Quantum BV theories on manifolds with boundary. Part 1

I plan to give an introduction to the programme of perturbative quantization of gauge theories (and in particular, topological field theories of AKSZ type) on manifolds with boundary. The machine we build eats an action functional of a gauge theory and gives as an output a quantum field theory satisfying certain set of axioms which include compatibility with gluing of source manifolds (a variant of Atiyah-Segal functorial gluing), compatibility with gauge symmetry (a variant of Batalin-Vilkovisky master equation) and compatibility with a version of Wilson's renormalization flow. One motivation for this programme is to reunite the perturbative treatment of topological field theory (e.g. Axelrod-Singer's perturbative partition function of Chern-Simons theory on a rational homology 3-sphere, given in terms of configuration space integrals), with the "non-perturbative" idea of functorial field theory. This is a report on joint work with Alberto S. Cattaneo and Nicolai Reshetikhin.

In this mini-course, I plan to first introduce Batalin-Vilkovisky formalism in finite-dimensional setting, then discuss classical field theory with boundary, and finally dive into our construction and give explicit examples (from abelian BF theory to, time permitting, Poisson sigma model). Also, time permitting, I will mention "cellular" examples, where all spaces of fields are finite-dimensional and all integrals have measure theoretic meaning (as opposed to stationary phase evaluation of path integrals).