String Theory and M-Theory

8.1 Low-energy effective actions 303
elf is German for 11. The indices M, N, . . . are used for base-space (curved)
vectors in 11 dimensions, and the indices A, B, . . . are used for tangent-space
(flat) vectors. The former transform nontrivially under general coordinate
transformations, and the latter transform nontrivially under local Lorentz
transformations. 2
The gauge field for local supersymmetry is the gravitino field ΨM, which
has an implicit spinor index in addition to its explicit vector index. For
each value of M, it is a 32-component Majorana spinor. When spinors
are included, the little group becomes the covering group of SO(9), which
is Spin(9). It has a real spinor representation of dimension 16. Group
theoretically, the Spin(9) Kronecker product of a vector and a spinor is
9 × 16 = 128 + 16. The analogous construction in four dimensions gives
spin 3/2 plus spin 1/2. As Rarita and Schwinger showed in the case of a
free vector-spinor field in four dimensions, there is a local gauge invariance
of the form δΨM = ∂Mε, which ensures that the physical degrees of freedom
are pure spin 3/2. The kinetic term for a free gravitino field ΨM in any
dimension has the structure
SΨ ∼
ΨMΓ MNP ∂NΨP d D x.
Due to the antisymmetry of Γ MNP , for δΨM = ∂Mε this is invariant up to
a total derivative.
In the case of 11 dimensions this local symmetry implies that the physical
degrees of freedom correspond only to the 128. Therefore, this is the
number of physical polarization states of the gravitino in 11 dimensions. In
the interacting theory this local symmetry is identified as local supersymmetry.
This amount of supersymmetry gives 32 conserved supercharges, which
form a 32-component Majorana spinor. This is the dimension of the minimal
spinor in 11 dimensions, so there couldn’t be less supersymmetry than that
in a Lorentz-invariant vacuum. Also, if there were more supersymmetry, the
representation theory of the algebra would require the existence of massless
states with spin greater than two. It is believed to be impossible to construct
consistent interacting theories with such higher spins in Minkowski spacetime.
For this reason, one does not expect to find nontrivial supersymmetric
theories for D > 11.
In order for the D = 11 supergravity theory to be supersymmetric, there
must be an equal number of physical bosonic and fermionic degrees of freedom.
The missing bosonic degrees of freedom required for supersymmetry
2 The reader not familiar with these concepts can consult the appendix of Chapter 9 for some
basics. These also appeared in the anomaly analysis of Chapter 5.