Abstract :
As generalizations of the original Volkov–Akulov action in four-dimensions, actions are found for all space-time dimensions D invariant under N non-linear realized global supersymmetries. We also give other such actions invariant under the global non-linear supersymmetry. As an interesting consequence, we find a non-linear supersymmetric Born–Infeld action for a non-Abelian gauge group for arbitrary D and N , which coincides with the linearly supersymmetric Born–Infeld action in D=10 at the lowest order. For the gauge group U(N) for M(atrix)-theory, this model has N2 -extended non-linear supersymmetries, so that its large N limit corresponds to the infinitely many ( ℵ0 ) supersymmetries. We also perform a duality transformation from Fμν into its Hodge dual Nμ1⋯μD−2 . We next point out that any Chern–Simons action for any (super)groups has the non-linear supersymmetry as a hidden symmetry. Subsequently, we present a superspace formulation for the component results. We further find that as long as superspace supergravity is consistent, this generalized Volkov–Akulov action can further accommodate such curved superspace backgrounds with local supersymmetry, as a super p -brane action with fermionic kappa-symmetry. We further elaborate these results to what we call “simplified” (Supersymmetry) 2 -models, with both linear and non-linear representations of supersymmetries in superspace at the same time. Our result gives a proof that there is no restriction on D or N for global non-linear supersymmetry. We also see that the non-linear realization of supersymmetry in “curved” space–time can be interpreted as “non-perturbative” effect starting with the “flat” space–time.
