signature. The Hodge -operator acting on p-forms is de¬ned as

µµ1 ···µp µp+1 ···µd

µ1 µp

· · · gµd νd dxνp+1 § · · · § dxνd .

(dx § · · · § dx ) = g

1/2 µp+1 νp+1

(d ’ p)!|g|

(9.241)

The Levi“Civita symbol µ transforms as a tensor density, while µ/|g|1/2 is a

tensor. A p-form A is said to be harmonic if and only if

∆p A = 0. (9.242)

Harmonic p-forms are in one-to-one correspondence with the elements of the

group H p (M ). Indeed, from the de¬nition of the Laplace operator it follows

that if Ap is harmonic

(dd† + d† d)Ap = 0, (9.243)

and as a result

(Ap , (dd† + d† d)Ap ) = 0 ’ (d† Ap , d† Ap ) + (dAp , dAp ) = 0. (9.244)

Using a positive-de¬nite scalar product it follows that Ap is closed and co-

closed. The Hodge theorem states that on a compact manifold that has a

positive de¬nite metric a p-form has a unique decomposition into harmonic,

exact and co-exact pieces

Ap = Ah + dAe. + d† Ac.e. . (9.245)

p p’1 p’1

As a result, a closed form can always be written in the form

Ap = Ah + dAe. . (9.246)

p p’1

Since the Hodge dual turns a closed p-form into a co-closed (d ’ p)-form

and vice versa, it follows that the Hodge dual provides an isomorphism

between the space of harmonic p-forms and the space of harmonic (d ’ p)-

forms. Therefore,

bp = bd’p . (9.247)

The connection

Another fundamental geometric concept is the connection. There are ac-

tually two of them: the a¬ne connection and the spin connection, though

they are related (via the vielbein). Connections are not tensors, though the

arbitrariness in their de¬nitions corresponds to adding a tensor. Also, they

are used in forming covariant derivatives, which are constructed so that they

446 String geometry

map tensors to tensors. The expressions for the connections can be deduced

from the fundamental requirement that the vielbein is covariantly constant

±

= ‚µ e± ’ “ρ e± + ωµ ± β eβ = 0.

µ eν (9.248)

ν µν ρ ν

This equation determines the a¬ne connection “ and the spin connection ω

up to a contribution characterized by a torsion tensor, which is described in

Chapter 10. The a¬ne connection, for example, is given by the Levi“Civita

connection plus a torsion contribution

ρ

“ρ = + Kµν ρ , (9.249)

µν

µν

where the Levi“Civita connection is

1

ρ

= g ρ» (‚µ gν» + ‚ν gµ» ’ ‚» gµν ), (9.250)

µν 2

and K is called the contortion tensor. The formula for the spin connection,

given by solving Eq. (9.248), is

ωµ ± β = ’eν (‚µ e± ’ “» e± ). (9.251)

ν µν »

β

Curvature tensors

The curvature tensor can be constructed from either the a¬ne connection “

or the spin connection ω. Let us follow the latter route. The spin connection

is a Lie-algebra valued one-form ω ± β = ωµ ± β dxµ . The algebra in question is

SO(d), or a noncompact form of SO(d) in the case of inde¬nite signature.

Thus, it can be regarded as a Yang“Mills gauge ¬eld. The curvature two-

form is just the corresponding ¬eld strength,

R± β = dω ± β + ω ± γ § ω γ β , (9.252)

which in matrix notation becomes

R = dω + ω § ω. (9.253)

Its components have two base-space and two tangent-space indices Rµν ± β .

One can move indices up and down and convert indices from early Greek

to late Greek by contracting with metrics, vielbeins and their inverses. In

particular, one can form Rµ νρ» , which coincides with the Riemann curvature

tensor that is usually constructed from the a¬ne connection. Contracting

a pair of indices gives the Ricci tensor

Rν» = Rµ νµ» , (9.254)

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

µ ’ U µ, (9.256)

where U is an element of Spin(d) in the spinor representation appropriate

to µ. 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,

µ ’ U1 U2 µ = U3 µ. (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 di-

mension d that admits spinors is Spin(d). Now one can consider di¬erent

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 K¨hler.

a

• H ⊆ SU (d/2) if and only if M is Calabi“Yau.

• H ⊆ Sp(d/4) if and only if M is hyper-K¨hler.

a

• H ⊆ Sp(d/4) · Sp(1) if and only if M is quaternionic K¨hler.

a

In the ¬rst two cases d must be a multiple of two, and in the last two

cases it must be a multiple of four. K¨hler manifolds and Calabi“Yau man-

a

ifolds are discussed later in this appendix. Hyper-K¨hler and quaternionic

a

K¨hler manifolds will not be considered further. There are two other cases

a

of special holonomy. In seven dimensions the exceptional Lie group G2 is

33 Re¬‚ections are avoided by assuming that the manifold is oriented.

448 String geometry

a possible holonomy group, and in eight dimensions Spin(7) is a possible

holonomy group. The G2 case is of possible physical interest in the context

of compactifying M-theory to four dimensions.