Green-Schwarz, Nambu-Goto Actions, and Cayley's Hyperdeterminant

TL;DR

This paper demonstrates that the Green-Schwarz superstring action in D=2+2 dimensions can be expressed using Cayley's hyperdeterminant, revealing a deep symmetry structure and connecting to self-dual supersymmetric Yang-Mills theory.

Contribution

It shows that the Green-Schwarz superstring action in D=2+2 shares the hyperdeterminant structure with Nambu-Goto action, highlighting its SL(2,R)^3 symmetry and fermionic kappa-symmetry.

Findings

Superstring action expressed via Cayley's hyperdeterminant.

Manifest SL(2,R)^3 symmetry in the superstring formulation.

Presence of fermionic kappa-symmetry ensuring light-cone equivalence.

Abstract

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) X SL(2,R) X SL(2,R) symmetry. In the present paper, we show that the same feature is shared by Green-Schwarz sigma-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 \sqrt{\hyperdet(\Pi_{i \a\Dot\a})} manifestly invariant under SL(2,R) X SL(2,R) X SL(2,R). The target space-time D=2+2 can accommodate self-dual supersymmetric Yang-Mills theory. Our action has also fermionic kappa-symmetry, satisfying the criterion for its light-cone equivalence to Neveu-Schwarz-Ramond formulation for N=2 superstring.