Higher gauge theory, BV formalism and self-dual theories from twistor space

In this Thesis we investigate higher homotopy structures arising in ordinary classical field theory, as well as in string and M-theory. First, we review L∞-algebras and we discuss their homotopy Maurer-Cartan theory. Our perspective is adapted to an application towards higher gauge theory from the outset. We observe that homotopy Maurer-Cartan theory always allows for a supersymmetric extension by auxiliary fields, just as ordinary Chern-Simons theory does. Then, we review in detail the Batalin-Vilkovisky formalism for Lagrangian field theories and its mathematical foundations, with an emphasis on higher algebraic structures and classical field theories. We explain that, with the help of this formalism, any classical field theory admitting an action can be fully described by an L∞-algebra, encoding its symmetry structure, the field contents, the equations of motion, as well as the Noether identities. Moreover, the classical action is given by the homotopy Maurer-Cartan action of its L∞-algebra. We employ the L∞- perspective of the Batalin-Vilkovisky formalism, with an eye to gauge theory and twistor theory. In particular, we show how quasi-isomorphisms between L∞-algebras correspond to classical equivalences of field theories. As examples, we explore Yang-Mills theory and we discuss in great detail higher (categorified) Chern-Simons theory, providing some useful shortcuts in usually rather involved computations. Moreover, we employ the fact that the ideas of higher gauge theory can be combined with those of twistor geometry to formulate self-dual higher gauge theory. We propose a twistor space action for non-Abelian self-dual tensor field theory in six-dimensions in terms of holomorphic higher Chern-Simons theory for a Lie 2-algebra. We explicitly show how both the Abelian and non-Abelian twistor space actions descend to six-dimensional Euclidean space-time and we comment about possible advantages of the L∞-perspective in this setting.