AKSZ-BV Formalism and Courant Algebroid-induced Topological Field Theories

Abstract. We give a detailed exposition of the Alexandrov-Kontsevich-Schwarz-Zaboronsky superfield formalism using the language of graded manifolds. As a main illustarting example, to every Courant algebroid structure we associate canonically a three-dimensional topological sigma-model. Using the AKSZ formalism, we construct the Batalin-Vilkovisky master action for the model. 1. Intro and Brief History. The standard procedure for quantizing classical field theories in the Lagrangian approach is by using the Feynman path integral. From the mathematical standpoint this is somewhat problematic, as it involves “integration ” over the infinitedimensional space of field configurations, on which no sensible measure has been found to exist. Nevertheless, the procedure can be made rigorous in the perturbative approach, provided the classical theory does not have too many symmetries (“too many ” means, roughly speaking, an infinite-dimensional space). In the presence of these gauge symmetries, however, the procedure needs to be modified, as one has to integrate over the space of gauge-equivalence classes of field configurations.