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Kodaira-Spencer theory for Courant algebroids

Julian Kupka, Ingmar Saberi, Charles Strickland-Constable, Fridrich Valach

TL;DR

The work develops a local, dg-geometric framework for Courant algebroids on smooth dg ringed manifolds, introducing the Roytenberg–Weinstein local L_inf algebra and a Courant contact model that yields a (2-n)-shifted symplectic structure with BV-type behavior when n is odd. Through reduction and extension of scalars, the authors connect Courant algebroid data to holomorphic structures and Dolbeault resolutions, enabling a precise description of twisted supergravity backgrounds and their reductions. In the Calabi–Yau fivefold case, the Courant contact model is shown to be equivalent to minimal type I BCOV theory, thereby extending Costello–Li’s holomorphic twist to generalized geometry and flux backgrounds. The results lay groundwork for a BV formulation of type I supergravity in twisted backgrounds and suggest deep links between generalized geometry, moduli of orientations, and holomorphic BCOV-type theories. Overall, the paper bridges shifted geometric structures, Courant algebroids, and twisted supergravity through explicit algebraic and geometric constructions.

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.

Kodaira-Spencer theory for Courant algebroids

TL;DR

The work develops a local, dg-geometric framework for Courant algebroids on smooth dg ringed manifolds, introducing the Roytenberg–Weinstein local L_inf algebra and a Courant contact model that yields a (2-n)-shifted symplectic structure with BV-type behavior when n is odd. Through reduction and extension of scalars, the authors connect Courant algebroid data to holomorphic structures and Dolbeault resolutions, enabling a precise description of twisted supergravity backgrounds and their reductions. In the Calabi–Yau fivefold case, the Courant contact model is shown to be equivalent to minimal type I BCOV theory, thereby extending Costello–Li’s holomorphic twist to generalized geometry and flux backgrounds. The results lay groundwork for a BV formulation of type I supergravity in twisted backgrounds and suggest deep links between generalized geometry, moduli of orientations, and holomorphic BCOV-type theories. Overall, the paper bridges shifted geometric structures, Courant algebroids, and twisted supergravity through explicit algebraic and geometric constructions.

Abstract

Studying Courant algebroids on dg ringed manifolds, we observe that the associated Roytenberg-Weinstein algebra admits a local structure reminiscent of a shifted contact structure. On a dg ringed manifold with an -orientation, its symplectification produces a sheaf of -shifted symplectic formal moduli problems, which we call the Courant contact model. This construction can be interpreted as a (-graded) theory in the Batalin-Vilkovisky formalism whenever 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.
Paper Structure (29 sections, 16 theorems, 94 equations)

This paper contains 29 sections, 16 theorems, 94 equations.

Table of Contents

  1. Introduction
  2. First motivation: Courant algebroids and shifted symplectic and contact structures
  3. Second motivation: twisted supergravity
  4. Outline
  5. Acknowledgments
  6. Notation and conventions
  7. Background
  8. Dg ringed manifolds
  9. Orientations
  10. Local Lie algebras
  11. Shifted symplectic and contact structures
  12. Courant algebroids
  13. Courant algebroids on dg ringed manifolds
  14. Characterizing Courant algebroids
  15. Two constructions
  16. ...and 14 more sections

Key Result

Lemma 3.6

Let $\mathcal{E}$ be a Courant algebroid over $\mathcal{O}$. Then $\rho$ preserves the brackets, i.e. Furthermore, the following is a (self-dual) complex of $\mathcal{O}$-modules: \begin{tikzcd} 0 \arrow[r] & \Der\cO^\vee \arrow[r, "\eta^{-1} \rho^\vee"] & \cE \arrow[r, "\rho"] & \Der\cO \arrow[r] & 0. \end{tikzcd}

Theorems & Definitions (72)

  • Definition 2.1: KevinWitten
  • Example 2.2
  • Example 2.3
  • Example 2.4
  • Remark 2.5
  • Definition 2.6
  • Example 2.7
  • Example 2.8
  • Example 2.9
  • Remark 2.10
  • ...and 62 more