Heterotic geometry without isometries

TL;DR

This paper explores properties of four-dimensional hyperkahler torsion (heterotic) geometries, showing their equivalence to hypercomplex structures and presenting two simplified formalisms for their analysis, including explicit examples.

Contribution

It introduces two equivalent, simplified formalisms for analyzing heterotic geometries, making them more tractable than hyperkahler counterparts.

Findings

Hypercomplex structures and weak torsion hyperkahler geometries are equivalent in 4D.

Two formal frameworks for heterotic geometries are established, involving potential functions and Ashtekar-like systems.

Examples include Callan-Harvey-Strominger type metrics and others.

Abstract

We present some properties of hyperkahler torsion (or heterotic) geometry in four dimensions that make it even more tractable than its hyperkahler counterpart. We show that in $d=4$ hypercomplex structures and weak torsion hyperkahler geometries are the same. We present two equivalent formalisms describing such spaces, they are stated in the propositions of section 1. The first is reduced to solve a non-linear system for a doublet of potential functions, first found by Plebanski and Finley. The second is equivalent to finding the solutions of a quadratic Ashtekar-Jacobson-Smolin like system, but without a volume preserving condition. This is why heterotic spaces are simpler than usual hyperkahler ones. We also analyze the strong version of this geometry. Certain examples are presented, some of them are metrics of the Callan-Harvey-Strominger type and others are not. In the conclusion we discuss the benefits and disadvantages of both formulations in detail.