An alternative approach to matter coupling in Minimal Massive 3D Gravity and spinor-matter interactions

In this work, using differential forms, an alternative approach to matter coupling in Minimal Massive 3D Gravity (MMG) is presented. In the first part, we consider the minimal coupling of matter Lagrangian assuming that matter Lagrangian 3-form depends on the metric co-frame fields and some matter fields but not on the connections. We construct additional source 2-form to obtain a consistent matter-coupled MMG theory. We see that additional source 2-form involves terms that are quadratic in stress-energy 2-forms. In addition, we derive consistency relation in the language of differential forms. Next, we consider minimal coupling of Dirac Lagrangian where for this case massive spinor-matter Lagrangian depends on both metric co-frames and connection fields. To get a consistent spinor-matter coupled MMG field equation, we obtain additional source 2-form as well. It is shown that with spinor coupling, source 2-form involves terms which are quadratic in covariant derivatives of spinor fields.