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The BV double of a Courant algebroid

Anton M. Zeitlin

Abstract

We characterize a Courant algebroid with a Calabi-Yau structure as a homotopy BV algebra with certain properties. We explain how it fits into recent double copy constructions relating Yang-Mills homotopy algebras to the ones of Double Field Theory and Gravity.

The BV double of a Courant algebroid

Abstract

We characterize a Courant algebroid with a Calabi-Yau structure as a homotopy BV algebra with certain properties. We explain how it fits into recent double copy constructions relating Yang-Mills homotopy algebras to the ones of Double Field Theory and Gravity.

Paper Structure

This paper contains 30 sections, 32 theorems, 276 equations.

Table of Contents

  1. Introduction
  2. Complex $(\mathcal{F}^{\bullet}, Q)$ and its properties
  3. The structure of differential $Q$
  4. Complex $(\mathcal{F}^{\bullet}_{1/2},Q)$ and an odd pairing
  5. Courant algebroids and Leibniz algebras on $(\mathcal{F}^{\bullet}_{1/2},Q)$
  6. Courant algebroids and homotopy algebras
  7. Calabi-Yau structures and Courant $\mathcal{O}_X$-algebroids
  8. Extending the Leibniz algebra to $(\mathcal{F}^{\bullet}_{1/2},Q)$
  9. Homotopy BV algebra on $(\mathcal{F}^{\bullet},Q)$ and the BV double
  10. Homotopy BV-LZ algebra
  11. The BV double of a Courant algebroid
  12. $C_\infty$-subalgebra
  13. $\mathcal{O}_X$-Courant algebroid and the cyclic structure
  14. $L_\infty$-subalgebra
  15. Vertex algebra realization
  16. ...and 15 more sections

Key Result

Theorem 2.1

The differential $Q$ on the complex $(\mathcal{F}, Q)$ admits the following decomposition: so that where $[{\rm d}, {\bf b}]=[{\rm d}, {\bf c}]=0$, $[{\rm d^*}, {\bf b}]=[{\rm d^*}, {\bf c}]=0$ so that ${\rm d^* ~d}=0$ and ${\rm d}~ {\rm d^*}=0$:

Theorems & Definitions (49)

  • Theorem 2.1
  • proof
  • Proposition 2.2
  • proof
  • Proposition 3.1
  • Theorem 3.2
  • Proposition 3.3
  • Proposition 4.1
  • Theorem 4.2
  • proof
  • ...and 39 more