Matter coupling in minimal massive 3D gravity and spinor-matter interactions in exterior algebra formalism

Abstract

In this work, by employing the exterior algebra formalism, we study the matter coupling in Minimal Massive 3D Gravity (MMG) by first considering that the matter Lagrangian is connection-independent and then considering that the matter coupling is connection-dependent. The matter coupling in MMG has been previously investigated in the work \cite{arvanitakis_2} in tensorial notation where the matter Lagrangian is considered to be connection-independent. In the first part of the present paper, we revisit the connection-independent matter coupling by using the language of differential forms. We derive the MMG field equation and construct the related source 2-form. We also obtain the consistency relation within this formalism. Next, we examine the case where the matter Lagrangian is connection-dependent. In particular, we concentrate on the spinor-matter coupling and obtain the MMG field equation by explicitly constructing the source term. We also get the consistency relation that the source term should satisfy in order that spinor-matter coupled MMG equation be consistent.

A note on the pp-wave solution of minimal massive 3D gravity coupled with Maxwell–Chern–Simons theory

“…Then by solving auxiliary one-form from the field equations, we have obtained the MMG gravitational field equation with the effective source two-form that contains quadratic terms in the stress-energy two-forms associated with the Maxwell field as well as their covariant derivative. As was mentioned and explicitly shown in [11,24], the source term of MMG completely differs from the source term of TMG where for the latter the source is just expressed by the stress-energy two-form τ c associated with the matter fields. Furthermore, unlike TMG, the covariant exterior derivative of the source term does not vanish but it has to satisfy a consistency relation given in (2.34).…”

“…expressed in terms of redefined Maxwell stress-energy two-forms tc . We further note that the source includes terms that are quadratic in stress-energy forms as is also discussed in [11,24]. It also involves the exterior covariant derivative of redefined Maxwell stress-energy two-form.…”

“…In this section, following [24], let us first introduce the MMG theory coupled with Maxwell-Chern-Simons theory and the formalism used. In the language of differential forms, the model can be described by the action…”

“…Note that this consistency relation holds for any matter source obtained from the connectionindependent matter coupling 6 . The equality of left-and right-hand sides of the consistency expression for any connection-independent matter coupling has been explicitly illustrated in [24] by using the exterior algebra formalism. This has been performed explicitly by recalling that stress-energy two-forms t c are covariantly conserved i.e.…”

“…In this work, motivated by [19], we aim to investigate the pp-wave type solutions of MMG minimally coupled with Maxwell-Chern-Simons theory within the exterior algebra formalism. It should also be noted that using the same formalism, MMG has been studied in the works [20][21][22][23][24] where a detailed investigation of the matter-coupling in the exterior algebra formalism is performed in the work [24]. Now in the first part of the present study, we obtain the field equations by making variations of the matter-coupled MMG Lagrangian with respect to co-frames, connections, auxiliary one-form and Maxwell potential one-form fields.…”