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

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.