Holonomy approach to supersymmetric supergravity backgrounds

In Breaking the M-waves I initiated the holonomy analysis of supersymmetric M-theory backgrounds (in the supergravity approximation). I focused on the simpler case of solutions without flux, for which the holonomy group is a subgroup of the structure group SO(1,10) of the tangent bundle. This required studying the holonomy groups of the Levi-Cività connection of a lorentzian manifold, which is a more delicate problem than the riemannian case, solved by Berger and others in what is now a classic result in differential geometry. The reason why this is a more delicate problem is that the lorentzian version of the de Rham decomposition theorem (due to Wu) is weaker and does not reduce the problem to irreducible holonomy representations, but one is forced to study reducible yet indecomposable holonomy representations — a problem which is still unsolved. My aim was simpler, though, in that I was interested only in those holonomy groups of (Ricci-flat) manifolds admitting parallel spinors; that is, the lorentzian analogue of the results of Wang in riemannian signature. With some help from Robert Bryant, who put on firmer footing my initial experiments in computing spinor isotropy subgroups in Spin(1,10), I framed a conjecture for the lorentzian analogue of the Wang list in dimension ≤ 11 and constructed examples of local metrics with each of these holonomy groups in terms of warped products. This conjecture was eventually proved and generalised to arbitrary dimension by Thomas Leistner, a student of Helga Baum's.

These lorentzian holonomy groups admitting parallel spinors also govern the worldvolume geometries of supersymmetric brane solutions, as was pointed out in More Ricci-flat branes, commenting on a paper of Malcolm Perry and his student Dominic Brecher.

In February 2001, George Papadopoulos visited Edinburgh to give a geometry seminar on supersymmetric supergravity backgrounds. During his visit we decided to apply a holonomy approach to the classification of supersymmetric supergravity backgrounds: the holonomy representation is now that of the connection defined by the gravitino variation, which in Maximal supersymmetry in ten and eleven dimensions, I imprudently claimed that it was in GL(32,ℝ), not realising, as was later observed by Chris Hull, that it is actually in SL(32,ℝ).

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