the B ¬elds transform linearly by the rule

B2 ’ ΛB2 . (8.63)

Since the parameters in Λ are constants, H3 transforms in the same way.

The complex scalar ¬eld „ , de¬ned by

„ = C0 + ie’¦ , (8.64)

is useful because it transforms nonlinearly by the familiar rule

a„ + b

„’ . (8.65)

c„ + d

The ¬eld C0 is sometimes referred to as an axion, because of the shift sym-

metry C0 ’ C0 +constant of the theory (in the supergravity approximation),

and then the complex ¬eld „ is referred to as an axion“dilaton ¬eld.

The action can be conveniently written in terms of the symmetric SL(2, )

¡

matrix

|„ |2 ’C0

M = e¦ , (8.66)

’C0 1

which transforms by the simple rule

M ’ (Λ’1 )T MΛ’1 . (8.67)

E

The canonical Einstein-frame metric gµν and the four-form C4 are SL(2, ) ¡

invariant. Note that since the dilaton transforms, the type IIB string-frame

metric gµν in the action (8.53), which is related to the canonical Einstein

metric by

gµν = e¦/2 gµν ,

E

(8.68)

8.1 Low-energy e¬ective actions 317

is not SL(2, ) invariant. The transformation of the scalar curvature term

¡

under this change of variables is given by

√ √

1 1 9

d10 x ’g e’2¦ R ’ 2 d10 x ’g(R ’ ‚ µ ¦‚µ ¦), (8.69)

2κ2 2κ 2

where the string-frame metric is used in the ¬rst expression and the Einstein-

frame metric is used in the second one.

Using the quantities de¬ned above, the type IIB supergravity action can

be recast in the form

√

1 1T 1

d10 x ’g R ’ Hµνρ MH µνρ + tr(‚ µ M‚µ M’1 )

S=

2κ2 12 4

√

1 (i) (j)

d10 x ’g|F5 |2 +

’ µij C4 § H3 § H3 , (8.70)

8κ2

where the metric g E is used throughout. This action is manifestly invariant

under global SL(2, ) transformations.

¡

The self-duality equation, F5 = F5 , which is imposed as a constraint in

this formalism, is also SL(2, ) invariant. To see this, ¬rst note that the

¡

Hodge dual that de¬nes F5 is invariant under a Weyl rescaling, so that it

doesn™t matter whether it is de¬ned using the string-frame metric or the

Einstein-frame metric. The de¬nition of F5 in Eq. (8.57) can be recast in

the manifestly SL(2, ) invariant form

¡

1 (i) (j)

F5 = F5 + µij B2 § H3 . (8.71)

2

The invariance of the self-duality equation then follows.

Type I supergravity

Field content

As explained in Chapter 6, type I superstring theory arises as an orientifold

projection of the type IIB superstring theory. This involves a truncation

of the type IIB closed-string spectrum to the left“right symmetric states

as well as the addition of a twisted sector consisting of open strings. The

massless closed-string sector is N = 1 supergravity in ten dimensions and

the massless open-string sector is N = 1 super Yang“Mills theory with gauge

group SO(32) in ten dimensions. Therefore, the low-energy e¬ective action

should describe the interactions of these two supermultiplets to leading order

in the ± expansion.

318 M-theory and string duality

Restricting to the bosonic sector of the theory, the massless ¬elds of type

I superstring theory in ten dimensions consist of

gµν , ¦, C2 and Aµ . (8.72)

Here gµν is the graviton, ¦ is the dilaton, C2 is the R“R two-form and Aµ

is the SO(32) Yang“Mills gauge ¬eld coming from the twisted sector.

Action

In the string frame, the bosonic part of the supersymmetric Lagrangian

describing the low-energy limit of the type I superstring is

κ2 ’¦

√

1 1

’2¦

10

(R + 4‚µ ¦‚ ¦) ’ |F3 | ’ 2 e tr(|F2 |2 ) .

µ 2

d x ’g e

S= 2

2κ 2 g

(8.73)

Here F2 = dA + A § A is the Yang“Mills ¬eld strength corresponding to the

gauge ¬eld A = Aµ dxµ . Moreover,

2

s

F3 = dC2 + ω3 , (8.74)

4

as explained in the anomaly analysis of Chapter 5.8 In the full string theory

the Chern“Simons term is

ω3 = ωL ’ ωYM , (8.75)

where

2

ωL = tr(ω § dω + ω § ω § ω) (8.76)

3

and

2