strings also transform as a doublet. In general, they transform into (p, q)

strings, which carry both kinds of charge. The restriction to the SL(2, )

subgroup is essential to ensure that these charges are integers, as is required

by the Dirac quantization conditions.

Symmetry under gs ’ 1/gs

Recall that in type IIB supergravity the complex ¬eld

„ = C0 + ie’¦ (8.96)

transforms nonlinearly under SL(2, ) transformations. This remains true

¡

in the full string theory, but only for the discrete subgroup SL(2, ). In

particular, the transformation „ ’ ’1/„ , evaluated at C0 = 0, changes the

sign of the dilaton, which implies that the string coupling constant maps

to its inverse. This is an S-duality transformation like the one that relates

the type I superstring and SO(32) heterotic string theories. In this case it

relates the type IIB superstring theory to itself. Moreover, it is only one

element of the in¬nite duality group SL(2, ). This duality group bears a

striking resemblance to that of the N = 4 SYM theory discussed at the

beginning of this section. In Chapter 12 it is shown that this is not an

accident.

(p, q) strings

The (p, q) strings are all on an equal footing, so they are all supersymmetric,

in particular. This implies that each of their tensions saturates a BPS bound

given by supersymmetry, and this uniquely determines what their tensions

328 M-theory and string duality

are. In the string frame, the result turns out to be

2

q2

θ0

T(p,q) = |p ’ q„B |TF1 = TF1 p’q + 2, (8.97)

2π gs

where we have de¬ned the vev

θ0 i

„B = „ = + (8.98)

2π gs

and

1

TF1 = T(1,0) = . (8.99)

2π 2

s

This result can be derived by constructing the (p, q) strings as solitonic

solutions of the type IIB supergravity ¬eld equations. The fact that these

equations are only approximations to the superstring equations doesn™t mat-

ter for getting the tension right, since it is a consequence of supersymmetry.

Later, we con¬rm this tension formula by deriving it from a duality that

relates the type IIB theory to M-theory.

Note that the F-string tension formula is valid for all values of θ0 , but the

usual D-string tension formula

TF1

TD1 = T(0,1) = (8.100)

gs

is only valid for θ0 = 0. Note also that a (p, q) string with θ0 = 2π is

equivalent to a (p ’ q, q) string with θ0 = 0.

These (p, q) string tensions satisfy a triangle inequality

T(p1 +p2 ,q1 +q2 ) ¤ T(p1 ,q1 ) + T(p2 ,q2 ) , (8.101)

and equality requires that the vectors (p1 , q1 ) and (p2 , q2 ) are parallel. One

way of stating the conclusion is that a (p, q) string can be regarded as a

bound state of p F-strings and q D-strings. It has lower tension than any

other con¬guration with the same charges if and only if p and q are coprime.

If there is a common divisor, there exists a multiple-string con¬guration with

the same charges and tension.

Other BPS states

Let us brie¬‚y consider the SL(2, ) properties of the other BPS type IIB

branes:

• The D3-brane carries a charge that couples to the SL(2, ) singlet ¬eld

C4 . Therefore, it transforms as an SL(2, ) singlet, as well. This fact has

the interesting consequence that any (p, q) string can end on a D3-brane,

8.3 M-theory 329

since an SL(2, ) transformation that turns an F-string into a (p, q) string

leaves the D3-brane invariant.

• There exist stable supersymmetric (p, q) 5-branes, which are the magnetic

duals of (p, q) strings. Their SL(2, ) properties are quite similar to those

of the (p, q) strings.

• The D7-brane couples magnetically to C0 . This ¬eld transforms in a

rather complicated way under SL(2, ), so it is not immediately obvious

how to classify 7-branes. Although this issue won™t be pursued here, the

classi¬cation is important, because certain nonperturbative vacua of type

IIB superstring theory (described by F-theory) contain various 7-branes.

This is addressed later.

The de¬nition of a D-brane as a p-brane on which an F-string can end

has to be interpreted carefully for p = 1. A naive interpretation of “a

fundamental string ending on a D-string” would suggest a junction of three

string segments, one of which is (1, 0) and two of which are (0, 1). This

is not correct, however, because the charge on the end of the fundamental

string results in ¬‚ux that must go into one or the other of the attached

string segments, changing the string charge in the process. In short, the

three-string junction must satisfy charge conservation. This means that an

allowed junction of three strings with charges (p(i) , q (i) ) with i = 1, 2, 3 has

to satisfy

p(i) = q (i) = 0. (8.102)