M-branes at angles

In contrast to the current rapid evolution in our understanding of the microscopic degrees of freedom of M-theory, the macroscopic picture has changed very little over the past year. The ‘basic’ ingredients remain D=11 supergravity and its 1/2-supersymmetic solutions: the M-wave, the M-2-brane, the M-5-brane and, in the S KaluzaKlein vacuum, the M-monopole (i.e. EuclideanTaub-Nut times 7-dimensional Minkowski). The novelty has been in the way that these ingredients have been combined to form new ‘intersectingbrane’ configurations, preserving some smaller fraction of supersymmetry, and in the remarkable insights into quantum field theory that these configurations have led to. I shall have nothing to say here about these applications of intersecting brane configurations. Instead, I shall discuss one aspect of attempts towards a classification of them: the conditions for partial preservation of supersymmetry by configurations of nonorthogonally intersecting M-branes. While it is not difficult to see why orthogonal intersections of branes can partially preserve supersymmetry, it is less obvious that this is also true for non-orthogonal intersections. This possibility was first pointed out by Berkooz, Douglas and Leigh [1], who also provided several D-brane examples, which they interpreted as rotations within an SU(n) subgroup of SO(2n). Further examples, interpretable as rotations within an Sp(2) subgroup of SO(8), were given by Gauntlett, Gibbons, Papadopoulos and the author [2] and shown to be related via M-theory dualities to 8dimensional hyper-Kähler manifolds. I shall review both cases here although the interpretation as rotations within subgroups of reduced holonomy and the connection with (hyper)complex geometry will not be explained. Instead I shall concentrate on clarifying some aspects of the Dirac matrix algebra needed to determine the fraction of supersymmetry preserved by rotated M-brane configurations, i.e. ‘M-branes at angles’. It is convenient to start with two parallel M-5branes. Essentially no generality is lost by considering only M-5-branes, and any non-parallel configuration of two intersecting M-5-branes can obtained by some rotation of one of them from the parallel configuration. We may assume that the two parallel M-5-branes lie in the 12345 5-plane. This configuration can be represented by the array