Abstract
In this work we study a non-relativistic three dimensional Chern-Simons gravity theory based on an enlargement of the Extended Bargmann algebra. A finite nonrelativistic Chern-Simons gravity action is obtained through the non-relativistic contraction of a particular U(1) enlargement of the so-called AdS-Lorentz algebra. We show that the non-relativistic gravity theory introduced here reproduces the Maxwellian Exotic Bargmann gravity theory when a flat limit ℓ → ∞ is applied. We also present an alternative procedure to obtain the non-relativistic versions of the AdS-Lorentz and Maxwell algebras through the semigroup expansion method.
Highlights
Of a constant electromagnetic field background in a Minkowski space
In this work we study a non-relativistic three dimensional Chern-Simons gravity theory based on an enlargement of the Extended Bargmann algebra
We show that the non-relativistic gravity theory introduced here reproduces the Maxwellian Exotic Bargmann gravity theory when a flat limit l → ∞ is applied
Summary
We review the construction of a three-dimensional CS gravity based on a semi-simple enlargement of the Poincare group. As we are interested in finding the NR Maxwell CS gravity [42] in the limit l → ∞, we will consider the [AdSLorentz] ⊕ u(1) ⊕ u(1) ⊕ u(1) theory as our initial relativistic theory, i.e, we add three new extra fields to the field content In this way, the NR contraction should lead to a finite Lagrangian and to a NR algebra with a non-degenerate bilinear form. Before studying the non-relativistic limit of the AdS-Lorentz gravity theory, let us first introduce the extra U(1) gauge fields in the one-form gauge connection (2.4). Considering the new enlarged one-form gauge connection (2.11) and invariant tensor, the relativistic CS action is written as IR =. Let us remark that in this work we are interested in the contraction of the [AdS-Lorentz] ⊕ u(1) ⊕ u(1) ⊕ u(1) algebra since, as we will show the resulting NR algebra admits on one hand a non-degenerate invariant tensor, and on the other hand, leads to the MEB algebra introduced in [42] in the l → ∞ limit
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