metry transformations by truncating to an N = 1 subsector and keeping

only the NS“NS ¬elds. A nice feature of this formulation is that the H3

contribution to δΨµ can be interpreted as torsion.

EXERCISES

EXERCISE 8.3

The previous section described the global symmetry of the type IIB super-

gravity action using a matrix M. Verify the identities

‚ µ „ ‚µ „

¯

1 1

’1

µ

= ’ ‚ µ ¦‚µ ¦ + e2¦ ‚ µ C0 ‚µ C0 .

tr(‚ M‚µ M ) = ’

2(Im„ )2

4 2

Verify the SL(2, ) invariance of this expression.

¡

322 M-theory and string duality

SOLUTION

By de¬nition „ = C0 + ie’¦ and

|„ |2 ’C0

¦

M=e .

’C0 1

As a result,

1 C0

M’1 = e¦ .

C0 |„ |2

So

1 1 1

tr(‚ µ M‚µ M’1 ) = ‚µ e¦ |„ |2 ‚ µ e¦ ’ ‚µ C0 e¦ ‚ µ C0 e¦

4 2 2

1µ

‚ ¦‚µ ¦ + e2¦ ‚ µ C0 ‚µ C0 .

=’

2

Also,

‚ µ „ ‚µ „

¯ 1

= ’ e2¦ ‚ µ C0 + ie’¦ ‚µ C0 ’ ie’¦

’

2(Im„ )2 2

1µ

‚ ¦‚µ ¦ + e2¦ ‚ µ C0 ‚µ C0 .

=’

2

This establishes the required identities. The SL(2, ) symmetry is manifest

¡

for tr(‚ µ M‚µ M’1 ), because when one substitutes M ’ (Λ’1 )T MΛ’1 the

constant Λ factors cancel using the cyclicity of the trace. 2

EXERCISE 8.4

Verify that the action in Eq. (8.70) agrees with Eq. (8.53).

SOLUTION

First we need the action (8.53) in the Einstein frame. Using Eqs (8.68) and

(8.69), it is given by S = SNS + SR + SCS , where

√

1 1 1

d10 x ’g R ’ ‚µ ¦‚ µ ¦ ’ e’¦ |H3 |3

SNS =

2κ2 2 2

√

1 1

d10 x ’g e2¦ |F1 |2 + e¦ |F3 |2 + |F5 |2

SR = ’

4κ2 2

1

SCS = ’ C4 § H3 § F3 .

4κ2

We only need to rewrite the ¬rst two terms in Eq. (8.70) and compare them

8.2 S-duality 323

with the corresponding terms in the above actions, since the last two terms

obviously agree. These terms are

1 1

’ 12 Hµνρ MH µνρ + 4 tr(‚ µ M‚µ M’1 )

T

= ’ 1 e¦ |„ |2 |H3 |2 + |F3 |2 ’ 2C0 F · H ’ 1

‚ µ ¦‚µ ¦ + e2¦ ‚ µ C0 ‚µ C0

2 2

= ’ 1 e’¦ |H3 |2 + e¦ (F3 ’ C0 H3 )2 ’ 1

‚ µ ¦‚µ ¦ + e2¦ ‚ µ C0 ‚µ C0 .

2 2

Using F3 = F3 ’ C0 H3 , it becomes manifest that all terms match. 2

8.2 S-duality

S-duality is a transformation that relates a string theory with coupling con-

stant gs to a (possibly) di¬erent theory with coupling constant 1/gs . This is

analogous to the way that T-duality relates a circular dimension of radius

R to one of radius 2 /R. In each case the parameter is given by the vacuum

s

expectation value of a scalar ¬eld. Thus the duality, at a more fundamental

level, can be understood in terms of ¬eld transformations.

The symmetry of Maxwell™s equation under the interchange of electric

and magnetic quantities, combined with the Dirac quantization condition,

already hints at the possibility of such a duality in ¬eld theory. This

strong“weak (or electric“magnetic) duality symmetry generalizes to non-

abelian gauge theories. The cleanest example is N = 4 supersymmetric

Yang“Mills (SYM) theory, which is a conformally invariant quantum the-

ory, a fact that plays an important role in Chapter 12. In fact, when one

includes a θ term

θ

Fa § Fa

Sθ = (8.86)

16π 2

in the de¬nition of the N = 4 SYM theory (as one should), this theory has

an SL(2, ) duality under which the complex coupling constant

θ 4π

„= +i 2 (8.87)

2π gYM

transforms as a modular parameter. The fact that the theory is conformally

invariant ensures that „ is a constant independent of any renormalization

scale. The simple electric“magnetic duality gYM ’ 4π/gYM corresponds to

the special case „ ’ ’1/„ evaluated for θ = 0. There has been extensive

progress in recent times in understanding electric“magnetic dualities of other

324 M-theory and string duality

supersymmetric gauge theories, starting with the important work of Seiberg

and Witten in 1994 for N = 2 gauge theories.

A double expansion

In order to understand the various string dualities and their relationships it is

useful to view string theory as a simultaneous expansion in two parameters:11