Kodaira-Spencer theory for Courant algebroids

Studying Courant algebroids on dg ringed manifolds, we observe that the associated Roytenberg-Weinstein $L_\infty$ algebra admits a local structure reminiscent of a shifted contact structure. On a dg ringed manifold with an $n$-orientation, its symplectification produces a sheaf of $(2-n)$-shifted symplectic formal moduli problems, which we call the Courant contact model. This construction can be interpreted as a ($\mathbb{Z}/2\mathbb{Z}$-graded) theory in the Batalin-Vilkovisky formalism whenever $n$ is odd. After developing the procedure of reduction and extension of scalars, we show how twisted backgrounds in type I supergravity naturally lead to Courant algebroids over the Dolbeault complex. Specialising to the case of a Calabi-Yau fivefold, we show that the Courant contact model for that Courant algebroid is equivalent to a central extension of minimal type I BCOV theory. Inspired by this, we extend the conjecture of Costello and Li and place it within the setting of generalized geometry, conjecturing a description of the BV formulation of type I supergravity in general twisted backgrounds.

💡 Research Summary

The paper develops a new bridge between Courant algebroids on differential‑graded (dg) ringed manifolds and shifted symplectic/contact geometry, with an eye toward applications in twisted type I supergravity. After reviewing the necessary background on dg ringed manifolds, orientations, local Lie algebras, and AKSZ‑type constructions, the authors define a strict Courant algebroid over a dg ringed manifold and recall the Roytenberg–Weinstein L∞‑algebra R W(E) that encodes infinitesimal generalized diffeomorphisms.

A central observation is that, once an n‑orientation (a volume form in the derived sense) is chosen, the “symplectification” of the total space T*