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Para-Holomorphic Algebroids and Para-Complex Connections

Aidan Patterson

TL;DR

The paper develops a para-Hermitian extension of Courant algebroids by leveraging Lie bialgebroid structures to define para-holomorphic algebroids and para-complex connections. It establishes when para-Hermitian algebroids admit para-holomorphic anchors, and demonstrates that exact para-Hermitian algebroids with flat para-complex connections realize para-holomorphic algebroid structures via a split-para-complex framework. The text provides concrete constructions and examples on $oldsymbol{T}M$, quadratic Lie groups, and Drinfeld doubles, showing how para-Hermitian geometry subsumes para-Kähler geometry and Poisson–Lie theory. These results generalize known structures and yield new perspectives on the interaction between para-complex and Courant-type objects, with potential extensions via supermanifold formalisms and broader classes of examples.

Abstract

The goal of this paper is to develop the theory of Courant algebroids with integrable para-Hermitian vector bundle structures by invoking the theory of Lie bialgebroids. We consider the case where the underlying manifold has an almost para-complex structure, and use this to define a notion of para-holomorphic algebroid. We investigate connections on para-holomorphic algebroids and determine an appropriate sense in which they can be para-complex. Finally, we show through a series of examples how the theory of exact para-holomorphic algebroids with a para-complex connection is a generalization of both para-Kähler geometry and the theory of Poisson-Lie groups.

Para-Holomorphic Algebroids and Para-Complex Connections

TL;DR

The paper develops a para-Hermitian extension of Courant algebroids by leveraging Lie bialgebroid structures to define para-holomorphic algebroids and para-complex connections. It establishes when para-Hermitian algebroids admit para-holomorphic anchors, and demonstrates that exact para-Hermitian algebroids with flat para-complex connections realize para-holomorphic algebroid structures via a split-para-complex framework. The text provides concrete constructions and examples on , quadratic Lie groups, and Drinfeld doubles, showing how para-Hermitian geometry subsumes para-Kähler geometry and Poisson–Lie theory. These results generalize known structures and yield new perspectives on the interaction between para-complex and Courant-type objects, with potential extensions via supermanifold formalisms and broader classes of examples.

Abstract

The goal of this paper is to develop the theory of Courant algebroids with integrable para-Hermitian vector bundle structures by invoking the theory of Lie bialgebroids. We consider the case where the underlying manifold has an almost para-complex structure, and use this to define a notion of para-holomorphic algebroid. We investigate connections on para-holomorphic algebroids and determine an appropriate sense in which they can be para-complex. Finally, we show through a series of examples how the theory of exact para-holomorphic algebroids with a para-complex connection is a generalization of both para-Kähler geometry and the theory of Poisson-Lie groups.
Paper Structure (21 sections, 32 theorems, 74 equations)

This paper contains 21 sections, 32 theorems, 74 equations.

Table of Contents

  1. Introduction
  2. Para-Complex Vector Bundles
  3. Basic Properties
  4. The Case of the Tangent Bundle
  5. Para-Holomorphic Bundles
  6. Para-Hermitian Manifolds
  7. Lie Algebroids, Courant Algebroids and Lie-Bialgebroids
  8. Lie Algebroids
  9. Courant Algebroids
  10. Lie Bialgebroids
  11. Para-Hermitian Algebroids
  12. Integrability and the Para-Käher Condition
  13. Exact Almost Para-Hermitian Algebroids
  14. Para-Holomorphic Algebroids
  15. Connections on Para-Hermitian Algebroids
  16. ...and 6 more sections

Key Result

Theorem 1.1

If $(E,J)$ is an exact para-Hermitian algebroid over a para-Hermitian manifold, and admits a flat para-complex connection $A$, then $E$ is a para-holomorphic algebroid.

Theorems & Definitions (74)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.1
  • Definition 2.2
  • Theorem 2.1
  • Theorem 2.2
  • Lemma 2.1
  • ...and 64 more