These notes collect several years of calculations, observations, and ideas on classical and supersymmetric field theory. Their perspective is deliberately specific: they are not intended as a standard set of notes on field theory, but rather approach the subject systematically through the Batalin–Vilkovisky formalism.
The guiding idea is simple: sophisticated mathematical structures arise naturally once one tries to describe, in an adequate and intrinsic way, the physics of field theories, particularly in the presence of gauge symmetries.
A recurring theme throughout the notes can be summarized schematically as follows:
equations lead naturally to derived geometry;
gauge symmetries lead to stacks;
fermionic degrees of freedom lead to families of supermanifolds.
In this sense, derived geometry, stack theory, and supergeometry are not introduced as external mathematical refinements, but emerge from the basic geometric requirements of the field theories themselves.
Although the notation has recently been made substantially more uniform — thanks, A.I. — these notes remain very much a work in progress. I would be grateful to anyone who points out errors, omissions, or places where the exposition could be improved.
My interest in these topics goes back to my postdoctoral years in Heidelberg. My understanding of them has been deeply shaped by many discussions and interactions with my collaborators Ingmar Saberi, Fabian Hahner, Raphael Senghaas, and Johannes Walcher, whom I would like to thank warmly. Any errors or inaccuracies, however, are entirely my own.