An Action for Extended String Newton-Cartan Gravity

TL;DR

This work constructs a four-dimensional action for extended string Newton-Cartan (ESNC) gravity by taking a nonrelativistic limit of General Relativity augmented with a BF term that involves a zero-flux two-form and a central extension vector. The resulting ESNC action, based on the ESNC algebra, generalizes string NC gravity and yields a second-order formulation after imposing a tailored set of geometric constraints. The authors show that, via longitudinal dimensional reduction and truncation, the 4D ESNC action reduces to the known 3D extended NC (Bargmann-like) gravity action, establishing a concrete link between 4D ESNC gravity and 3D CS formulations. They also discuss extensions to extended p-brane NC gravity and note algebraic caveats, such as the lack of a nondegenerate invariant bilinear form, highlighting directions for future holographic and spectrum analyses.

Abstract

We construct an action for four-dimensional extended string Newton-Cartan gravity which is an extension of the string Newton-Cartan gravity that underlies nonrelativistic string theory. The action can be obtained as a nonrelativistic limit of the Einstein-Hilbert action in General Relativity augmented with a term that contains an auxiliary two-form and one-form gauge field that both have zero flux on-shell. The four-dimensional extended string Newton-Cartan gravity is based on a central extension of the algebra that underlies string Newton-Cartan gravity. The construction is similar to the earlier construction of a three-dimensional Chern-Simons action for extended Newton-Cartan gravity, which is based on a central extension of the algebra that underlies Newton-Cartan gravity. We show that this three-dimensional action is naturally obtained from the four-dimensional action by a reduction over the spatial isometry direction longitudinal to the string followed by a truncation of the extended string Newton-Cartan gravity fields. Our construction can be seen as a special case of the construction of an action for extended p-brane Newton-Cartan gravity in p+3 dimensions.