Green-Schwarz, Nambu-Goto actions, and Cayley's hyperdeterminant

It has been recently shown that Nambu-Goto action can be re-expressed in terms of Cayley's hyperdeterminant with the manifest SL(2,R)×SL(2,R)×SL(2,R) symmetry. In the present Letter, we show that the same feature is shared by Green-Schwarz σ-model for N=2 superstring whose target space-time is D=2+2. When its zweibein field is eliminated from the action, it contains the Nambu-Goto action which is nothing but the square root of Cayley's hyperdeterminant of the pull-back in superspace Det(Πiαα˙) manifestly invariant under SL(2,R)×SL(2,R)×SL(2,R). The target space-time D=2+2 can accommodate self-dual supersymmetric Yang-Mills theory. Our action has also fermionic κ-symmetry, satisfying the criterion for its light-cone equivalence to Neveu-Schwarz-Ramond formulation for N=2 superstring.