Heterotic flux compactifications and their moduli

We study supersymmetric compactification to four dimensions with non-zero H-flux in heterotic string theory. The background metric is generically conformally balanced and can be conformally Kähler if the primitive part of the H-flux vanishes. Analyzing the linearized variational equations, we write down necessary conditions for the existence of moduli associated with the metric. In a heterotic model that is dual to a IIB compactification on an orientifold, we find the metric moduli in a fixed H-flux background via duality and check that they satisfy the required conditions. We also discuss expressing the conditions for moduli in a fixed flux background using twisted differential operators.