On Supersymmetric M-brane configurations with an R * 1,1 /Z2 submanifold

Abstract

We obtain new examples of partially supersymmetric M -brane solutions defined on products of Ricci-flat manifolds, which contain two-dimensional Lorentzian submanifold R 1,1 * /Z 2 with one parallel spinor. The examples belong to the following configurations: M 2, M 5, M 2 ∩ M 5 and M 5 ∩ M 5. Among them a M 2 solution with N = 1/32 fractional number of preserved supersymmetries is presented. The examples with three M -branes were considered earlier in our paper with A.A. Golubtsova. In this letter we present new examples of (partially) supersymmetric solutions in 11dimensional supergravity [1]. The renewed interest to supersymmetric solutions may be motivated by the developments of [2,3,4] (for recent publications see [5] and refs. therein) and also by proving new supergravity/gauge (or AdS/CFT) correspondences: for a review see [6] and refs. therein, e.g. pioneering papers: [7] and [8,9].We deal with the following M-brane configurations: M2, M5, M2 ∩ M5 and M5 ∩ M5 defined on products of (spin) Ricci-flat manifolds and use our general approach to Mbrane solutions (e.g. supersymmetric ones) from [10,11,12,13,14,15], which was inspired by earlier (e.g. pioneering) papers [16]- [29].The examples of supersymmetric solutions, presented here, are defined on elevendimensional manifold which contain a two-dimensional Lorentzian submanifold R 1,1 * /Z 2 with one parallel spinor.Here and in what follows R 1,1 is the 2-dimensional pseudo-Euclidean space (R 2 , η) with the metric η = −dy 0 ⊗ dy 0 + dy 1 ⊗ dy 1 . we denote a spin manifold [30], which has one of two inequivalent (Spin(1, 1)) spin structures labeled by index ε = +1, −1. We use (e.g. as a base of the spin manifold) the quotient space R 1,1 * = R 1,1 \ {(0, 0)} (which as a set is coinciding with R 2 * = R 2 \ {(0, 0)}) with the metric induced from R 1,1 . The generator of Z 2 acts on R 1,1 * (or R 2 * ) by (x 1 , x 2 ) → (−x 1 , −x 2 ). The spin manifold (R 1,1 * /Z 2 ) ε has one parallel (i.e. covariantly constant) chiral spinor, or more rigorously, the pair of dimensions of spaces of chiral parallel spinors (n(1), n(−1)) is either (1, 0) (for ε = +1), or (0, 1) (for ε = −1).It should be noted that supersymmetric configurations with three M-branes, were considered earlier in [14,15]; some examples which contain R 1,1 * /Z 2 submanifold were presented in [15].In this paper we use notations from [13,15]. For any (spin) manifold M i (i = 0, ..., n, where n is the number of factor spaces) we denote by n i (c (i) ) the number of (linear independent) chiral parallel spinors η i with chiral number c (i) , satisfyingΓ (i) η i = c (i) η i , whereΓ (i) is the product of (frame) gamma-matrices corresponding to M i and by n i -the number of parallel spinors on M i . For oriented (e.g. spin) manifold M i with the metric g i we denote by τ i = τ i (g i ) the volume form corresponding to g i .

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“…An open problem here is to find exact solutions for a FCY cosmological model with small enough variations of G and α obeying modern observational restrictions. (Multidimensional cosmological models with small enough variations of G were considered previously in [20,21,22,23] while variations of α and G were obtained recently in [24,25,26] in the framework of nonlinear multidimensional gravity.) This can be a subject of a separate study.…”

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