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Generalised Complex and Spinor Relations

Thomas C. De Fraja, Vincenzo Emilio Marotta, Richard J. Szabo

Abstract

Courant algebroid relations are used to define notions of relations between Dirac structures and spinors. It is shown under which circumstances a spinor relation gives a Courant algebroid relation and how it descends to a relation between Dirac structures. A converse to this result is proved: a T-duality relation induces a spinor relation that links the Dirac generating operators defining T-dual Courant algebroids, generalising the isomorphism of twisted cohomology associated with topological T-duality. We introduce the notion of relation between generalised complex structures and characterise their reduction. We also define relations between generalised Kähler structures, and rephrase them in terms of bi-Hermitian structures which induce T-duality relations between $\mathcal{N}=(2,2)$ supersymmetric sigma-models. We prove existence results for T-dual structures, and demonstrate compatibility of T-duality relations with Type II supergravity equations.

Generalised Complex and Spinor Relations

Abstract

Courant algebroid relations are used to define notions of relations between Dirac structures and spinors. It is shown under which circumstances a spinor relation gives a Courant algebroid relation and how it descends to a relation between Dirac structures. A converse to this result is proved: a T-duality relation induces a spinor relation that links the Dirac generating operators defining T-dual Courant algebroids, generalising the isomorphism of twisted cohomology associated with topological T-duality. We introduce the notion of relation between generalised complex structures and characterise their reduction. We also define relations between generalised Kähler structures, and rephrase them in terms of bi-Hermitian structures which induce T-duality relations between supersymmetric sigma-models. We prove existence results for T-dual structures, and demonstrate compatibility of T-duality relations with Type II supergravity equations.
Paper Structure (41 sections, 51 theorems, 395 equations, 1 table)

This paper contains 41 sections, 51 theorems, 395 equations, 1 table.

Table of Contents

  1. Introduction
  2. Courant Algebroids
  3. Generalised Complex Geometry
  4. Courant Algebroid Relations
  5. The Fourier-Mukai Transform
  6. Outline and Summary of Main Results
  7. Acknowledgements
  8. Linear Algebra of Spinors and Relations
  9. Linear Relations
  10. Clifford Algebras
  11. Linear Clifford Relations
  12. Courant Algebroids, Dirac Structures and Generalised Complex Geometry
  13. Spinors and Courant Algebroids
  14. Reduction of Courant Algebroids
  15. Dirac Structures
  16. ...and 26 more sections

Key Result

Proposition 1.14

Under the assumptions of DeFraja:2023fhe, let $\underline{L}{}_1$ be an invariant Dirac structure on $\underline{E}{}_1$. Then there exists a unique Dirac structure $\underline{L}{}_2$ on $\underline{E}{}_2$ such that the T-duality relation $R:\underline{E}{}_1\dashrightarrow\underline{E}{}_2$ is a

Theorems & Definitions (171)

  • Definition 1.11: Definition \ref{['def:Diracrel']}
  • Proposition 1.14: Proposition \ref{['prop:TdualDirac']}
  • Definition 1.15: Definition \ref{['def:Rcliffrelation']}
  • Proposition 1.16: Proposition \ref{['prop:LreliffUrelsmooth']}
  • Proposition 1.17: Proposition \ref{['prop:DGOimpliesCArel']}
  • Proposition 1.18: Proposition \ref{['prop:regCAonB']}
  • Theorem 1.20: Theorem \ref{['prop:canonicalDGOforR']}
  • Proposition 1.21: Proposition \ref{['prop:TdualitytypeII']}
  • Definition 1.23: Definition \ref{['def:generalisedcomplexrel']}
  • Proposition 1.24: Proposition \ref{['prop:UreliffJrel']}
  • ...and 161 more