Two-Dimensional Gauge Theories in BV Formalism and Gluing-Cutting

The thesis is based on three publications. In the first paper we discuss the A-model as a gauge fixing of the Poisson Sigma Model with target a symplectic structure. We complete the comparison between Witten’s A-model and the PSM by showing how to recover the A-model hierarchy of observables in terms of the AKSZ observables. Moreover, we discuss the off-shell supersymmetry of the A-model as a residual BV symmetry of the gauge fixed PSM action. In the second publication we discuss observables of an equivariant extension of the A-model in the framework of the AKSZ construction. We introduce the A-model observables, a class of observables that are homotopically equivalent to the canonical AKSZ observables but are better behaved in the gauge fixing. We discuss them for two different choices of gauge fixing: the first one is conjectured to compute the correlators of the A-model with target the Marsden-Weinstein reduced space; in the second one we recover the topological Yang-Mills action coupled with A-model so that the A-model observables are closed under supersymmetry. In the third publication we recover the non-perturbative partition function of 2D Yang-Mills theory from the perturbative path-integral. To achieve this goal, we study the perturbative path-integral quantization for 2D Yang-Mills theory on surfaces with boundaries and corners in the Batalin-Vilkovisky formalism (or, more precisely, in its adaptation to the setting with boundaries, compatible with gluing and cutting – the BV-BFV formalism). We prove that cutting a surface (e.g. a closed one) into simple enough pieces – building blocks – choosing a convenient gauge-fixing on the pieces and assembling back the partition function on the surface, one recovers the known non-perturbative answers for 2D Yang-Mills theory.