The purpose of this paper is to solve a problem posed by Strominger in constructing smooth models of superstring theory with flux. These are given by non-Kähler manifolds with torsion.
"The theory of superstring with flux on non-Kähler manifolds and the complex Monge-Ampère equation." J. Differential Geom. 78 (3) 369 - 428, March 2008. https://doi.org/10.4310/jdg/1207834550