Local heterotic geometry in holomorphic coordinates

TL;DR

This paper investigates (2,0) and (4,0) heterotic geometries in holomorphic coordinates, analyzing torsion properties, conformal equivalences, and providing new examples in four dimensions, challenging previous assumptions about their relation to Kähler Ricci-flat geometries.

Contribution

It offers a detailed analysis of heterotic geometries in holomorphic coordinates, showing their relation to conformal equivalence and providing new examples, especially in four dimensions.

Findings

(2,0) geometry can be conformally equivalent to (2,2) geometry under certain conditions.

(4,0) heterotic geometry is not necessarily conformally equivalent to (4,4) Kähler Ricci-flat geometry.

Examples include new four-dimensional manifolds with torsion, such as Eguchi-Hanson and Taub-NUT metrics.

Abstract

In the same spirit as done for N=2 and N=4 supersymmetric non-linear $\si$ models in 2 space-time dimensions by Zumino and Alvarez- Gaum\'e 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) K\"ahler 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.