Higher structures
Loosely speaking, higher structures is short for higher homotopy structures, and their appearance is simply the consequence of the gauge principle in physics. Thus, understanding such structures properly is a key to understanding quantum physics.
Research leads
Dr Jan Gutowski
Associate Professor
Dr Martin Wolf
Associate Professor
Overview
Categorified or higher mathematical structures appear very naturally within string theory and M-theory. For instance, the Kalb-Ramond two-form B-field is part of the connection on what is called a higher principal bundle, which is the categorification of the notion of a connection on a circle bundle. Moreover, string field theory, the second quantisation of string theory, is fundamentally based on homotopy Maurer—Cartan theory, the generalisation of Chern—Simons theory to L∞-algebras which are ∞-categorifications of the notion of a Lie algebra.
The standard approach to quantising a theory makes use of either the path integral approach or the Becchi—Rouet—Stora—Tyutin formalism. While in normal circumstances this is sufficient, when gauge symmetries only close on-shell, one has to use a more general quantisation procedure known as the Batalin—Vilkovisky quantisation. Importantly, the Batalin—Vilkovisky formalism is extremely useful in analysing mathematical structures of quantum systems, even when standard quantisation methods are applicable.
The homotopy algebra perspective
The Batalin—Vilkovisky formalism associates to a field theory an L∞-algebra. This, in turn, implies that the field theory can be understood as a higher homotopy Maurer—Cartan theory.
Recently, it has been realised that the L∞-algebra encodes not only classical physics such the field content, the equations of motion, and all the symmetries but also everything about perturbative quantum physics via homological perturbation theory. We have demonstrated how powerful recursive methods of constructing scattering amplitudes can be understood in terms of homotopy algebras, and in fact, exist for any theory. Ultimately, this leads to the homotopy algebraic perspective on perturbative quantum field theory, which implies a dictionary between physical concepts and algorithms and mathematical notions and constructions.
Generally, our research focuses on understanding all aspects of perturbative quantum field theory in terms of homotopy algebras.
Higher gauge theory
Principal bundles with connections are mathematical gadgets that are central to formulating gauge theories. The prime example in this context is Yang—Mills theory which, in turn, is the fundamental building block of the Standard Model of Particle Physics.
Higher principal bundles are certain generalisations, also known as categorifications, of principal bundles. In particular, the connections on such bundles are described by higher differential forms, contrary to principal bundles where they are described by differential one-forms. Higher principal bundles with connections are central to formulating of what is known as higher gauge theory.
We develop and analyse higher structures, such as higher differential geometry and homotopy algebras, with an aim of gaining a better understanding of higher gauge theory. We also make use of twistor geometry in our approach to higher gauge theory.