Evidence for the classical integrability of the complete AdS(4) x CP(3) superstring

TL;DR

This work addresses classical integrability of the complete Green–Schwarz superstring on AdS$_4\times$CP$^3$, where standard $Z_4$-graded coset methods fail in sectors with eight broken-supersymmetry fermions. By analyzing the theory to quadratic order and then in a gauge-fixed, worldsheet T-dual AdS$_4$ sub-sector, the authors construct zero-curvature Lax connections that incorporate the broken-fermion sector and demonstrate their flatness using Noether currents, equations of motion, and Killing-vector structure. They show that the complete coset Lax connection emerges from a gauge transform in the appropriate limit, and provide a Lax pair for the AdS$_4$ sub-sector to all orders in fermions, offering strong evidence for the classical integrability of the full AdS$_4\times$CP$^3$ superstring. The results suggest a path to extend integrability analyses beyond coset models, potentially via worldsheet superfield or superembedding formalisms, with implications for other backgrounds with reduced or broken supersymmetry.

Abstract

We construct a zero-curvature Lax connection in a sub-sector of the superstring theory on AdS(4) x CP(3) which is not described by the OSp(6|4)/U(3) x SO(1,3) supercoset sigma-model. In this sub-sector worldsheet fermions associated to eight broken supersymmetries of the type IIA background are physical fields. As such, the prescription for the construction of the Lax connection based on the Z_4-automorphism of the isometry superalgebra OSp(6|4) does not do the job. So, to construct the Lax connection we have used an alternative method which nevertheless relies on the isometry of the target superspace and kappa-symmetry of the Green-Schwarz superstring.