Eduardo Rodríguez

Expanding Lie (super)algebras through Abelian semigroups

We propose an outgrowth of the expansion method introduced by de Azcarraga et al. [Nucl. Phys. B 662, 185 (2003)]. The basic idea consists in considering the direct product between an Abelian semigroup S and a Lie algebra g. General conditions under which relevant subalgebras can systematically be extracted from S×g are given. We show how, for a particular choice of semigroup S, the known cases of expanded algebras can be reobtained, while new ones arise from different choices. Concrete examples, including the M algebra and a D’Auria-Fre-like superalgebra, are considered. Finally, we find explicit, nontrace invariant tensors for these S-expanded algebras, which are essential ingredients in, e.g., the formulation of supergravity theories in arbitrary space-time dimensions.

Standard General Relativity from Chern-Simons Gravity

Abstract Chern–Simons models for gravity are interesting because they provide a truly gauge-invariant action principle in the fiber-bundle sense. So far, their main drawback has largely been its 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 goal, 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.

A generalized action for(2 + 1)-dimensional Chern–Simons gravity

It is shown that the expansion methods developed in refs. arXiv:hep-th/0212347 and arXiv:hep-th/0401033v2 can be generalized so that they permit to study the expansion of algebras of loops, both when the compact finite-dimensional algebra and the algebra of loops have a decomposition into two subspaces.We show that the so-called semi-simple extended Poincare (SSEP) algebra in D dimensions can be obtained from the anti-de Sitter algebra by means of the S-expansion procedure with an appropriate semigroup S. A general prescription is given for computing Casimir operators for S-expanded algebras, and the method is exemplified for the SSEP algebra. The S-expansion method also allows us to extract the corresponding invariant tensor for the SSEP algebra, which is a key ingredient in the construction of a generalized action for Chern–Simons gravity in (2 + 1) dimensions.

Eleven-dimensional gauge theory for the M-algebra as an Abelian semigroup expansion of \mathfrak{osp} ( 32 | 1 )

A new Lagrangian realizing the symmetry of the M-algebra in eleven-dimensional space-time is presented. By means of the novel technique of Abelian semigroup expansion, a link between the M-algebra and the orthosymplectic algebra

Even-dimensional General Relativity from Born-Infeld gravity

\mathfrak{osp}(32|1)

Chern–Simons and Born–Infeld gravity theories and Maxwell algebras type

is established, and an M-algebra-invariant symmetric tensor of rank six is computed. This symmetric invariant tensor is a key ingredient in the construction of the new Lagrangian. The gauge-invariant Lagrangian is displayed in an explicitly Lorentz-invariant way by means of a subspace separation method based on the extended Cartan homotopy formula.

Maxwell superalgebras and Abelian semigroup expansion

It is an accepted fact that requiring the Lovelock theory to have the maximun possible number of degree of freedom, fixes the parameters in terms of the gravitational and the cosmological constants. In odd dimensions, the Lagrangian is a Chern-Simons form for the (A)dS group. In even dimensions, the action has a Born-Infeld-like form. Recently was shown that standard odd-dimensional General Relativity can be obtained from Chern-Simons Gravity theory for a certain Lie algebra B. Here we report on a simple model that suggests a mechanism by which standard even-dimensional General Relativity may emerge as a weak coupling constant limit of a Born-Infeld theory for a certain Lie subalgebra of the algebra B. Possible extension to the case of even-dimensional supergravity is briefly discussed.

Dual formulation of the Lie algebra S-expansion procedure

Recently it was shown that standard odd- and even-dimensional general relativity can be obtained from a

N = 1 supergravity and Maxwell superalgebras

(2n+1)