José Figueroa-O’Farrill

Plücker-type relations for orthogonal planes

Abstract We explore a Plucker-type relation which occurs naturally in the study of maximally supersymmetric solutions of certain supergravity theories. This relation generalises at the same time the classical Plucker relation and the Jacobi identity for a metric Lie algebra and coincides with the Jacobi identity of a metric n -Lie algebra. In low dimension we present evidence for a geometric characterisation of the relation in terms of middle-dimensional orthogonal planes in Euclidean or Lorentzian inner product spaces.

On the Lie-Algebraic Origin of Metric 3-Algebras

Since the pioneering work of Bagger–Lambert and Gustavsson, there has been a proliferation of three-dimensional superconformal Chern–Simons theories whose main ingredient is a metric 3-algebra. On the other hand, many of these theories have been shown to allow for a reformulation in terms of standard gauge theory coupled to matter, where the 3-algebra does not appear explicitly. In this paper we reconcile these two sets of results by pointing out the Lie-algebraic origin of some metric 3-algebras, including those which have already appeared in three-dimensional superconformal Chern–Simons theories. More precisely, we show that the real 3-algebras of Cherkis–Sämann, which include the metric Lie 3-algebras as a special case, and the hermitian 3-algebras of Bagger–Lambert can be constructed from pairs consisting of a metric real Lie algebra and a faithful (real or complex, respectively) unitary representation. This construction generalises and we will see how to construct many kinds of metric 3-algebras from pairs consisting of a real metric Lie algebra and a faithful (real, complex or quaternionic) unitary representation. In the real case, these 3-algebras are precisely the Cherkis–Sämann algebras, which are then completely characterised in terms of this data. In the complex and quaternionic cases, they constitute generalisations of the Bagger–Lambert hermitian 3-algebras and anti-Lie triple systems, respectively, which underlie N = 6 and N = 5 superconformal Chern–Simons theories, respectively. In the process we rederive the relation between certain types of complex 3-algebras and metric Lie superalgebras.

The homogeneity theorem for supergravity backgrounds

A bstractWe prove the strong homogeneity conjecture for eleven- and ten-dimensional (Poincaré) supergravity backgrounds. In other words, we show that any backgrounds of 11- dimensional, type I/heterotic or type II supergravity theories preserving a fraction

Symmetric M-theory backgrounds

\nu >\frac{1}{2}

Symmetric backgrounds of type IIB supergravity

of the supersymmetry of the underlying theory are necessarily locally homogeneous. Moreover we show that the homogeneity is due precisely to the supersymmetry, so that at every point of the spacetime one can find a frame for the tangent space made out of Killing vectors constructed out of the Killing spinors.

The homogeneity theorem for supergravity backgrounds II: the six-dimensional theories

We classify symmetric backgrounds of eleven-dimensional supergravity up to local isometry. In other words, we classify triples (M, g, F), where (M,g) is an eleven-dimensional lorentzian locally symmetric space and F is an invariant 4-form, satisfying the equations of motion of eleven-dimensional supergravity. The possible (M,g) are given either by (not necessarily nondegenerate) Cahen-Wallach spaces or by products AdSd × M11−d for 2 ⩽ d ⩽ 7 and M11−d a not necessarily irreducible riemannian symmetric space. In most cases we determine the corresponding F-moduli spaces.

Higher-dimensional kinematical Lie algebras via deformation theory

In this paper, we study homogeneous backgrounds of type IIB supergravity where the underlying geometry is that of a symmetric space. We determine which ten-dimensional Lorentzian symmetric spaces (up to local isometry) admit such backgrounds, and in about two-thirds of the cases we determine fully their moduli space.

Kinematical Lie algebras via deformation theory

A bstractWe prove that supersymmetric backgrounds of (1,0) and (2,0) six-dimensional supergravity theories preserving more than one half of the supersymmetry are locally homogeneous. As a byproduct we also establish that the Killing spinors of such a background generate a Lie superalgebra.

Kinematical lie algebras in 2 + 1 dimensions

We classify kinematical Lie algebras in dimension

Killing superalgebras for Lorentzian four-manifolds

D \geq 4