Classification of supersymmetric backgrounds of string theory

We review the recent progress made towards the classification of supersymmetric solutions in ten and eleven dimensions with emphasis on those of IIB supergravity. In particular, the spinorial geometry method is outlined and adapted to nearly maximally supersymmetric backgrounds. We then demonstrate its effectiveness by classifying the maximally supersymmetric IIB G-backgrounds and by showing that N=31 IIB solutions do not exist.Comment: 8 pages, contribution to the proceedings of the RTN project `Constituents, Fundamental Forces and Symmetries of the Universe' conference in Napoli, October 9 - 13, 200

Supersymmetric geometries of IIA supergravity I

IIA supergravity backgrounds preserving one supersymmetry locally admit four types of Killing spinors distinguished by the orbits of $Spin(9,1)$ on the space of spinors. We solve the Killing spinor equations of IIA supergravity with and without cosmological constant for Killing spinors representing two of these orbits, with isotropy groups $Spin(7)$ and $Spin(7)\ltimes\mathbb{R}^8$. In both cases, we identify the geometry of spacetime and express the fluxes in terms of the geometry. We find that the geometric constraints of backgrounds with a $Spin(7)\ltimes\mathbb{R}^8$ invariant Killing spinor are identical to those found for heterotic backgrounds preserving one supersymmetry.Comment: 21 page

GAMMA: A Mathematica package for performing gamma-matrix algebra and Fierz transformations in arbitrary dimensions

We have developed a Mathematica package capable of performing gamma-matrix algebra in arbitrary (integer) dimensions. As an application we can compute Fierz transformations.Comment: 12 pages, LaTeX, v2: some clarifications added, updated package availabl

Monotone-light factorisation systems and torsion theories

Given a torsion theory (Y,X) in an abelian category C, the reflector I from C to the torsion-free subcategory X induces a reflective factorisation system (E, M) on C. It was shown by A. Carboni, G.M. Kelly, G. Janelidze and R. Par\'e that (E, M) induces a monotone-light factorisation system (E',M*) by simultaneously stabilising E and localising M, whenever the torsion theory is hereditary and any object in C is a quotient of an object in X. We extend this result to arbitrary normal categories, and improve it also in the abelian case, where the heredity assumption on the torsion theory turns out to be redundant. Several new examples of torsion theories where this result applies are then considered in the categories of abelian groups, groups, topological groups, commutative rings, and crossed modules.Comment: 12 page

Higher commutator conditions for extensions in Mal'tsev categories

We define a Galois structure on the category of pairs of equivalence relations in an exact Mal'tsev category, and characterize central and double central extensions in terms of higher commutator conditions. These results generalize both the ones related to the abelianization functor in exact Mal'tsev categories, and the ones corresponding to the reflection from the category of internal reflexive graphs to the subcategory of internal groupoids. Some examples and applications are given in the categories of groups, precrossed modules, precrossed modules of Lie algebras, and compact groups.Comment: 32 page

Some aspects of semi-abelian homology and protoadditive functors

In this note some recent developments in the study of homology in semi-abelian categories are briefly presented. In particular the role of protoadditive functors in the study of Hopf formulae for homology is explained.Comment: 7 page

Non-Commutative Open (p,q)-String Theories

In this paper we make an SL(2,Z)-covariant generalisation of the noncommutative theories, NCYM and NCOS on the D3-brane, and NCOS on the D5-brane in type IIB. Usually, the noncommutative theories are obtained by studying perturbative F-string theory, and the parameters governing the noncommutative theories are given by the open string data. The S-duality of NCYM and NCOS on the D3-brane has been seen by dualising the background, keeping the F-string under study fixed. We give an SL(2,Z)-covariant generalisation of the open string data relevant when one instead studies perturbative (p,q)-string theory. The S-duality of NCYM and NCOS on the D3-brane is reproduced by instead keeping the background fixed and studying different (p,q)-string theories. We also obtain new noncommutative open (p,q)-string theories on the D3-brane and the D5-brane which are S-dual to ordinary NCOS. The theories are studied using the supergravity duals of the D3-brane and the D5-brane, corresponding to a probe brane in the relevant background.Comment: AMSLaTeX. 23 pages. Minor clarifications added. To be published in JHE

Some remarks on pullbacks in Gumm categories

We extend some properties of pullbacks which are known to hold in a Mal'tsev context to the more general context of Gumm categories. The varieties of universal algebras which are Gumm categories are precisely the congruence modular ones. These properties lead to a simple alternative proof of the known property that central extensions and normal extensions coincide for any Galois structure associated with a Birkhoff subcategory of an exact Goursat category.Comment: 12 page