Non-Linear Realization of ${\aleph_0}$-Extended Supersymmetry

摘要

As generalizations of the original Volkov-Akulov action in four-dimensions,actions are found for all space-time dimensions D invariant under N non-linearrealized global supersymmetries. We also give other such actions invariantunder the global non-linear supersymmetry. As an interesting consequence, wefind a non-linear supersymmetric Born-Infeld action for a non-Abelian gaugegroup for arbitrary D and N, which coincides with the linearly supersymmetricBorn-Infeld action in D=10 at the lowest order. For the gauge group U({\cal N})for M(atrix)-theory, this model has {\cal N}^2-extended non-linearsupersymmetries, so that its large {\cal N} limit corresponds to the infinitelymany (\aleph_0) supersymmetries. We also perform a duality transformation fromF_{\mu\nu} into its Hodge dual N_{\mu_1...\mu_{D-2}}. We next point out thatany Chern-Simons action for any (super)groups has the non-linear supersymmetryas a hidden symmetry. Subsequently, we present a superspace formulation for thecomponent results. We further find that as long as superspace supergravity isconsistent, this generalized Volkov-Akulov action can further accommodate suchcurved superspace backgrounds with local supersymmetry, as a super p-braneaction with fermionic kappa-symmetry. We further elaborate these results towhat we call `simplified' (Supersymmetry)^2-models, with both linear andnon-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 globalnon-linear supersymmetry. We also see that the non-linear realization ofsupersymmetry in `curved' space-time can be interpreted as `non-perturbative'effect starting with the `flat' space-time.