± = ’1/6. This gives a result that is correct for large r, namely

1

A ∼ ’ log r. (9.199)

6

By the change of variables ρ = r 11/12 this brings the two-dimensional metric

to the asymptotic form

2

11

2 2 2

ds ∼ dρ + ρ dθ , (9.200)

12

which shows that there is a de¬cit angle of π/6 in the Einstein frame.

A more accurate solution, applicable for multiple 7-branes at positions

zi , i = 1, . . . , N , can be constructed as follows. The general solution of

Eq. (9.198) is

eA = |f (z)|2 Im „ (9.201)

9.11 K3 compacti¬cations and more string dualities 431

where f (z) is holomorphic. This function is determined by requiring mod-

ular invariance and r ’1/6 singularities at the cores of 7-branes. The result

is

N

(z ’ zi )’1/12 .

2

f (z) = [·(„ )] (9.202)

i=1

The Dedekind · function is

∞

·(„ ) = q 1/24 (1 ’ q n ), (9.203)

n=1

where

q = e2πi„ . (9.204)

Under a modular transformation the Dedekind · function transforms as

√

·(’1/„ ) = ’i„ ·(„ ). (9.205)

Thus, |·(„ )|4 Im „ is modular invariant.

Since all 7-branes are related by modular transformations that leave the

Einstein-frame metric invariant, it follows that in Einstein frame they all

have a de¬cit angle of π/6. Suppose that 7-branes (of various types) are

localized at (¬nite) points on the transverse space such that the total de¬cit

angle is

φi = 4π. (9.206)

Then the transverse space acquires the topology of a sphere with its cur-

vature localized at the positions of the 7-branes, and the z-plane is bet-

ter described as a projective space P 1 . Since every de¬cit angle is π/6,

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Eq. (9.206) requires that there are a total of 24 7-branes. However, the

choice of which types of 7-branes to use, and how to position them, is not

completely arbitrary. For one thing, it is necessary that the monodromy

associated with a circle that encloses all of them should be trivial, since the

circle can be contracted to a point on the other side of the sphere without

crossing any 7-branes.

The „ parameter is well de¬ned up to an SL(2, ) transformation every-

where except at the positions of the 7-branes, where it becomes singular. A

nicer way of expressing this is to say that one can associate a torus with

complex-structure modulus „ (z) with each point in the z-plane. This gives

a T 2 ¬bration with base space P 1 , where the 24 singular ¬bers correspond

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to the positions of the 7-branes. Such a T 2 ¬bration is also called an elliptic

432 String geometry

¬bration. Only the complex structure of the torus is speci¬ed by the mod-

ulus „ . Its size (or K¨hler structure) is not a dynamical degree of freedom.

a

Recall that the type IIB theory can be obtained by compactifying M-theory

on a torus and letting the area of the torus shrink to zero. In this limit

the modular parameter of the torus gives the „ parameter of the type IIB

theory. Therefore, the best interpretation is that the torus in the F-theory

construction has zero area.

A nice way of describing the complex structure of a torus is by an algebraic

equation of the form

y 2 = x3 + ax + b. (9.207)

This describes the torus as a submanifold of 2 , which is parametrized by

£

complex numbers x and y. The constants a and b determine the complex

structure „ of the torus. There is no metric information here, so the area

is unspeci¬ed. The torus degenerates, that is, „ is ill-de¬ned, whenever the

discriminant of this cubic vanishes. This happens for

27a3 ’ 4b2 = 0. (9.208)

Thus, the positions of the 7-branes correspond to the solutions of this equa-

tion. To ensure that z = ∞ is not a solution, we require that a3 and b2 are

polynomials of the same degree.

Since there should be 24 7-branes, the equation should have 24 solutions.

Thus, a = f8 (z) and b = f12 (z), where fn denoted a polynomial of degree

n. The total space can be interpreted as a K3 manifold that admits a

T 2 ¬bration. The only peculiar feature is that the ¬bers have zero area.

Let us now count the number of moduli associated with this construction.

The polynomials f8 and f12 have arbitrary coe¬cients, which contribute

9 + 13 = 22 complex moduli. However, four of these are unphysical because

of the freedom of an SL(2, ) transformation of the z-plane and a rescaling

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f8 ’ »2 f8 , f12 ’ »3 f12 . This leaves 18 complex moduli. In addition there

is one real modulus (a K¨hler modulus) that corresponds to the size of the

a

1 base space. The complex moduli parametrize the positions of the 7-

P

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branes (modulo SL(2, )) in the z-plane. The fact that there are fewer

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than 21 such moduli shows that the positions of the 7-branes (as well as

their monodromies) is not completely arbitrary.

Remarkably, there is a dual theory that has the same properties. The

heterotic string theory compacti¬ed on a torus to eight dimensions has 16

unbroken supersymmetries and the moduli space

+

— M18,2 . (9.209)

¡

9.12 Manifolds with G2 and Spin(7) holonomy 433

The real modulus is the string coupling constant, which therefore corre-

sponds to the area of the P 1 in the F-theory construction. The second

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