Field content

Chapter 5 derived the massless spectrum of the type IIB superstring, which

gives the particle content of type IIB supergravity. The fermionic part of the

spectrum consists of two left-handed Majorana“Weyl gravitinos (or, equiv-

alently, one Weyl gravitino) and two right-handed Majorana“Weyl dilatinos

(or, equivalently, one Weyl dilatino). The NS“NS bosons consist of the met-

ric (or zehnbein), the two-form B2 (with ¬eld strength H3 = dB2 ) and the

dilaton ¦. The R“R sector consists of form ¬elds C0 , C2 and C4 . The latter

has a self-dual ¬eld strength F5 .

The self-dual ¬ve-form

The presence of the self-dual ¬ve-form introduces a signi¬cant complication

for writing down a classical action for type IIB supergravity. The basic issue,

which also exists for analogous self-dual tensors in two and six dimensions,

is that an action of the form

|F5 |2 d10 x (8.52)

does not incorporate the self-duality constraint, and therefore it describes

twice the desired number of propagating degrees of freedom. The introduc-

tion of a Lagrange multiplier ¬eld to implement the self-duality condition

does not help, because the Lagrange multiplier ¬eld itself ends up reintro-

ducing the components it was intended to eliminate.

314 M-theory and string duality

There are several di¬erent ways of dealing with the problem of the self-

dual ¬eld. The original approach is to not construct an action, but only

the ¬eld equations and the supersymmetry transformations. This is entirely

adequate for most purposes, since the supergravity theory is only an e¬ec-

tive theory, and not a quantum theory that one inserts in a path integral.

The basic idea is that the supersymmetric variation of an equation of mo-

tion should give another equation of motion (or combination of equations

of motion). By pursuing this systematically, it turns out to be possible to

determine the supersymmetry transformations and the ¬eld equations simul-

taneously. In fact, the equations are highly overconstrained, so one obtains

many consistency checks.

It is possible to formulate a manifestly covariant action with the correct

degrees of freedom if, following Pasti, Sorokin, and Tonin (PST), one in-

troduces an auxiliary scalar ¬eld and a compensating gauge symmetry in a

suitable manner. The extra gauge symmetry can be used to set the auxiliary

scalar ¬eld equal to one of the space-time coordinates as a gauge choice, but

then the resulting gauge-¬xed theory does not have manifest general coordi-

nate invariance in one of the directions. Nonetheless, it is a correct theory,

at least for space-time topologies for which this gauge choice is globally well

de¬ned.

An action

We do not follow the PST approach here, but instead present an action

that gives the correct equations of motion when one imposes the self-duality

condition as an extra constraint. Such an action is not supersymmetric, how-

ever, because (without the constraint) it has more bosonic than fermionic

degrees of freedom. Moreover, the constraint cannot be incorporated into

the action for the reasons discussed above.

The way to discover this action is to ¬rst construct the supersymmetric

equations of motion, and then to write down an action that reproduces those

equations when the self-duality condition is imposed by hand. The bosonic

part of the type IIB supergravity action obtained in this way takes the form

S = SNS + SR + SCS . (8.53)

Here SNS is the same expression as for the type IIA supergravity theory in

Eq. (8.40), while the parts of the action describing the massless R“R sector

¬elds are given by

√

1 1

d10 x ’g |F1 |2 + |F3 |2 + |F5 |2 ,

SR = ’ (8.54)

4κ2 2

8.1 Low-energy e¬ective actions 315

1

SCS = ’ C4 § H3 § F3 . (8.55)

4κ2

In these formulas Fn+1 = dCn , H3 = dB2 and

F3 = F 3 ’ C0 H3 , (8.56)

1 1

F5 = F5 ’ C2 § H3 + B2 § F3 . (8.57)

2 2

These are the gauge-invariant combinations analogous to F4 in the type

IIA theory. In each case the R“R ¬elds that appear here di¬er by ¬eld

rede¬nitions from the ones that couple simply to the D-brane world volumes,

as described in Chapter 6. The ¬ve-form satisfying the self-duality condition

is F5 , that is,

F5 = F5 . (8.58)

This condition has to be imposed as a constraint that supplements the equa-

tions of motion that follow from the action.

Supersymmetry transformations

Even though the action we presented is not the bosonic part of a supersym-

metric action, the ¬eld equations, including the constraint, are. In other

words, as explained earlier, the supersymmetry variations of these equa-

tions vanish if after the variation one imposes the equations themselves.

The supersymmetry transformations of type IIB supergravity are required

in later chapters, so we present them here.

Let us represent the dilatino and gravitino ¬elds by Weyl spinors » and

Ψµ , respectively. Similarly, the in¬nitesimal supersymmetry parameter is

represented by a Weyl spinor µ. The supersymmetry transformations of the

fermi ¬elds of type IIB supergravity (to leading order in fermi ¬elds) are

1 1

‚µ ¦ ’ ie¦ ‚µ C0 “µ µ + ie¦ F(3) ’ H(3) µ

δ» = (8.59)

2 4

and

i i 1

+ e¦ F(1) “µ + e¦ F(5) “µ µ ’ 2H(3) + ie¦ F(3) “µ µ .

δΨµ = µ µ

8 16 8

(8.60)

Global SL(2, ) symmetry

¡

Type IIB supergravity has a noncompact global symmetry SL(2, ). This

¡

is not evident in the equations above, so let us sketch what is required to

make it apparent. The theory has two two-form potentials, B2 and C2 , which

316 M-theory and string duality

transform as a doublet under the SL(2, ) symmetry group. Therefore, to

¡

rewrite the action in a way that the symmetry is manifest, let us rename

(1) (2)

the two-form potentials B2 = B2 and C2 = B2 and introduce a two-

component vector notation

(1)

B2

B2 = . (8.61)

(2)

B2

Similarly, H3 = dB2 is also a two-component column vector. Under a trans-

formation by

dc

∈ SL(2,

Λ= ), (8.62)

ba

¡