J. A. de Azcárraga

Superconformal mechanics, black holes, and nonlinear realizations

The OSp(2|2)-invariant planar dynamics of a D = 4 superparticle near the horizon of a large mass extreme black hole is described by an N = 2 superconformal mechanics, with the SO(2) charge being the superparticle’s angular momentum. The non-manifest superconformal invariance of the superpotential term is shown to lead to a shift in the SO(2) charge by the value of its coefficient, which we identify asthe orbital angular momentum. The full SU(1,1|2)-invariant dynamics is found from an extension to N = 4 superconformal mechanics.

The geometry of branes and extended superspaces

Abstract We argue that a description of supersymmetric extended objects from a unified geometric point of view requires an enlargement of superspace. To this aim we study in a systematic way how superspace groups and algebras arise from Grassmann spinors when these are assumed to be the only primary entities. In the process, we recover generalized space-time superalgebras and extensions of supersymmetry found earlier. The enlargement of ordinary superspace with new parameters gives rise to extended superspace groups, on which manifestly supersymmetric actions may be constructed for various types of p -branes, including D-branes (given by Chevalley-Eilenberg cocycles) with their Born-Infeld fields. This results in a field/extended superspace democracy for superbranes: all brane fields appear as pull-backs from a suitable target superspace. Our approach also clarifies some facts concerning the origin of the central charges for the different p -branes.

Invariant tensors for simple groups

The forms of the invariant primitive tensors for the simple Lie algebras Al, Bl, Cl, and Dl are investigated. A new family of symmetric invariant tensors is introduced using the non-trivial cocycles for the Lie algebra cohomology. For the Al algebra it is explicitly shown that the generic forms of these tensors become zero except for the l primitive ones and that they give rise to the l primitive Casimir operators. Some recurrence and duality relations are given for the Lie algebra cocycles. Tables for the 3- and 5-cocycles for su(3) are su(4) are also provided. Finally, new relations involving the d and f su(n) tensors are given.

Higher order simple Lie algebras

It is shown that the non-trivial cocycles on simple Lie algebras may be used to introduce antisymmetric multibrackets which lead to higher-order Lie algebras, the definition of which is given. Their generalised Jacobi identities turn out to be satisfied by the antisymmetric tensors (or higher-order “structure constants”) which characterise the Lie algebra cocycles. This analysis allows us to present a classification of the higher-order simple Lie algebras as well as a constructive procedure for them. Our results are synthesised by the introduction of a single, complete BRST operator associated with each simple algebra.Abstract. It is shown that the non-trivial cocycles on simple Lie algebras may be used to introduce antisymmetric multibrackets which lead to higher-order Lie algebras, the definition of which is given. Their generalised Jacobi identities turn out to be satisfied by the antisymmetric tensors (or higher-order “structure constants”) which characterise the Lie algebra cocycles. This analysis allows us to present a classification of the higher-order simple Lie algebras as well as a constructive procedure for them. Our results are synthesised by the introduction of a single, complete BRST operator associated with each simple algebra.

Galilean Superconformal Symmetries

Abstract We consider the non-relativistic c → ∞ contraction limit of the ( N = 2 k ) -extended D = 4 superconformal algebra su ( 2 , 2 ; N ) , introducing in this way the non-relativistic ( N = 2 k ) -extended Galilean superconformal algebra. Such a Galilean superconformal algebra has the same number of generators as su ( 2 , 2 | 2 k ) . The usp ( 2 k ) algebra describes the non-relativistic internal symmetries, and the generators from the coset u ( 2 k ) usp ( 2 k ) become central charges after contraction.

Extensions, expansions, Lie algebra cohomology and enlarged superspaces

After briefly reviewing the methods that allow us to derive consistently new Lie (super)algebras from given ones, we consider enlarged superspaces and superalgebras, their relevance and some possible applications.

On the higher-order generalizations of Poisson structures

The characterization of the Nambu - Poisson n-tensors as a subfamily of the generalized Poisson ones recently introduced (and here extended to the odd-order case) is discussed. The homology and cohomology complexes of both structures are compared, and some physical considerations are made.

Group theoretical foundations of fractional supersymmetry

Fractional supersymmetry denotes a generalization of supersymmetry which may be constructed using a single real generalized Grassmann variable, θ=θ,θn=0, for arbitrary integer n=2,3,.... An explicit formula is given in the case of general n for the transformations that leave the theory invariant, and it is shown that these transformations possess interesting group properties. It is shown also that the two generalized derivatives that enter the theory have a geometric interpretation as generators of left and right transformations of the fractional supersymmetry group. Careful attention is paid to some technically important issues, including differentiation, that arise as a result of the peculiar nature of quantities such as θ.

Expansions of Algebras and Superalgebras and Some Applications

Abstract After reviewing the three well-known methods to obtain Lie algebras and superalgebras from given ones, namely, contractions, deformations and extensions, we describe a fourth method recently introduced, the expansion of Lie (super)algebras. Expanded (super)algebras have, in general, larger dimensions than the original algebra, but also include the İnönü–Wigner and generalized IW contractions as a particular case. As an example of a physical application of expansions, we discuss the relation between the possible underlying gauge symmetry of eleven-dimensional supergravity and the superalgebra osp(1|32).

Geometrical foundations of fractional supersymmetry

A deformed q-calculus is developed on the basis of an algebraic structure involving graded brackets. A number operator and left and right shift operators are constructed for this algebra, and the whole structure is related to the algebra of a q-deformed boson. The limit of this algebra when q is an nth root of unity is also studied in detail. By means of a chain rule expansion, the left and right derivatives are identified with the charge Q and covariant derivative D encountered in ordinary/fractional supersymmetry, and this leads to new results for these operators. A generalized Berezin integral and fractional superspace measure arise as a natural part of our formalism. When q is a root of unity the algebra is found to have a nontrivial Hopf structure, extending that associated with the anyonic line. One-dimensional ordinary/fractional superspace is identified with the braided line when q is a root of unity, so that one-dimensional ordinary/fractional supersymmetry can be viewed as invariance under translation along this line. In our construction of fractional supersymmetry the q-deformed bosons play a role exactly analogous to that of the fermions in the familiar supersymmetric case.