February , 1997 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, m) 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) super-symmetry, while for signature (2,2) there is a hypersymplectic structure consisting of a complex structure squaring to − and two 'real structures' squaring to. 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.