Standard General Relativity from Chern-Simons Gravity

Chern-Simons models for gravity are interesting because they provide with a truly gauge-invariant action principle in the fiber-bundle sense. So far, their main drawback has largely been the perceived remoteness from standard General Relativity, based on the presence of higher powers of the curvature in the Lagrangian (except, remarkably, for three-dimensional spacetime). Here we report on a simple model that suggests a mechanism by which standard General Relativity in five-dimensional spacetime may indeed emerge at a special critical point in the space of couplings, where additional degrees of freedom and corresponding "anomalous" Gauss-Bonnet constraints drop out from the Chern-Simons action. To achieve this result, both the Lie algebra g and the symmetric g-invariant tensor that define the Chern-Simons Lagrangian are constructed by means of the Lie algebra S-expansion method with a suitable finite abelian semigroup S. The results are generalized to arbitrary odd dimensions, and the possible extension to the case of eleven-dimensional supergravity is briefly discussed.Comment: 6 pages, no figures; v2: published versio

Chern-Simons--Antoniadis-Savvidy forms and standard supergravity

In the context of the so called the Chern--Simons--Antoniadis--Savvidy (ChSAS) forms, we use the methods for FDA decomposition in 1-forms to construct a four-dimensional ChSAS supergravity action for the Maxwell superalgebra. On the another hand, we use the Extended Cartan Homotopy Formula to find a method that allows the separation of the ChSAS action into bulk and boundary contributions and permits the splitting of the bulk Lagrangian into pieces that reflect the particular subspace structure of the gauge algebra.Comment: 14 page

Minimal AdS-Lorentz supergravity in three-dimensions

The $\mathcal{N}=1$ AdS-Lorentz superalgebra is studied and its relationship to semigroup expansion developed. Using this mathematical tool, the invariant tensors and Casimir operators are found. In terms of these invariants, a three-dimensionnal Chern--Simons supergravity action with AdS-Lorentz symmetry is constructed. The Killing spinors for a BTZ black-hole like solution of the theory are discussed.Comment: 18 pages, matches published versio

Dual Formulation of the Lie Algebra S-expansion Procedure

The expansion of a Lie algebra entails finding a new, bigger algebra G, through a series of well-defined steps, from an original Lie algebra g. One incarnation of the method, the so-called S-expansion, involves the use of a finite abelian semigroup S to accomplish this task. In this paper we put forward a dual formulation of the S-expansion method which is based on the dual picture of a Lie algebra given by the Maurer-Cartan forms. The dual version of the method is useful in finding a generalization to the case of a gauge free differential algebra, which in turn is relevant for physical applications in, e.g., Supergravity. It also sheds new light on the puzzling relation between two Chern-Simons Lagrangians for gravity in 2+1 dimensions, namely the Einstein-Hilbert Lagrangian and the one for the so-called "exotic gravity".Comment: 12 pages, no figure

Euler Chern Simons Gravity from Lovelock Born Infeld Gravity

In the context of a gauge theoretical formulation, higher dimensional gravity invariant under the AdS group is dimensionally reduced to Euler-Chern-Simons gravity. The dimensional reduction procedure of Grignani-Nardelli [Phys. Lett. B 300, 38 (1993)] is generalized so as to permit reducing D-dimensional Lanczos Lovelock gravity to d=D-1 dimensions.Comment: 6 pages, no figures, accepted for publication in Phys. Lett.

A Chern–Simons gravity action in d=4

AbstractRecently, Antoniadis, Konitopoulos and Savvidy have introduced in Refs. [1–4] a procedure to construct background-free gauge invariants, using non-abelian gauge potentials described by forms of higher degree. Their construction is particularly useful because it can be used in both, odd- and even-dimensional spacetimes. Using their technique, we generalize the Chern–Weil theorem and construct a gauge-invariant, (2n+2)-dimensional transgression form, and study its relationship with the generalized Chern–Simons forms introduced in Refs. [1,2].Using the methods for FDA manipulation and decomposition in 1-forms developed in Ref. [5] and applied in Refs. [6] and [7], we construct a four-dimensional Chern–Simons gravity action, which is off-shell gauge invariant under the Maxwell algebra

Einstein-Chern-Simons equations on the 3-brane world

In this article it is studied the 3-brane world in the context of five-dimensional Einstein-Chern-Simons gravity. We started by considering Israel's junction condition for AdS-Chern-Simons gravity. Using the S-expansion procedure, we mapped the AdS-Chern-Simons junction conditions to Einstein-Chern-Simons gravity, allowing us to derive effective four-dimensional Einstein-Chern-Simons field equations

Higher dimensional gravity invariant under the AdS group

A higher dimensional gravity invariant both under local Lorentz rotations and under local Anti de Sitter boosts is constructed. It is shown that such a construction is possible both when odd dimensions and when even dimensions are considered. It is also proved that such actions have the same coefficients as those obtained by Troncoso and Zanelli.Comment: 5 pages, no figures; final version to appear in Phys. Lett.

Effectively four-dimensional spacetimes emerging from d=5 Einstein-Gauss-Bonnet Gravity

Einstein-Gauss-Bonnet gravity in five-dimensional spacetime provides an excellent example of a theory that, while including higher-order curvature corrections to General Relativity, still shares many of its features, such as second-order field equations for the metric. We focus on the largely unexplored case where the coupling constants of the theory are such that no constant-curvature solution is allowed, leaving open the question of what the vacuum state should then be. We find that even a slight deviation from the anti-de Sitter Chern-Simons theory, where the vacuum state is five-dimensional AdS spacetime, leads to a complete symmetry breakdown, with the fifth dimension either being compactified into a small circle or shrinking away exponentially with time. A complete family of solutions, including duality relations among them, is uncovered and shown to be unique within a certain class. This dynamical dimensional reduction scenario seems particularly attractive as a means for higher-dimensional theories to make contact with our four-dimensional world.Comment: 9 pages, 4 figures. v2: New section on geometrical significance of solutions. Final version for CQ