String Theory and M-Theory

9.2 Calabi–Yau manifolds: mathematical properties 363
Then, in general, the only components of Qα that are invariant under g are
those whose weights εα have the same sign for all n components, since then
ε a αφ a = 0. In special cases, other components may also be invariant. For
each value of α for which the supercharge is not invariant, the amount of
unbroken supersymmetry is cut in half. Thus, if there is invariance for only
one value of α, the fraction of the supersymmetry that is unbroken is 2 1−n .
This chapter shows that the same fraction of supersymmetry is preserved
by compactification on a Calabi–Yau n-fold. In fact, some orbifolds of this
type are singular limits of smooth Calabi–Yau manifolds.
9.2 Calabi–Yau manifolds: mathematical properties
Definition of Calabi–Yau manifolds
By definition, a Calabi–Yau n-fold is a Kähler manifold having n complex
dimensions and vanishing first Chern class
c1 = 1
[R] = 0. (9.7)
2π
A theorem, conjectured by Calabi and proved by Yau, states that any compact
Kähler manifold with c1 = 0 admits a Kähler metric of SU(n) holonomy.
As we will see below a manifold with SU(n) holonomy admits a spinor
field which is covariantly constant and as a result is necessarily Ricci flat.
This theorem is only valid for compact manifolds. In order for it to be valid
in the noncompact case, additional boundary conditions at infinity need to
be imposed. As a result, metrics of SU(n) holonomy correspond precisely
to Kähler manifolds of vanishing first Chern class.
We will motivate the above theorem by showing that the existence of a
covariantly constant spinor implies that the background is Kähler and has
c1 = 0. A fundamental theorem states that a compact Kähler manifold has
c1 = 0 if and only if the manifold admits a nowhere vanishing holomorphic
n-form Ω. In local coordinates
Ω(z 1 , z 2 , . . . , z n ) = f(z 1 , z 2 , . . . , z n )dz 1 ∧ dz 2 · · · ∧ dz n . (9.8)
In section 9.5 we will establish the vanishing of c1 by explicitly constructing
Ω in backgrounds of SU(n) holonomy.
Hodge numbers of a Calabi–Yau n-fold
Betti numbers are fundamental topological numbers associated with a manifold.
4 The Betti number bp is the dimension of the pth de Rham cohomology
4 There is more discussion of this background material in the appendix of this chapter.