, including tail effects. Although there are residual integrals due to the functional degrees of freedom in the background space-time, our formulae have fewer such integrals than expected from space-time perturbation theory. In highly symmetric special cases, such as self-dual plane waves, the number of residual integrals can be further reduced, resulting in even simpler expressions for the scattering amplitudes. If there is time I will discuss connections with celestial holography. This is joint work with Tim Adamo and Atul Sharma based on 2203.02238, 2110.06066, 2103.16984, 2010.14996.