No N=4 strings on Wolf spaces

We generalize the standard N=2 supersymmetric Kazama-Suzuki coset construction to the N=4 case by requiring the nonlinear (Goddard-Schwimmer) N=4 quasisuperconformal algebra to be realized on cosets. The constraints that we find allow a very simple geometrical interpretation and have the Wolf spaces as their natural solutions. Our results obtained by using component-level superconformal field theory methods are fully consistent with standard results about N=4 supersymmetric two-dimensional nonlinear σ models and N=4 WZWN models on Wolf spaces. We construct the actions for the latter and express the quaternionic structure, appearing in the N=4 coset solution, in terms of the symplectic structure associated with the underlying Freudenthal triple system. Next, we gauge the N=4 QSCA and build a quantum BRST charge for the N=4 string propagating on a Wolf space. Surprisingly, the BRST charge nilpotency conditions rule out the nontrivial Wolf spaces as consistent string backgrounds.