Kodaira-Spencer theory for Courant algebroids

TL;DR

This paper explores the geometric and algebraic structures of Courant algebroids on dg ringed manifolds, revealing their relation to shifted symplectic structures, supergravity backgrounds, and BCOV theory, with implications for generalized geometry.

Contribution

It introduces the Courant contact model as a shifted symplectic structure, connects Courant algebroids to supergravity and BCOV theory, and extends conjectures in generalized geometry.

Findings

Courant algebroids admit a local shifted contact structure.

The Courant contact model relates to minimal type I BCOV theory.

Twisted backgrounds in supergravity induce Courant algebroids over the Dolbeault complex.

Abstract

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.