Forty-Fifth Meeting of the North British Mathematical Physics Seminar

We recover the classification of the maximally supersymmetric bosonic backgrounds of eleven-dimensional supergravity by purely Lie algebraic means, our point of departure being the supertranslation ideal. Along the way we compute some cohomological groups and recover exactly the zeroth order terms of the supercovariant connection associated with the four-form flux. This is a joint work with José Figueroa-O’Farrill.

It is possible for some classical 1+1-dimensional integrable field theories to accommodate discontinuities in the fields and yet remain integrable, with the fields on either side of the defect related by some set of defect conditions. In this talk momentum conserving defects in the ATFTs based on the A series of Lie algebras and in the Tzitzeica model are reviewed, and the fact that the defect conditions give a Backlund transformation for the bulk theory is noted. A more general form of the defect is then considered, which is momentum conserving for ATFTs based on the B, C and D series of Lie algebras.

A diagrammatic calculus for finite real spectral triples is introduced. This calculus relates the axioms of non-commutative geometry with geometric properties of the diagrams. The diagram calculus determines a two-dimensional topological quantum field theory with defects. Conversely, defects in a TQFT determine a generalised notion of non-commutative geometry.

Higher gauge theory is an interesting generalization of ordinary gauge theory. This requires the introduction of higher form potentials and categorified mathematical structures. In this talk, we discuss the proof of the non- abelian Poincare lemma in higher gauge theory. That is, we show that every flat local connective structure is gauge trivial. Moreover, we show how higher flatness appears as a necessary integrability condition of a linear system involving connective structures on principal 2-bundles.

After introducing the quantum inverse scattering method I will review the construction of Baxter Q-operators for closed rational spin chains. Furthermore, I will combine the ideas of the Q-operator construction with Sklyanin's formulation of the quantum inverse scattering method for systems with boundaries to construct Q-operators for the open Heisenberg spin chains with diagonal boundary conditions.

Studies in holographic contexts, such as holographic QCD and holographic superconductors, have motivated the investigation of solitons in AdS and AdS-like spacetimes. In this talk we will investigate the baby Skyrme model in a pure AdS background without a pion mass term, and numerically find soliton and multi-soliton solutions. We find that ring-like structures appear to be energetically favourable, and we construct a point-particle approximation to predict the energy-minimising configurations at higher topological charges.