manifolds: 2 = — , — T 2 .

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Compact examples: T 4 , K3

Requiring a covariantly constant spinor is very restrictive in four real di-

mensions. In fact, K3 and T 4 are the only two examples of four-dimensional

compact K¨hler manifolds for which they exist. As a result, these mani-

a

folds are the only examples of Calabi“Yau two-folds. If one requires the

holonomy to be SU (2), and not a subgroup, then only K3 survives. By con-

trast, there are very many (possibly in¬nitely many) Calabi“Yau three-folds.

Since K3 and T 4 are Calabi“Yau manifolds, they admit a Ricci-¬‚at K¨hler

a

metric. Moreover, since SU (2) = Sp(1), it turns out that they are also

hyper-K¨hler.6 The explicit form of the Ricci-¬‚at metric of a smooth K3

a

6 In general, a 4n-dimensional manifold of Sp(n) holonomy is called hyper-K¨hler. The notation

a

U Sp(2n) is also used for the same group when one wants to emphasize that the compact form

is being used. Both notations are used in this book.

9.3 Examples of Calabi“Yau manifolds 367

is not known. However, K3 can be described in more detail in the orbifold

limit, which we present next.

Orbifold limit of K3

A singular limit of K3, which is often used in string theory, is an orbifold

of the T 4 . This has the advantage that it can be made completely explicit.

Consider the square T 4 constructed by taking 2 and imposing the following

£

four discrete identi¬cations:

za ∼ za + 1 z a ∼ z a + i, a = 1, 2. (9.21)

There is a isometry group generated by

2

I : (z 1 , z 2 ) ’ (’z 1 , ’z 2 ). (9.22)

This 2 action has 16 ¬xed points, for which each of the z a takes one of the

following four values

1 i 1+i

0, , , . (9.23)

22 2

Therefore, the orbifold T 4 / 2 has 16 singularities. These singularities can

be repaired by a mathematical operation called blowing up the singularities

of the orbifold.

Blowing up the singularities

The singular points of the orbifold described above can be “repaired” by

the insertion of an Eguchi“Hanson space. The way to do this is to excise a

small ball of radius a around each of the ¬xed points. The boundary of each

ball is S 3 / 2 since opposite points on the sphere are identi¬ed, according to

Eq. (9.22). One excises each ball and replaces it by a smooth noncompact

Ricci-¬‚at K¨hler manifold whose boundary is S 3 / 2 . The unique manifold

a

that has an S 3 / 2 boundary and all the requisite properties to replace each

of the 16 excised balls is an Eguchi“Hanson space. The metric of the Eguchi“

Hanson space is

1 1

ds2 = ∆’1 dr2 + r2 ∆(dψ + cos θdφ)2 + r2 d„¦2 , (9.24)

2

4 4

with ∆ = 1 ’ (a/r)4 and d„¦2 = dθ2 + sin2 θdφ2 . The radial coordinate is in

2

the range a ¤ r ¤ ∞, where a is an arbitrary constant and ψ has period 2π.

Repairing the singularities in this manner gives a manifold with the desired

topology, but the metric has to be smoothed out to give a true Calabi“Yau

geometry. The orbifold then corresponds to the limit a ’ 0. The nonzero

368 String geometry

Hodge numbers of the Eguchi“Hanson space are h0,0 = h1,1 = h2,2 = 1.

Moreover, the (1, 1)-form is anti-self-dual and is given by

1 1

rdr § (dψ + cos θdφ) ’ r2 sin θdθ § dφ,

J= (9.25)

2 4

as you are asked to verify in a homework problem. In terms of the complex

coordinates

i i

(ψ ’ φ) ,

z1 = r cos (θ/2) exp (ψ + φ) and z2 = r sin (θ/2) exp

2 2

(9.26)

the metric is K¨hler with K¨hler potential

a a

r2 exp(r4 + a4 )1/2

K = log . (9.27)

a2 + (r4 + a4 )1/2

Hodge numbers of K3

The cohomology of K3 can be computed by combining the contributions

of the T 4 and the Eguchi“Hanson spaces. The result obtained in this way

remains correct after the metric has been smoothed out.

The Eguchi“Hanson spaces contribute a total of 16 generators to H 1,1 ,

one for each of the 16 spaces used to blow up the singularities. Moreover, on

the T 4 the following four representatives of H 1,1 cohomology classes survive

the 2 identi¬cations:

dz 1 § d¯1 , dz 2 § d¯2 , dz 1 § d¯2 , dz 2 § d¯1 .

z z z z (9.28)

This gives in total h1,1 = 20. In addition, there is one H 2,0 class represented

by dz 1 § dz 2 and one H 0,2 class represented by d¯1 § d¯2 . As a result, the

z z