pansion in ± is an expansion in “stringiness” about the point-particle

limit. Mathematically, it is the perturbation expansion that corresponds

to quantum-mechanical treatment of the string world-sheet theory, even

though it concerns the classical physics of a string. (Recall that the world-

sheet action has a coe¬cient 1/± , so that ± plays a role analogous to

Planck™s constant.) Since ± has dimensions of (length)2 , the dimension-

less expansion parameter can be ± p2 , where p is a characteristic momen-

tum or energy, or ± /L2 , where L is a characteristic length scale, such as

the size of a compact dimension.

• The second expansion is the one in the string coupling constant gs , which

is the expectation value of the exponentiated dilaton ¬eld. This is the

expansion in the number of string loops or, equivalently, the genus of the

string world sheet.

S-duality and T-duality are quite analogous. However, S-duality seems

deeper in that it is nonperturbative in the string loop expansion, whereas

T-duality holds order by order in the loop expansion. In particular, it is

valid in the leading (tree or classical) approximation.

Type I superstring “ SO(32) heterotic string duality

The low-energy e¬ective actions for the type I and SO(32) heterotic theories

are very similar. In particular, they are mapped into one another by the

simple transformation

¦ ’ ’¦ (8.88)

combined with a Weyl rescaling of the metric

gµν ’ e’¦ gµν . (8.89)

Thus the canonical Einstein metric gµν = e’¦/2 gµν is an invariant combina-

E

tion. All other bosonic ¬elds remain unchanged (A ” A and B2 ” C2 ).

11 The discussion that follows applies to any of the superstring theories.

8.2 S-duality 325

This leads to the conjecture that the two string theories (not just their

low-energy limits) are actually dual to one another, which means that they

are descriptions in two di¬erent regions of the parameter space of one and

the same quantum theory. Since the string coupling constant is the vev of

exp(¦) in each case, Eq. (8.88) implies that the type I superstring coupling

constant is the reciprocal of the SO(32) heterotic string coupling constant,

IH

gs gs = 1. (8.90)

Thus, when one of the two theories is weakly coupled, the other one is

strongly coupled. This, of course, makes proving the type I“heterotic duality

di¬cult. Some checks, beyond the analysis of the e¬ective actions described

above, can be made and no discrepancy has been found. More signi¬cantly,

this is one link in an intricate overconstrained web of dualities. If any of

them were wrong, the whole story would fall apart.

Nonperturbative test

As an example of a nonperturbative test of the duality, consider the D-string

of the type I theory, whose tension is

11

TD1 = . (8.91)

gs 2π 2

s

Let us test the conjecture that this string actually is the SO(32) heterotic

string, whose tension is

1

TF1 = , (8.92)

2π 2

s

continued from weak coupling to strong coupling. The D-string is a super-

symmetric object that saturates a BPS bound, and therefore the tension

formula ought to be exact for all values of gs . To compare these formulas

one must realize that although the physical values of s are the same in the

two cases, they are being measured in di¬erent metrics, as a consequence of

the Weyl rescaling in Eq. (8.89). Thus

√

’ gs . (8.93)

s s

Combined with the rule gs ’ 1/gs , this indeed implies that the tensions TD1

√

and TF1 agree. Note that the transformation gs ’ 1/gs , s ’ s gs squares

to the identity, and so it is the same as its inverse.

The tensions of the magnetically-charged 5-branes that are dual to these

strings can be compared in similar fashion. This is guaranteed to work by

326 M-theory and string duality

what has already been said, but let™s check it anyway. In the type I theory

1

TD5 = , (8.94)

gs (2π)5 6

s

and in the heterotic theory

1

TNS5 = . (8.95)

(gs )2 (2π)5 6

s

Once again, these map into one another in the required fashion.

The fundamental type I string

Having seen that the SO(32) heterotic string can be identi¬ed with the type

I D-string, one might wonder whether one can also identify a counterpart

for the fundamental type I string in the SO(32) heterotic theory. To answer

this it is important to understand the essential di¬erence between the two

types of strings. The type I F-string does not carry a conserved charge, and

it is not supersymmetric. The two-form B2 , which is the ¬eld that couples

to a fundamental type IIB string, is removed from the spectrum by the

orientifold projection. There are two ways of thinking about the reason that

a type I F-string can break, both of which are correct. One is that there

are space-time-¬lling D9-branes, and fundamental strings can break on D-

branes. The other one is that since it does not carry a conserved charge,

and it is not supersymmetric, there is no conservation law that prevents it

from breaking. The amplitude for breaking a type I string is proportional

√

to gs , so these strings can be long-lived for su¬ciently small coupling

constant. This is good enough for making them the fundamental objects

on which to base a perturbation expansion. However, if the type I coupling

constant is large, the type I F-strings are no longer a useful concept, since

they disintegrate as shown in Fig. 8.1. Accordingly, there is no trace of them

in the weakly-coupled heterotic description.

Fig. 8.1. The fundamental type I string disintegrates at strong coupling.

8.2 S-duality 327

Type IIB S-duality

Type IIB supergravity has a global SL(2, ) symmetry that was described

¡

earlier. However this symmetry of the low-energy e¬ective action is not

shared by the full type IIB superstring theory. Indeed, it is broken by

a variety of stringy and quantum e¬ects to the in¬nite discrete subgroup

SL(2, ). One way of seeing this is to think about stable strings in this

theory. Since there are two two-form gauge ¬elds B2 (NS“NS two-form) and

C2 (R“R two-form) there are two types of charge that a string can carry.

The F-string (or fundamental string) has charge (1, 0), which means that it

has one unit of the charge that couples to B2 and none of the charge that

couples to C2 . In similar fashion, the D-string couples to C2 and has charge

(0, 1). Since the two-forms form a doublet of SL(2, ) it follows that these

¡