Extended Bargmann FDA and non-relativistic gravity

In this paper we consider the construction of a free differential algebra as an extension of the extended Bargmann algebra in arbitrary dimensions. This is achieved by introducing a new Maurer-Cartan equation for a three-form gauge multiplet in the adjoint representation of the extended Bargmann algebra. The new Maurer-Cartan equation is provided of non-triviality by means of the introduction of a four-form cocycle, representative of a Chevalley-Eilenberg cohomology class. We derive the corresponding dual $L_{\infty}$ algebra and, by using the formalism of non-linear realizations, propose a five-dimensional gauge invariant action principle. Then, we derive the corresponding equations of motion and study how the presence of the three-form gauge fields and the four-cocycle modify the corresponding non-relativistic dynamics.