3

Here ωL is the Lorentz Chern“Simons term (ω is the spin connection) and

ωYM is the Yang“Mills Chern“Simons term. However, the Lorentz Chern“

Simons term is higher-order in derivatives, so only the Yang“Mills Chern“

Simons term is part of the low-energy e¬ective supergravity theory.

The parameter g, introduced in Eq. (8.73), is related to the ten-dimensional

Yang“Mills coupling constant gYM by

2

g2

gYM

gs = (2π s )6 gs .

= (8.78)

4π 4π

In type I superstring theory, gYM is an open-string coupling, and therefore

8 The conventions here correspond to setting the parameter µ that was introduced in Section 5.4

equal to 8/ 2 . The gauge ¬eld A is antihermitian as in Chapter 5.

s

8.1 Low-energy e¬ective actions 319

√

it is proportional to gs . As discussed in Chapter 3, this is a consequence of

the fact that open strings couple to world-sheet boundaries, whereas closed

strings couple to interior points of the string world sheet.9 In the heterotic

string theory, considered in the next section, the counting is a bit di¬erent.

There gYM is a closed-string coupling, and therefore it is proportional to gs .

Note that the ¬rst two terms of Eq. (8.73) come from a spherical world

sheet (with χ = ’2), whereas the last term comes from a disk world sheet

(with χ = ’1). The third term involves an R“R ¬eld and therefore is

independent of ¦, as discussed earlier.

The action (8.73) describes N = 1 supergravity coupled to SO(32) super

Yang“Mills theory in ten dimensions. As such, it only contains the leading

terms in the low-energy expansion of the e¬ective action of the type I super-

string theory. In this particular case, some of the higher-order corrections

to this action are already known from the anomaly analysis. Speci¬cally,

as mentioned above, the Chern“Simons term in the de¬nition of F3 con-

tains both a Yang“Mills and a Lorentz contribution in the full theory, but

the Lorentz Chern“Simons term is higher-order in derivatives, and there-

fore it is not included in the leading low-energy e¬ective action. A local

counterterm proportional to

C2 § Y8 , (8.79)

also required by anomaly cancellation, consists entirely of terms of higher

dimension than are included in the action given above.10

Supersymmetry transformations

Let us now consider the supersymmetry transformations that leave the type

I e¬ective action invariant. The terms involving the supergravity multiplet

can be obtained by truncation of the type IIB supersymmetry transforma-

tions given earlier. The type IIB formulas used complex fermi ¬elds such as

» = »1 + i»2 , and similarly for Ψµ and the supersymmetry parameter µ. In

the truncation to type I the combinations that survive are Majorana“Weyl

¬elds given by sums such as » = »1 + »2 , and similarly for Ψµ and the

supersymmetry parameter µ. Using this rule, the type IIB formulas imply

that the transformations of the fermions in the supergravity multiplet are

9 This rule can be understood in terms of the genus of the relevant world-sheet diagrams.

10 The precise form of Y8 can be found in Chapter 5.

320 M-theory and string duality

given in the type I case by

’ 1 e¦ F(3) “µ µ,

δΨµ = µµ 8

= 2 ‚ ¦µ + 1 e¦ F(3) µ,

1 (8.80)

δ» / 4

1

= ’ 2 F(2) µ.

δχ

The last equation represents the supersymmetry transformation of the ad-

joint fermions χ in the super Yang“Mills multiplet. As always, there are

corrections to these formulas that are quadratic in fermi ¬elds, but these

are not needed to construct Killing spinor equations.

Heterotic supergravity

Chapter 7 derived the particle spectrum of the heterotic string theories in

ten-dimensional Minkowski space-time. The massless ¬eld content of the

SO(32) heterotic string theory is exactly the same as that of the type I

superstring theory. The massless ¬elds of the E8 — E8 heterotic string di¬er

only by the replacement of the gauge group, though the di¬erences are more

substantial for the massive excitations.

Action

The bosonic part of the low-energy e¬ective action of both of the heterotic

theories in the ten-dimensional string frame is given by

κ2

√

1 1

’2¦

10 µ 2

Tr(|F2 |2 ) .

d x ’ge R + 4‚µ ¦‚ ¦ ’ |H3 | ’

S= 2 2

2κ 2 30g

(8.81)

Note that the entire action comes from a spherical world sheet in this case,

and heterotic theories have no R“R ¬elds, which explains why every term

contains a factor of exp(’2¦). F2 is the ¬eld strength corresponding to the

gauge groups SO(32) or E8 — E8 and

2

s

H3 = dB2 + ω3 (8.82)

4

satis¬es the relation

2 1

s

trR § R ’ TrF § F

dH3 = . (8.83)

4 30

However, as noted in the type I context, the Lorentz term is not part of

the leading low-energy e¬ective theory. The gauge theory trace denoted Tr

8.1 Low-energy e¬ective actions 321

is evaluated using the 496-dimensional adjoint representation. As was dis-

cussed in Chapter 5, this can be re-expressed in terms of the 32-dimensional

fundamental representation of SO(32), for which the trace is denoted tr, by

using the identity

1

trF § F = TrF § F. (8.84)

30

Sometimes this notation is used in the E8 — E8 theory, as well, even though

this group doesn™t have a 32-dimensional representation. In this notation,

the cohomology classes of trR § R and trF § F must be equal, since dH3 is

exact.

Supersymmetry transformations

The heterotic string e¬ective action has N = 1 local supersymmetry in ten

dimensions, which means that the gravitino ¬eld Ψµ is a Majorana“Weyl

spinor. There is also a Majorana“Weyl dilatino ¬eld ». The bosonic parts

of the transformation formulas of the fermi ¬elds, which is what is required

to read o¬ the Killing spinor equations, are

(3)

1

’ 4 Hµ µ,

δΨµ = µµ

1 1

= ’ 2 “µ ‚µ ¦µ + 4 H(3) µ, (8.85)

δ»

1

= ’ 2 F(2) µ.

δχ