String Theory and M-Theory

Appendix: Some basic geometry and topology 447
and one more contraction gives the scalar curvature
R = g µν Rµν. (9.255)
Holonomy groups
The holonomy group of a Riemannian manifold M of dimension d describes
the way various objects transform under parallel transport around closed
curves. The objects that are parallel transported can be tensors or spinors.
For spin manifolds (that is, manifolds that admit spinors), spinors are the
most informative. The reason is that the most general transformation of
a vector is a rotation, which is an element of SO(d). 33 The corresponding
transformation of a spinor, on the other hand, is an element of the covering
group Spin(d). So let us suppose that a spinor is parallel transported around
a closed curve. As a result, the spinor is rotated from its original orientation
ε → Uε, (9.256)
where U is an element of Spin(d) in the spinor representation appropriate
to ε. Now imagine taking several consecutive paths each time leaving and
returning to the same point. The result for the spinor after two paths is, for
example,
ε → U1U2ε = U3ε. (9.257)
As a result, the U matrices build a group, called the holonomy group H(M).
The generic holonomy group of a Riemannian manifold M of real dimension
d that admits spinors is Spin(d). Now one can consider different
special classes of manifolds in which H(M) is only a subgroup of Spin(d).
Such manifolds are called manifolds of special holonomy.
• H ⊆ U(d/2) if and only if M is Kähler.
• H ⊆ SU(d/2) if and only if M is Calabi–Yau.
• H ⊆ Sp(d/4) if and only if M is hyper-Kähler.
• H ⊆ Sp(d/4) · Sp(1) if and only if M is quaternionic Kähler.
In the first two cases d must be a multiple of two, and in the last two
cases it must be a multiple of four. Kähler manifolds and Calabi–Yau manifolds
are discussed later in this appendix. Hyper-Kähler and quaternionic
Kähler manifolds will not be considered further. There are two other cases
of special holonomy. In seven dimensions the exceptional Lie group G2 is
33 Reflections are avoided by assuming that the manifold is oriented.