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.
Categorified or higher mathematical structures appear very naturally within string theory. For instance, the Kalb-Ramond two-form B-field is part of the connective structure 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 gauge 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 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. Generally, our research focusses on understanding quantum field theory in terms of homotopy algebras.