Note on the matter coupling in Exotic General Massive 3D Gravity

TL;DR

The paper develops a complete exterior-algebra treatment of matter coupling in Exotic General Massive Gravity, deriving the field equations and a closed source 2-form by solving auxiliary fields algebraically. It demonstrates that the matter source contains linear, quadratic, and derivative terms in the energy-momentum and hyper-momentum 2-forms, and derives a third-way consistency condition linking these matter bilinears. Through explicit examples of minimal (connection-independent and dependent) and non-minimal couplings, as well as higher-curvature extensions, the work provides a versatile framework for analyzing non-minimal couplings and extended gravity theories in 3D. The results pave the way for exploring exact solutions, unitarity properties, and supersymmetric generalizations of EGMG in future studies.

Abstract

In this work, within exterior algebra formalism, we revisit and study the matter coupling in Exotic General Massive 3D Gravity (EGMG), previously introduced in \cite{ozkan_1}. Considering that the matter Lagrangian is both metric co-frame and connection dependent in general, we explicitly construct associated source 2-form by algebraically solving auxiliary 1-form fields in terms of gravitational field variables, energy-momentum and hyper-momentum 2-forms. It is seen that that the source 2-form involves terms that are quadratic in energy-momentum and hyper-momentum 2-forms as well as terms that depend on covariant derivatives of them. In addition, we derive the consistency relation in order that matter-coupled EGMG field equation be third-way consistent. In general, the consistency requires a particular relation that should be satisfied by energy-momentum and hyper-momentum 2-forms. We note that our construction and formalism allows to investigate non-minimal matter couplings with gravitational fields. It also enables to construct the higher order curvature extensions of EGMG.