Supersymmetric M2-branes with Englert Fluxes and the simple group PSL(2,7)

A new class is introduced of M2-branes solutions of d=11 supergravity that include internal fluxes obeying Englert equation in 7-dimensions. A simple criterion for the existence of Killing spinors in such backgrounds is established. Englert equation is viewed as the generalization to d=7 of Beltrami equation defined in d=3 and it is trated accordingly. All 2-brane solutions of minimal d=7 supergracity can be uplifted to d=11 and have N > 4 or N = 4 supersymmetry. It is shown that the simple group PSL(2,7) is crystallographic in d=7 having an integral action on the A7 root lattice. By means of this point-group and of the T7 torus obtained quotiening R7 with the A7 root lattice we were able to construct new M2 branes with Englert fluxes and N < 4. In particular we exhibit here an N=1 solution depending on 4-parameters and admitting a large non abelian discrete symmetry, namely G21 = Z3 semidirect product with Z7 = subgroup of PSL(2,7). The dual d=3 field theories have the same symmetries and have complicated non linear interactions.Comment: 59 pages, two figures. Original Research Article. Few misprints corrected, a pair of sentences added in the aknowledgment

BPS D3-branes on smooth ALE manifolds

In this talk I review the recent construction of a new family of classical BPS solutions of type IIB supergravity describing 3-branes transverse to a 6-dimensional space with topology $\mathbb{R}^{2}\times$ALE. They are characterized by a non-trivial flux of the supergravity 2-forms through the homology 2-cycles of a generic smooth ALE manifold. These solutions have two Killing spinors and thus preserve $\mathcal{N}=2$ supersymmetry. They are expressed in terms of a quasi harmonic function $H$ (the ``warp factor''), whose properties was studied in detail in the case of the simplest ALE, namely the Eguchi-Hanson manifold. The equation for $H$ was identified as an instance of the confluent Heun equation.Comment: Talk given at the Conference "New Trends in Particle Physics", September 2001, Yalta, Ukrain

BPS Black Holes in Superegravity

In these lectures we explain the concept of supergravity p-branes and BPS black holes. Introducing an audience of general relativists to all the necessary geometry related with extended supergravity (special geometry, symplectic embeddings and the like) we describe the general properties of N=2 black holes, the structure of central charges in extended supergravity and the description of black hole entropy as an invariant of the U duality group. Then, after explaining the concept and the use of solvable Lie algebras we present the detailed construction of 1/2, 1/4 and 1/8 supersymmetry preserving black holes in the context of N=8 supergravity. The Lectures are meant to be introductory and self contained for non supersymmetry experts but at the same time fully detailed and complete on the subject.Comment: LaTeX, 132 pages, Book.sty. Lecture Notes for the SIGRAV Graduate School in Contemporary Relativity, Villa Olmo, Como First Course, April 199

N=8 BPS black holes preserving 1/8 supersymmetry

In the context of N=8 supergravity we consider BPS black-holes that preserve 1/8 supersymmetry. It was shown in a previous paper that, modulo U-duality transformations of E_{7(7)} the most general solution of this type can be reduced to a black-hole of the STU model. In this paper we analize this solution in detail, considering in particular its embedding in one of the possible Special K\"ahler manifold compatible with the consistent truncations to N=2 supergravity, this manifold being the moduli space of the T^6/Z^3 orbifold, that is: SU(3,3)/SU(3)*U(3). This construction requires a crucial use of the Solvable Lie Algebra formalism. Once the group-theoretical analisys is done, starting from a static, spherically symmetric ans\"atz, we find an exact solution for all the scalars (both dilaton and axion-like) and for gauge fields, together with their already known charge-dependent fixed values, which yield a U-duality invariant entropy. We give also a complete translation dictionary between the Solvable Lie Algebra and the Special K\"ahler formalisms in order to let comparison with other papers on similar issues being more immediate. Although the explicit solution is given in a simplified case where the equations turn out to be more manageable, it encodes all the features of the more general one, namely it has non-vanishing entropy and the scalar fields have a non-trivial radial dependence.Comment: 29+1 pages, 1 Latex file; a misprint in the entropy formula, eq.(5.14), correcte