Actions for (2,1) sigma models and strings

Effective actions are derived for (2, 0) and (2, 1) superstrings by studying the corresponding sigma models. The geometry is a generalisation of Kahler geometry involving torsion and the field equations imply that the curvature with torsion is self-dual in four dimensions, or has SU(n, in) holonomy in other dimensions. The Yang-Mills fields are self-dual in four dimensions and satisfy a form of the Uhlenbeck-Yau equation in higher dimensions. In four dimensions with Euclidean signature, there is a hyperKahler structure and the sigma model has (4, 1) supersymmetry, while for signature (2, 2) there is a hypersymplectic structure consisting of a complex structure squaring to -1 and two "real structures" squaring to 1. The theory is invariant under a twisted form of the (4, 1) superconformal algebra which includes an SL(2, R) Kac-Moody algebra instead of an SU(2) Kac-Moody algebra. Kahler and related geometries are generalised to ones involving real structures. (C) 1998 Elsevier Science B.V.