noncompact space-time dimensions and wrapping (8 ’ d)-cycles in the com-

pact dimensions. Therefore, before explaining F-theory, it is necessary to

discuss the classi¬cation and basic properties of 7-branes. 7-branes in ten

dimensions are codimension two, and so they can be enclosed by a circle,

just as is the case for a point particle in three dimensions and a string in

four dimensions. Just as in those cases, the presence of the brane creates

a de¬cit angle in the orthogonal plane that is proportional to the tension

of the brane. Thus, a small circle of radius R, centered on the core of the

brane, has a circumference (2π ’ φ)R, where φ is the de¬cit angle. In fact,

this property is the key to searching for cosmic strings that might stretch

across the sky.

The fact that ¬elds must be single-valued requires that, when they are

analytically continued around a circle that encloses a 7-brane, they return

to their original values up to an SL(2, ) transformation. The reason for

this is that SL(2, ) is a discrete gauge symmetry, so that the con¬guration

space is the naive ¬eld space modded out by this gauge group. So the

requirement stated above means that ¬elds should be single-valued on this

quotient space. The ¬eld „ , in particular, can have a nontrivial monodromy

transformation like that in Eq. (9.188). Other ¬elds, such as B2 and C2 ,

must transform at the same time, of course.

Since 7-branes are characterized by their monodromy, which is an SL(2, )

transformation, there is an in¬nite number of di¬erent types. In the case of a

D7-brane, the monodromy is „ ’ „ +1. This implies that 2πC0 is an angular

coordinate in the plane perpendicular to the brane. More precisely, the 7-

brane is characterized by the conjugacy class of its monodromy. If there is

another 7-brane present the path used for the monodromy could circle the

other 7-brane then circle the 7-brane of interest, and ¬nally circle the other

7-brane in the opposite direction. This gives a monodromy described by a

di¬erent element of SL(2, ) that belongs to the same conjugacy class and

is physically equivalent. The conjugacy classes are characterized by a pair

of coprime integers (p, q). This is interpreted physically as labelling the type

9.11 K3 compacti¬cations and more string dualities 429

of IIB string that can end on the 7-brane. In this nomenclature, a D7-brane

is a (1, 0) 7-brane, since a fundamental string can end on it.

Let us examine the type IIB equations of motion in the supergravity

approximation. The relevant part of the type IIB action, described in Ex-

ercise 8.3, is

1√ ‚µ „ ‚ ν „

¯

’g R ’ g µν d10 x. (9.189)

2

2 (Im„ )

To describe a 7-brane, let us look for solutions that are independent of

the eight dimensions along the brane, which has a ¬‚at Lorentzian metric,

and parametrize the perpendicular plane as the complex plane with a local

coordinate z = reiθ . The idea is that the brane should be localized at the

origin of the z-plane. Now let us look for a solution to the equations of

motion in the gauge in which the metric in this plane is conformally ¬‚at

ds2 = eA(r,θ) (dr2 + r2 dθ2 ) ’ (dx0 )2 + (dx1 )2 + . . . + (dx7 )2 . (9.190)

Just as in the case of the string world sheet, the conformal factor cancels

out of the „ kinetic term. Therefore, its equation of motion is the same as

in ¬‚at space. The „ equation of motion is satis¬ed if „ is a holomorphic

function „ (z), as you are asked to verify in a homework problem.

The elliptic modular function j(„ ) gives a one-to-one holomorphic map

of the fundamental region of SL(2, ) onto the entire complex plane. It

is invariant under SL(2, ) modular transformations, and it has a series

expansion of the form

∞

cn e2πin„

j(„ ) = (9.191)

n=’1

with c’1 = 1. Its leading asymptotic behavior for Im „ ’ +∞ is given by

the ¬rst term

j(„ ) ∼ e’2πi„ . (9.192)

If we choose the holomorphic function „ (z) to be given by

j „ (z) = Cz, (9.193)

where C is a constant, then for large z

1

„ (z) ∼ ’

log z. (9.194)

2πi

This exhibits the desired monodromy „ ’ „ ’ 1 as one encircles the 7-

brane.31

31 To get „ ’ „ + 1 instead, one could replace z by z , which corresponds to replacing the brane

¯

by an antibrane.

430 String geometry

The tension of the 7-brane is given by

¯„ ¯ ¯

‚„ · ‚ „

1 ¯ 1 ‚„ ‚¯ + ‚„ ‚ „

d2 x d2 x

T7 = = . (9.195)

(Im „ )2 (Im „ )2

2 2

Now let us evaluate this for the solution proposed in Eq. (9.193). Since „ is

holomorphic

¯¯ d2 „

1 ‚„ ‚ „ 1 π

2

T7 = dx = =. (9.196)

(Im „ )2 2 F (Im „ )2

2 6

This has used the fact that the inverse image of the complex plane is the

fundamental region F. The volume of the moduli space was evaluated in

Exercise 3.9.

The integrand in Eq. (9.196) is the energy density that acts as a source

for the gravitational ¬eld in the Einstein equation

¯„

1 1 ’A ‚„ ‚¯

R00 ’ g00 R = ’ g00 e . (9.197)

(Im „ )2

2 2

Evaluating the curvature for the metric in Eq. (9.190), one obtains the equa-

tion

¯¯

¯ = ’ 1 ‚„ ‚ „ = ‚ ‚ log Im „.

¯

‚ ‚A (9.198)

2 („ ’ „ )2

¯

The energy density is concentrated within a string-scale distance of the

origin, where the supergravity equations aren™t reliable. The total energy is

reliable because of supersymmetry (saturation of the BPS bound), however.

So, to good approximation, we can take A = ± log r and use 2 log r =

2πδ 2 (x) to approximate the energy density by a delta function at the core.