#### 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-CivitÃ
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).