dAp = (9.226)

p!

A crucial property that follows from this de¬nition is that the operator d is

nilpotent, which means that d2 = 0. This can be illustrated by applying d2

442 String geometry

to a zero form

‚ 2 A0

‚A0 µ

dxµ § dxν ,

ddA0 = d dx = (9.227)

‚xµ µ ‚xν

‚x

which vanishes due to antisymmetry of the wedge product. A p-form is

called closed if

dAp = 0, (9.228)

and exact if there exists a globally de¬ned (p ’ 1)-form Ap’1 such that

Ap = dAp’1 . (9.229)

A closed p-form can always be written locally in the form dAp’1 , but this

may not be possible globally. In other words, a closed form need not be

exact, though an exact form is always closed.

Let us denote the space of closed p-forms on M by C p (M ) and the space

of exact p-forms on M by Z p (M ). Then the pth de Rham cohomology group

H p (M ) is de¬ned to be the quotient space

H p (M ) = C p (M )/Z p (M ). (9.230)

H p (M ) is the space of closed forms in which two forms which di¬er by an

exact form are considered to be equivalent. The dimension of H p (M ) is

called the Betti number. Betti numbers are very basic topological invariants

characterizing a manifold. The Betti numbers of S 2 and T 2 are described in

Fig. 9.14. Another especially important topological invariant of a manifold

is the Euler characteristic, which can be expressed as an alternating sum of

Betti numbers

d

(’1)i bi (M ).

χ(M ) = (9.231)

i=0

The Betti numbers of a manifold also give the dimensions of the homology

groups, which are de¬ned in a similar way to the cohomology groups. The

analog of the exterior derivative d is the boundary operator δ, which acts

on submanifolds of M . Thus, if N is a submanifold of M , then δN is its

boundary. This operator associates with every submanifold its boundary

with signs that take account of the orientation. The boundary operator is

also nilpotent, as the boundary of a boundary is zero. Therefore, it can

be used to de¬ne homology groups of M in the same way that the exterior

derivative was used to de¬ne cohomology groups of M . Arbitrary linear

combinations of submanifolds of dimension p are called p-chains. Here again,

to be more precise, one should say what type of coe¬cients is used to form

Appendix: Some basic geometry and topology 443

Fig. 9.14. The Betti numbers bp count the number of p-cycles which are not bound-

aries. For the sphere all one-cycles can be contracted to a point and the Betti

numbers are b0 = b2 = 1 and b1 = 0. The torus supports nontrivial one-cycles and

as a result the Betti numbers are b0 = b2 = 1 and b1 = 2.

the linear combinations. A chain that has no boundary is called closed, and

a chain that is a boundary is called exact. A closed chain zp , also called a

cycle, satis¬es

δzp = 0. (9.232)

The simplicial homology group Hp (M ) is de¬ned to consist of equivalence

classes of p-cycles. Two p-cycles are equivalent if and only if their di¬erence

is a boundary.

Poincar´ duality

e

A fundamental theorem is Stokes™ theorem. Given a real manifold M , let

A be an arbitrary p-form and let N be an arbitrary (p + 1)-chain. Then

Stokes™ theorem states

dA = A. (9.233)

N δN

This formula provides an isomorphism between H p (M ) and Hd’p (M ) that

is called Poincar´ duality. To every closed p-form A there corresponds a

e

(d ’ p)-cycle N with the property

A§B = B, (9.234)

M N

for all closed (d ’ p)-forms B. The fact that the left-hand side only depends

on the cohomology class of A and the right-hand side only depends on the

homology class of N is an immediate consequence of Stokes™ theorem and

the fact that M has no boundary. Poincar´ duality allows us to determine

e

the Betti numbers of a manifold by counting the nontrivial cycles of the

manifold. For example, S N has Betti numbers b0 = 1, b1 = 0, . . . , bN = 1.

444 String geometry

Riemannian geometry

Metric tensor

The manifolds described so far are entirely characterized by their topol-

ogy. Next, we consider manifolds endowed with a metric. If the metric

is positive de¬nite, the manifold is called a Riemannian manifold. If it

has inde¬nite signature, as in the case of general relativity, it is called a

pseudo-Riemannian manifold. In either case the metric is a symmetric ten-

sor characterized by an in¬nitesimal line element

ds2 = gµν (x)dxµ dxν , (9.235)

which allows one to compute the length of a curve by integration. The

line element itself is coordinate independent. This fact allows one to com-

pute how the metric components gµν (x) transform under general coordinate

transformations (di¬eomorphisms).

The metric tensor can be expressed in terms of the frame. This consists

of d linearly independent one-forms e± that are de¬ned locally on M . In

terms of a basis of one-forms

e± = e± dxµ . (9.236)

µ

The components e± form a matrix called the vielbein. Let · ±β and ·±β

µ

denote the ¬‚at metric whose only nonzero entries are ±1 on the diagonal.

In the Riemannian case (Euclidean signature) · is the unit matrix. In the

Lorentzian case, there is one ’1 corresponding to the time direction. The

metric tensor is given in terms of the frame by

g = ·±β e± — eβ . (9.237)

In terms of components this corresponds to

gµν = ·±β e± eβ . (9.238)

µν

The inverse vielbein and metric are denoted eµ and g µν .

±

Harmonic forms

The metric is needed to de¬ne the Laplace operator acting on p-forms on a

d-dimensional space given by

∆p = d† d + dd† = (d + d† )2 , (9.239)

where

d† = (’1)dp+d+1 d (9.240)

Appendix: Some basic geometry and topology 445