Academic Year 2008/2009

Non-Abelian Berry phase is a generic feature in quantum mechanics, where states undergo a holonomy as one changes parameters. I will give an introduction to this phenomenon and explain why it is particularly natural in supersymmetric quantum mechanics. I will also show how supersymmetry allows us to find exact expressions for the Berry phase in strongly coupled, or many-body, systems.

All we know about the structure of physical spacetime, we ultimately infer from the behaviour of matter inhabiting it. Restricting attention to classical matter, Einstein showed in this fashion how the properties of electromagnetic fields reveal an underlying metric structure of spacetime. General relativity then builds on taking the Lorentzian metric seriously as a dynamical object in its own right, providing a phenomenally successful theory of gravity. But all matter is quantum, and in this talk I will show how this leads to the emergence of a refined `area metric' geometry. Gravitational dynamics for four-dimensional area metric manifolds then provide a theory of spacetime and matter, which surprisingly addresses the observed late-time accelerated expansion of the universe without a need for dark energy. The emergence of an area metric spacetime structure has further remarkable consequences for singularity theory and our understanding of physics in strong gravitational fields.

Superconformal Chern-Simons + matter theories in three dimensions can be formulated in terms of 3-algebras, and this has intensified their study recently. I will introduce the notions of a metric 3-Lie and more generally 3-Leibniz algebra, the associated Leibniz algebra. I will then present the deformation problem for general 3-Leibniz algebras and show that they are governed by the cohomology of the associated Leibniz algebra with coefficients in a certain representation. I will then specialize to the 3-algebras of physical interest and end with an example. Although the topic is inherently technical, I hope to give an overview emphasising the ideas and not the details.