Gravity and Strings

482 From eleven to four dimensions
The n = 6, d = 4 case. To exhibit the new symmetry, we first go to the Einstein frame by
rescaling the metric,
gµν = e 2φ gEµν. (16.158)
Using the formulae in Appendix E to rescale the Ricci scalar and defining the complex
scalar field τ that parametrizes the coset space SL(2, R)/SO(2),
τ ≡ a + ie −2φ , a ≡ ˜B, (16.159)
we obtain the action of N = 4, d = 4 SUEGRA coupled to p vector multiplets:
S =
1
16πG (4)
N
d 4 x √ |gE|
RE − 1 ∂µτ∂
2
µ τ 1
−
(Im(τ)) 2 8 Tr(∂Mη∂Mη)
− 1
4e−2φ (ηMη) F F + 1
8η 1
√ aɛF
|g| F
.
(16.160)
The truncation of the vector fields A = = 7,...,12 + p and the scalar fields M =
I(12+p)×(12+p) takes us to the pure supergravity action Eq. (12.58). The truncated action is
invariant under SO(6) rotations of the vector fields, which are associated with rotations 17
in the internal T 6 and, as discussed in Section 12.2, the equations of motion are covariant
under SL(2, R) transformations of τ, anon-perturbative symmetry that will not have a
simple interpretation until we introduce the solitonic 5-brane.
16.6 T duality, compactification, and supersymmetry
The hidden symmetries of supergravity theories that we have studied can be used to transform
11- or ten-dimensional solutions with the appropriate number of isometries using one
of the mechanisms we described in Chapter 11 to generate new solutions: reduce, use the
d-dimensional hidden symmetry transformation, and oxidize again. The T-duality Buscher
rules of the string common sector can be interpreted as the simplest application of this
mechanism, using the Z2 symmetry of the theory reduced on a circle.
On the other hand, these hidden symmetries of supergravity theories are evidently symmetries
of the supersymmetry transformation rules, which means that they preserve the unbroken
supersymmetries of the d-dimensional solutions, acting covariantly on their Killing
spinors.
This may lead us to think that the whole procedure of reduction–dualization–oxidation
preserves the unbroken supersymmetry of the 11- or 10-dimensional solutions. There is,
however, one loophole in all these arguments: unbroken supersymmetry has to be preserved
by dimensional reduction and this requires that the 11- or 10-dimensional Killing spinors
17 This SO(6) is part of the original T-duality group O(6, 6 + p), but does not contain any interchanges of
winding and KK vectors, which are constrained to be equal by the truncation conditions.