Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Darboux theorem for generalized complex structures on transitive Courant algebroids

Vicente Cortés, Liana David

TL;DR

This paper establishes a Darboux-type local normal form for integrable generalized complex structures on transitive Courant algebroids with neutral signature by reducing to untwisted models $E=TM\oplus T^*M\oplus (M\times\mathfrak g)$ and describing the $(1,0)$-bundle via data $(W,\sigma,\mathcal D,\epsilon)$. Integrability is translated into concrete conditions on these data (involutivity of $W$, Lagrangian and stability properties of $\mathcal D$, a flat partial connection, and curvature/3-form constraints), enabling a detailed local classification (regular case) and a study of regularity versus weak-regularity. The main result shows that any regular GC structure is locally equivalent to a structure on an untwisted CA with a product base $M=U\times V$ and explicit root-theoretic data, extending Gualtieri’s theorem and Wang’s invariant structures. Moreover, a weaker notion of regularity (weak-regularity) is shown to imply true regularity in a neighborhood of regular points, via a root-system analysis of Lagrangian subalgebras in $((\mathfrak g^{\oplus 2})^{\mathbb C},\langle\cdot,\cdot\rangle)$, providing a robust local classification framework. The work yields a comprehensive Darboux-type description for transitive CA and clarifies how Cartan normalization governs the local geometry of generalized complex structures in this setting.

Abstract

Let E be a transitive Courant algebroid with scalar product of neutral signature. A generalized almost complex structure \mathcal J on E is a skew-symmetric smooth field of endomorphisms of E which squares to minus the identity. We say that \mathcal J is integrable (or is a generalized complex structure) if the space of sections of its (1,0) bundle is closed under the Dorfman bracket of E. In this paper we determine, under certain natural conditions, the local form of \mathcal J around regular points. This result is analogous to Gualtieri's Darboux theorem for generalized complex structures on manifolds and extends Wang's description of skew-symmetric left-invariant complex structures on compact semisimple Lie groups.

Darboux theorem for generalized complex structures on transitive Courant algebroids

TL;DR

This paper establishes a Darboux-type local normal form for integrable generalized complex structures on transitive Courant algebroids with neutral signature by reducing to untwisted models and describing the -bundle via data . Integrability is translated into concrete conditions on these data (involutivity of , Lagrangian and stability properties of , a flat partial connection, and curvature/3-form constraints), enabling a detailed local classification (regular case) and a study of regularity versus weak-regularity. The main result shows that any regular GC structure is locally equivalent to a structure on an untwisted CA with a product base and explicit root-theoretic data, extending Gualtieri’s theorem and Wang’s invariant structures. Moreover, a weaker notion of regularity (weak-regularity) is shown to imply true regularity in a neighborhood of regular points, via a root-system analysis of Lagrangian subalgebras in , providing a robust local classification framework. The work yields a comprehensive Darboux-type description for transitive CA and clarifies how Cartan normalization governs the local geometry of generalized complex structures in this setting.

Abstract

Let E be a transitive Courant algebroid with scalar product of neutral signature. A generalized almost complex structure \mathcal J on E is a skew-symmetric smooth field of endomorphisms of E which squares to minus the identity. We say that \mathcal J is integrable (or is a generalized complex structure) if the space of sections of its (1,0) bundle is closed under the Dorfman bracket of E. In this paper we determine, under certain natural conditions, the local form of \mathcal J around regular points. This result is analogous to Gualtieri's Darboux theorem for generalized complex structures on manifolds and extends Wang's description of skew-symmetric left-invariant complex structures on compact semisimple Lie groups.
Paper Structure (17 sections, 47 theorems, 224 equations)

This paper contains 17 sections, 47 theorems, 224 equations.

Table of Contents

  1. Introduction
  2. Preliminary material
  3. Untwisted Courant algebroids
  4. The case $\mathfrak{g}$ semisimple of compact type
  5. Dirac structures of $V\oplus V^{*}\oplus \mathfrak{g}$
  6. Dirac structures of $(V \oplus V^{*} \oplus \mathfrak{g})^{\mathbb{C}}$
  7. The $(1,0)$-bundle of a generalized complex structure
  8. Integrability of generalized complex structures
  9. Generalized complex structures and equivalences
  10. Regular generalized complex structures
  11. On regularity of generalized complex structures
  12. Lagrangian subalgebras of $( ( \mathfrak{g}^{\oplus 2})^{\mathbb{C}}, \langle \cdot , \cdot \rangle )$
  13. Weak-regularity versus regularity
  14. Appendix
  15. Families of Cartan subalgebras
  16. ...and 2 more sections

Key Result

Theorem 1

Under a natural additional assumption, any regular generalized complex structure on a transitive Courant algebroid is equivalent, in a neighborhood of any regular point, to a generalized complex structure on an untwisted Courant algebroid $E = TM\oplus T^{*}M\oplus (M\times \mathfrak{g} )$ over $M where $( x^{i} )$ are coordinates on $\mathbb{R}^{n-2k}$, $(z^{i})$ are coordinates on $\mathbb{C}^

Theorems & Definitions (103)

  • Theorem 1
  • Definition 3
  • Remark 4
  • Remark 5
  • Remark 6
  • Proposition 8
  • proof
  • Definition 9
  • Lemma 10
  • proof
  • ...and 93 more