Gravity and Strings

14.1 Strings 415
14.1.2 Green–Schwarz Actions
Green–Schwarz-type actions can be constructed for particles, strings, and other extended
objects, as we will see. The simplest example describes a massless particle moving in flat
(target) superspace with supercoordinates Z M (ξ) = (X µ (ξ), θ α I (ξ)) where I = 1,...,N
numbers the supersymmetries, α is a spacetime spinorial index, and the θ I are anticommuting
spacetime spinors (but worldsheet scalars) [191]:
S =− p
2
dξe −1 ( ˙X µ δ a µ − i ¯θ I γ a ˙θ I )ηab( ˙X ν δ b ν − i ¯θ I γ b ˙θ I ). (14.28)
This action is invariant under worldline reparametrizations, under target-space Poincaré
transformations, and also under the standard global superspace transformations
δɛθ I = ɛ I , δɛ X µ = i ¯ɛ I γ a θ I δa µ . (14.29)
In principle, there is no worldline supersymmetry, a very desirable property. A necessary
condition for having linearly realized worldline supersymmetry is that the numbers of onshell
bosonic and fermionic degrees of freedom should be equal. 9 To find the numbers of
on-shell degrees of freedom, it is necessary to know all the local symmetries of the action.
The above action turns out to have worldline-reparametrization invariance, which can be
used to gauge away one of the X µ s, and a new local symmetry generated by a fermionic
infinitesimal parameter κ (κ-symmetry), which halves the number of fermionic degrees
of freedom [60, 61, 854], which has already been halved by the Dirac equation. Under
κ-symmetry
δκθ I =−iµγ a δa µ κ I , δκ X µ = i ¯θ I γ a δa µ δκθ I , δκe = 4e ˙¯θ I κ I , (14.30)
where µ = δS/δ ˙X µ is the momentum conjugate to X µ .
Taking into account this new symmetry, if we denote by M the number of real components
of the minimal spinor in the spacetime dimension d considered, then, the necessary
condition for having worldsheet supersymmetry reads
NM = 4(d − 1), (14.31)
and, taking into account the values of M in Table B.1, it can be satisfied for d = 2, 3, 4, 5,
and 9 with N = 4, 4, 3, 2, and 2, respectively. Thus, the dimensions in which these actions
can be consistent are restricted.
The condition can be generalized to objects with p extended dimensions. Using worldvolume
reparametrizations, we can always gauge away p + 1 X µ s. The condition of worldvolume
supersymmetry becomes now
NM = 4(d − p − 1), (14.32)
9 In just one dimension, talking about degrees of freedom does not make much sense. However, the same
reasoning carries over to higher-dimensional cases, which allows a classification of all the possible supersymmetric
extended objects [13].