Alberto Cattaneo (U Zürich):
A cohomological view of quantum field theory
Time and place
Thursday, 30.11.23, 15:00-16:00, Hörsaal II, Albertstr. 23b
Abstract
In physics, fields (e.g., the electromagnetic field) are quantities that depend on time. In mathematics, they correspond to functions, vector fields, sections of sheaves, depending on the context. Classical and quantum field theory describe the evolution of fields and study how to compute their properties. One useful approach takes as its starting point the action, a functional on the space of fields.\nIn the classical theory, the PDEs the fields have to satisfy (e.g., the Maxwell equations) are the critical points of this functional; in\nthe quantum theory, one also has to study fluctuations around them, and one heuristic approach is the functional integral, where one formally integrate over the space of fields.\n \nAn interesting situation occurs when there are symmetries: vector fields on the space of fields under which the action functional is invariant. In classical physics, one is then interested in the space of critical points modulo symmetries. In the functional integral approach to quantum theory, one is morally interested in integrating over the quotient of the space of fields by the symmetries, but this is too complicated (and too singular). Instead one considers a section, called a gauge fixing, i.e., a submanifold that intersects the symmetries transversally, and integrates over it. Invariance under the choice of this section is a fundamental question.\n \nThe BV formalism is a cohomological procedure that solves the two goals: it gives a resolution of the critical locus modulo symmetries and allows showing the formal independence of the functional integral from deformations of the gauge fixing. In addition to these properties, in this talk I will recall other important aspects. One is the study of field theories on manifolds with boundaries (or, more generally, higher-codimensional stratifications) where the BV formalism is nicely coupled with the BFV formalism—responsible for the cohomological resolution of the reduced phase space (roughly speaking, the space of initial conditions).\nAnother aspect is the BV pushforward (i.e., a partial integration) which plays a role in defining effective theories, in casting renormalization à la Wilson for gauge theories, and in constructing nontrivial observables.