When we derive various supermultiplets of states, at the noninteracting level, these states can easily be described in terms of local fields. But, at the interacting level, there are certain ambiguities that withdraw as a result of different field representations describing the same massless free states. So the proper choice of the field representation may be subtle. The supermultiplets can then be converted into supersymmetric actions, quadratic in the fields. For selfdual tensor fields, the action must be augmented by a duality constraint on the corresponding field strength. For the graviton field,

The linearized Einstein equation for g_{μν} = η_{μν} + κh_{μν} implies that (for D ≥ 3)

R_{μν} ∝ ∂^{2}h_{μν} + ∂_{μ}∂_{ν}h – ∂_{μ}∂^{ρ}h_{νρ} – ∂_{ν}∂^{ρ}h_{ρμ} = 0 —– (1)

where h ≡ h_{μμ} and R_{μν} is the Ricci tensor. To analyze the number of states implied by this equation, one may count the number of plane-wave solutions with given momentum q^{μ}. It then turns out that there are D arbitrary solutions, corresponding to the linearized gauge invariance h_{μν} → h_{μν} + ∂_{μ}ξ_{ν} + ∂_{ν}ξ_{μ}, which can be discarded. Many other components vanish and the only nonvanishing ones require the momentum to be lightlike. Thee reside in the fields h_{ij}, where the components i, j are in the transverse (D-2) dimensional subspace. In addition, the trace of h_{ij }must be zero. Hence, the relevant plane-wave solutions are massless and have polarizations (helicities) characterized by a symmetric traceless 2-rank tensor. This tensor comprises 1/2D(D-3), which transform irreducibly under the SO(D-2) helicity group of transverse rotations. For the special case of D = 6 spacetime dimensions, the helicity group is SO(4), which factorizes into two SU(2) groups. The symmetric traceless representation then transforms as a doublet under each of the SU(2) factors and it is thus denoted by (2,2). As for D = 3, there are obviously no dynamic degrees of freedom associated with the gravitational field. When D = 2 there are again no dynamic degrees of freedom, but here (1) should be replaced by R_{μν }= 1/2g_{μν}R.