  We report on some recent progress in understanding heterotic flux
compactifications, from a worldsheet perspective mainly. We consider local
models consisting in torus fibration over warped Eguchi-Hanson space and
non-K\"ahler resolved conifold geometries. We analyze the supergravity
solutions and define a double-scaling limit of the resolved singularities,
defined such that the geometry is smooth and weakly coupled. We show that,
remarkably, the heterotic solutions admit solvable worldsheet CFT descriptions
in this limit. This allows in particular to understand the important role of
worldsheet non-perturbative effects.
