Gravity and Strings

16.1 Dimensional reduction from d = 11 to d = 10 461
16.1.4 Reduction of fermions and the supersymmetry rules
Here we want to reduce to ten dimensions the fermions and the supersymmetry transformation
laws in Eq. (16.8). We will keep only terms up to second order in fermions.
First we need to decompose the 11-dimensional gamma matrices in terms of the tendimensional
ones. This is done in Appendix B.1.4. Next, we have to decompose the 11dimensional
spinors into ten-dimensional spinors. Eleven-dimensional Majorana spinors
are also ten-dimensional Majorana spinors. However, in ten-dimensional supergravity, the
elementary spinor is a Majorana–Weyl spinor. Thus, each 11-dimensional spinor can be
considered as a pair of Majorana–Weyl spinors with opposite chiralities (this is why this
theory is non-chiral and it is N = 2). In principle we could split all the spinors into their
chiral halves, but is is not worth doing it for the moment. Later on, we will have to do it in
order to relate the spinors to those of the type-IIB theory, which are of the same chirality
and cannot be considered the two halves of any Majorana spinor. As we will see, there
are two options in the type-IIB case: either we use indices i = 1, 2 for the spinors or we
combine them into a chiral complex spinor. We will use the first possibility.
We express the 11-dimensional spinors in terms of the ten-dimensional spinors (gravitino
ˆψ ˆµ, dilatino ˆλ, and the supersymmetry transformation parameter ˆɛ) asfollows: 8
ˆɛ = e − 1 6 ˆφ ˆɛ,
ˆψ â = e 1 6
ˆφ
2 ˆψâ − 1
ˆƔâ ˆλ , 3
ˆψ z = 2i
3 e 1 6 ˆφ ˆƔ11 ˆλ. (16.60)
Observe that, with these definitions, the gravitino ˆψ ˆµ is real but the dilatino ˆλ is purely
imaginary. We could use a purely real dilatino just by multiplying by i, but then its supersymmetry
rule would look unconventional.
We now want to use the relation between the 11- and ten-dimensional bosonic fields that
we have already obtained. However, we have performed the dimensional reduction working
in a special Lorentz gauge êâ z = 0and supersymmetry transformations do not preserve this
gauge. In fact,
δ ê ˆɛ â z = 1
3e 1 3 ˆφ ¯ˆɛ ˆƔ â ˆƔ11 ˆλ. (16.61)
We have to introduce a compensating local Lorentz transformation in order to preserve
our gauge choice. Then, the ten-dimensional supersymmetry transformation δˆɛ will
be a combination of an 11-dimensional supersymmetry transformation δ and an 11-
ˆɛ
dimensional compensating local Lorentz transformation δ such that
ˆσ
δˆɛê â z ≡ δ ˆɛ + δ ˆσ
ê â z = 1
3 e 1 3 ˆφ ¯ ˆɛ ˆƔ â ˆƔ11 ˆλ + 1
2 ˆσ ˆbĉ Ɣv
ˆˆM ˆbĉ
â
ˆˆd
ê ˆd
z = 0. (16.62)
Since the generators of the Lorentz group in the vector representation are given by
Eq. (A.60), the parameter of the compensating Lorentz transformation is given by
ˆσ â z =− 1
3 e 1 3 ˆφ ¯ ˆɛ ˆƔ â ˆƔ11 ˆλ. (16.63)
8 As a first step, we simply identify, up to factors involving the dilaton, the dilatino ˆλ with the (flat) component
ˆψz and the gravitino ˆψ â with the (flat) components ˆψ â . Then we see that it is natural to combine this dilatino
and this gravitino into a new one whose supersymmetry transformation rules are much simpler. The final
combinations are the ones we write.