LOCAL HETEROTIC GEOMETRY IN HOLOMORPHIC COORDINATES

In the same spirit as for N = 2 and N = 4 supersymmetric nonlinear sigma models in two spacetime dimensions by Zumino and by Alvarez-Gaume and Freedman, we analyse the (2, 0) and (4, 0) heterotic geometry in holomorphic coordinates. We study the properties of the torsion tensor and give the conditions under which (2, 0) geometry is conformally equivalent to a (2, 2) one. Using additional isometries, we show that it is difficult to equip a manifold with a closed torsion tensor, but for the real four-dimensional case where we exhibit new examples. We show that, contrarily to Callan et al's claim for real four-dimensional manifolds, (4, 0) heterotic geometry is not necessarily conformally equivalent to a (4,4) Kahler-Ricci flat geometry. We rather prove that, whatever the real dimension is, they are special quasi-Ricci flat spaces, and we exemplify our results on Eguchi-Hanson and Taub-NUT metrics with torsion.