Local heterotic geometry in holomorphic coordinates

In the same spirit as done for N=2 and N=4 supersymmetric non-linear σ models in 2 space-time dimensions by Zumino and AlvarezGaumé 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 4 dimensional case where we exhibit new examples. We show that, contrarily to Callan, Harvey and Strominger ’s claim for real 4 dimensional manifolds, (4,0) heterotic geometry is not necessarily conformally equivalent to a (4,4) Kähler Ricci flat geometry. We rather prove that, whatever the real dimension be, they are special quasi Ricci flat spaces, and we exemplify our results on Eguchi-Hanson and Taub-NUT metrics with torsion. PAR/LPTHE/93-56 December 1993 ∗Laboratoire de Physique Théorique et des Hautes Energies, Unité associée au CNRS URA 280, Université Paris 7, 2 Place Jussieu, 75251 Paris Cedex 05.