Kodaira-Spencer theory for flux backgrounds

We give an explicit description (in component fields) of a holomorphic theory associated to a general supersymmetric background of 𝒩 =1 supergravity in ten dimensions. Conjecturally, this provides a sought-for holomorphic realization of the supergravity twist in such backgrounds, generalizing the minimal type I BCOV theory for Calabi-Yau manifolds. Our theory unpacks the recently introduced Courant contact model associated to a holomorphic Courant algebroid. We also show that a coisotropic reduction of this model reproduces the recent model of Ashmore, Ibarra, Strickland-Constable, and Svanes [J. High Energy Phys. 03 (2026) 152]. which is formulated in terms of constrained fields.