Three-Dimensional Extended Bargmann Supergravity

TL;DR

The paper focuses on building a consistent 3D non-relativistic gravity theory with matter and supersymmetry by introducing Extended Bargmann Gravity (EBG), a Chern–Simons theory for the extended Bargmann algebra with central charges $M$ and $S$, obtained as a non-relativistic limit of a relativistic action. It extends to a supersymmetric version with generators $(Q^+,Q^-,R)$, yielding a CS action that includes fermionic curvatures and local SUSY transformations. Matter couplings lead to equations $e^{-1}\epsilon^{\mu\nu\rho} R_{\nu\rho}(S) = \frac{4\pi}{k} t^\mu$, $e^{-1}\epsilon^{\mu\nu\rho} R_{\nu\rho}(J) = \frac{4\pi}{k} j^\mu$, and $e^{-1}\epsilon^{\mu\nu\rho} R_{\nu\rho}(G_a) = -\frac{4\pi}{k} t^\mu{}_a$, with Bianchi identities ensuring consistency; unlike NC gravity, EBG allows nontrivial spatial curvature in the presence of matter. This framework connects non-relativistic gravity to holography, localization, and Hořava-Lifshitz-like structures, and opens avenues for higher-spin and cosmological-constant extensions, as well as a deeper understanding of non-relativistic supergravity backgrounds.

Abstract

We show that three-dimensional General Relativity, augmented with two vector fields, allows for a non-relativistic limit, different from the standard limit leading to Newtonian gravity, that results into a well-defined action which is of the Chern-Simons type. We show that this three-dimensional `Extended Bargmann Gravity', after coupling to matter, leads to equations of motion allowing a wider class of background geometries than the ones that one encounters in Newtonian gravity. We give the supersymmetric generalization of these results and point out an important application in the context of calculating partition functions of non-relativistic field theories using localization techniques.