J. A. de Azcarraga

n-ary algebras: a review with applications

This paper reviews the properties and applications of certain n-ary generalizations of Lie algebras in a self-contained and unified way. These generalizations are algebraic structures in which the two entries Lie bracket has been replaced by a bracket with n entries. Each type of n-ary bracket satisfies a specific characteristic identity which plays the r^ole of the Jacobi identity for Lie algebras. Particular attention will be paid to generalized Lie algebras, which are defined by even multibrackets obtained by antisymmetrizing the associative products of its n components and that satisfy the generalized Jacobi identity (GJI), and to Filippov (or n-Lie) algebras, which are defined by fully antisymmetric n-brackets that satisfy the Filippov identity (FI). Three-Lie algebras have surfaced recently in multi-brane theory in the context of the Bagger-Lambert-Gustavsson model. Because of this, Filippov algebras will be discussed at length, including the cohomology complexes that govern their central extensions and their deformations (Whiteheads lemma extends to all semisimple n-Lie algebras). When the skewsymmetry of the n-Lie algebra is relaxed, one is led the n-Leibniz algebras. These will be discussed as well, since they underlie the cohomological properties of n-Lie algebras. nThe standard Poisson structure may also be extended to the n-ary case. We shall review here the even generalized Poisson structures, whose GJI reproduces the pattern of the generalized Lie algebras, and the Nambu-Poisson structures, which satisfy the FI and determine Filippov algebras. Finally, the recent work of Bagger-Lambert and Gustavsson on superconformal Chern-Simons theory will be briefly discussed. Emphasis will be made on the appearance of the 3-Lie algebra structure and on why the A_4 model may be formulated in terms of an ordinary Lie algebra, and on its Nambu bracket generalization.

Generalizations of Maxwell (super)algebras by the expansion method

This paper has been supported by research grants from the Spanish MINECO (FIS2008-01980, FIS2009-09002, CONSOLIDER CPAN-CSD2007-00042), from the Polish Ministry of Science and Education (202332139) and from the Polish National Science Center (project 2011/01/B/ST2/0335).

Two-twistor particle models and free massive higher spin fields

A bstractWe present D = 3 and D = 4 world-line models for massive particles moving in a new type of enlarged spacetime, with D−1 additional vector coordinates, which after quantization lead to towers of massive higher spin (HS) free fields. Two classically equivalent formulations are presented: one with a hybrid spacetime/bispinor variables and a second described by a free two-twistor dynamics with constraints. After first quantization in the D = 3 and D = 4 cases, the wave functions satisfying a massive version of Vasiliev’s free unfolded equations are given as functions on the SL(2, ℝ) and SL(2, ℂ) group manifolds respectively, which describe arbitrary on-shell momenta and spin degrees of freedom. Further we comment on the D = 6 case, and possible supersymmetric extensions are mentioned as well. Finally, the description of interactions and the AdS/CFT duality are briefly considered for massive HS fields.

Minimal D=4 supergravity from the superMaxwell algebra

We show that the first-order D = 4, N = 1 pure supergravity lagrangian four-form can be obtained geometrically as a quadratic expression in the curvatures of the Maxwell superalgebra. This is achieved by noticing that the relative coefficient between the two terms of the lagrangian that makes the action locally supersymmetric also determines trivial field equations for the gauge fields associated with the extra generators of the Maxwell superalgebra. Along the way, a convenient geometric procedure to check the local supersymmetry of a class of lagrangians is developed

Supertwistors, massive superparticles and κ-symmetry

We consider a D = 4 two-twistor lagrangian for a massive particle that incorporates the mass-shell condition in an algebraic way, and extend it to a two-supertwistor model with N = 2 supersymmetry and central charge identified with the mass. In the purely supertwistorial picture the two D = 4 super twistors are coupled through a Wess-Zumino term in their fermionic sector. We demonstrate how the ?-gauge symmetry appears in the purely supertwistorial formulation and reduces by half the fermionic degrees of freedom of the two supertwistors; a formulation of the model in terms of ?-invariant degrees of freedom is also obtained. We show that the ?-invariant supertwistor coordinates can be obtained by dimensional (D = 6 ? D = 4) reduction from a D = 6 supertwistor. We derive as well by 6 ? 4 reduction the N = 2 D = 4 massive superparticle model with Wess-Zumino term introduced in 1982. Finally, we comment on general superparticle models constructed with more than two supertwistors.

Optimally defined Racah–Casimir operators for su(n) and their eigenvalues for various classes of representations

This paper deals with the striking fact that there is an essentially canonical path from the ith Lie algebra cohomology cocycle, i=1,2,…,l, of a simple compact Lie algebra g of rank l to the definition of its primitive Casimir operators C(i) of order mi. Thus one obtains a complete set of Racah–Casimir operators C(i) for each g and nothing else. The paper then goes on to develop a general formula for the eigenvalue c(i) of each C(i) valid for any representation of g, and thereby to relate c(i) to a suitably defined generalized Dynkin index. The form of the formula for c(i) for su(n) is known sufficiently explicitly to make clear some interesting and important features. For the purposes of illustration, detailed results are displayed for some classes of representation of su(n), including all the fundamental ones and the adjoint representation.

Superbranes, D = 11 CJS Supergravity and Enlarged Superspace Coordinates/Fields Correspondence

We discuss the role of enlarged superspaces in two seemingly different contexts, the structure of the p‐brane actions and that of the Cremmer‐Julia‐Scherk eleven‐dimensional supergravity. Both provide examples of a common principle: the existence of an enlarged superspaces coordinates/fields correspondence by which all the (worldvolume or spacetime) fields of the theory are associated to coordinates of enlarged superspaces. In the context of p‐branes, enlarged superspaces may be used to construct manifestly supersymmetry‐invariant Wess‐Zumino terms and as a way of expressing the Born‐Infeld worldvolume fields of D‐branes and the worldvolume M5‐brane two‐form in terms of fields associated to the coordinates of these enlarged superspaces. This is tantamount to saying that the Born‐Infeld fields have a superspace origin, as do the other worldvolume fields, and that they have a composite structure. In D=11 supergravity theory enlarged superspaces arise when its underlying gauge structure is investigated and, ...We discuss the role of enlarged superspaces in two seemingly different contexts, the structure of the p-brane actions and that of the Cremmer-Julia-Scherk eleven-dimensional supergravity. Both provide examples of a common principle: the existence of an enlarged superspaces coordinates/fields correspondence by which all the (worldvolume or spacetime) fields of the theory are associated to coordinates of enlarged superspaces. In the context of p-branes, enlarged superspaces may be used to construct manifestly supersymmetry-invariant Wess-Zumino terms and as a way of expressing the Born-Infeld worldvolume fields of D-branes and the worldvolume M5-brane two-form in terms of fields associated to the coordinates of these enlarged superspaces. This is tantamount to saying that the Born-Infeld fields have a superspace origin, as do the other worldvolume fields, and that they have a composite structure. In D=11 supergravity theory enlarged superspaces arise when its underlying gauge structure is investigated and, as a result, the composite nature of the A3 field is revealed: there is a full one-parametric family of enlarged superspace groups that solve the problem of expressing A3 in terms of spacetime fields associated to their coordinates. The corresponding enlarged supersymmetry algebras turn out to be deformations of an expansion of the osp(1 vertical barmorexa0» 32) algebra. The unifying mathematical structure underlying all these facts is the cohomology of the supersymmetry algebras involved.«xa0less

BPS preons in M-theory and supergravity

After introducing the notion of BPS preons as the basic constituents of Mtheory, we discuss the recent negative results in the search for solutions of the D=10 and D=11 supergravity equations preserving 31/32 supersymmetries i.e., of preonic solutions. The absence of these supergravity preonic solutions may point out to a pure quantum nature of BPS preons, manifesting itself in the need of incorporating quantum (stringy/M-theoretic) corrections to the supergravity equations.

Bosonic D = 11 supergravity from a generalized Chern–Simons action

Abstract It is shown that the action of the bosonic sector of D = 11 supergravity may be obtained by means of a suitable scaling of the originally dimensionless fields of a generalized Chern–Simons action. This follows from the eleven-form CS-potential of the most general linear combination of closed, gauge invariant twelve-forms involving the s p ( 32 ) -valued two-form curvatures supplemented by a three-form field. In this construction, the role of the skewsymmetric four-index auxiliary function needed for the first order formulation of D = 11 supergravity is played by the gauge field associated with the five Lorentz indices generator of the bosonic s p ( 32 ) subalgebra of o s p ( 1 | 32 ) .

137-01137-01137-01-deformed extended Galilei algebra

We apply the pseudoextension mechanism, which in the undeformed case gives the centrally extended Galilei group {ie137-02} as a contraction of a trivial extensionP×U(1) of the Poincaré group, to the case of theκ-Poincaré algebra. As a result, the four-dimensional {ie137-03}-deformed extended Galilei algebra is obtained.