AKSZ models of semistrict higher gauge theory

Abstract

In the first part of this paper, we work out a perturbative Lagrangian formulation of semistrict higher gauge theory, that avoids the subtleties of the relationship between Lie 2-groups and algebras by relying exclusively on the structure semistrict Lie 2-algebra $ \mathfrak{v} $ and its automorphism 2-group $ \mathrm{Aut}\left( \mathfrak{v} \right) $ . Gauge transformations are defined and shown to form a strict 2-group depending on $ \mathfrak{v} $ . Fields are $ \mathfrak{v} $ -valued and their global behaviour is controlled by appropriate gauge transformation gluing data organized as a strict 2-groupoid. In the second part, using the BV quantization method in the AKSZ geometrical version, we write down a 3-dimensional semistrict higher BF gauge theory generalizing ordinary BF theory, carry out its gauge fixing and obtain as end result a semistrict higher topological gauge field theory of the Witten type. We also introduce a related 4-dimensional semistrict higher Chern-Simons gauge theory.