In 1981, covariantly constant spinors were introduced into Kaluza-Klein
theory as a way of counting the number of supersymmetries surviving
compactification. These are related to the holonomy group of the compactifying
manifold. The first non-trivial example was provided in 1982 by D=11
supergravity on the squashed S7, whose G2 holonomy yields N=1 in D=4. In 1983,
another example was provided by D=11 supergravity on K3, whose SU(2) holonomy
yields half the maximum supersymmetry. In 2002, G2 and K3 manifolds continue to
feature prominently in the full D=11 M-theory and its dualities. In particular,
singular G2 compactifications can yield chiral (N=1,D=4) models with realistic
gauge groups. The notion of generalized holonomy is also discussed.Comment: Notes added on n, the number of allowed M-theory supersymmetries.
Asymmetric orbifold compactifications of Type II strings from D=10 to D=2
permit n=0,1,2,3,4,5,6,8,9,10,12,16,17,18,20,24,3
