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Non-principal T-duality, generalized complex geometry and blow-ups

Gil R. Cavalcanti, Aldo Witte

TL;DR

The paper develops a comprehensive framework to extend $T$-duality from principal torus bundles to manifolds with fixed points by using elliptic divisors and the elliptic tangent bundle $\\mathcal{A}_{|D|}$. This enables a smooth isomorphism of Courant algebroids on the level of the elliptic algebroids, allowing the transport of invariant generalized complex structures across duals and clarifying when such lifts exist via residue and Haefliger-Salem data. It also shows that $T$-duality commutes with blow-ups in the elliptic setting, yielding a robust toolkit for constructing and transporting GC geometries in the presence of singular torus actions. The work provides both a structural theory (divisors, algebroids, residues, lifts) and practical mechanisms (correspondence spaces, dualities, and blow-up compatibilities) for generating and analyzing dual GC manifolds, with several explicit examples. Overall, it significantly broadens the scope of T-duality in differential geometry and its interaction with generalized complex and symplectic structures in singular settings.

Abstract

We extend the notion of T-duality to manifolds endowed with non-principal torus actions. The singularities of the torus action are controlled by a certain Lie algebroid, called the elliptic tangent bundle. Using this Lie algebroid, we explain how certain invariant generalized complex structures can be transported via T-duality. Along the way, we use the elliptic tangent bundle to define connections for these torus action, and give new insight to the classification of such actions by Haefliger-Salem.

Non-principal T-duality, generalized complex geometry and blow-ups

TL;DR

The paper develops a comprehensive framework to extend -duality from principal torus bundles to manifolds with fixed points by using elliptic divisors and the elliptic tangent bundle . This enables a smooth isomorphism of Courant algebroids on the level of the elliptic algebroids, allowing the transport of invariant generalized complex structures across duals and clarifying when such lifts exist via residue and Haefliger-Salem data. It also shows that -duality commutes with blow-ups in the elliptic setting, yielding a robust toolkit for constructing and transporting GC geometries in the presence of singular torus actions. The work provides both a structural theory (divisors, algebroids, residues, lifts) and practical mechanisms (correspondence spaces, dualities, and blow-up compatibilities) for generating and analyzing dual GC manifolds, with several explicit examples. Overall, it significantly broadens the scope of T-duality in differential geometry and its interaction with generalized complex and symplectic structures in singular settings.

Abstract

We extend the notion of T-duality to manifolds endowed with non-principal torus actions. The singularities of the torus action are controlled by a certain Lie algebroid, called the elliptic tangent bundle. Using this Lie algebroid, we explain how certain invariant generalized complex structures can be transported via T-duality. Along the way, we use the elliptic tangent bundle to define connections for these torus action, and give new insight to the classification of such actions by Haefliger-Salem.
Paper Structure (35 sections, 39 theorems, 55 equations)

This paper contains 35 sections, 39 theorems, 55 equations.

Table of Contents

  1. Introduction
  2. $T$-duality
  3. Torus actions and connections
  4. Blow-ups
  5. Generalized complex structures
  6. Organisation of the paper
  7. Acknowledgements
  8. Divisors and Lie algebroids; generalized complex structures and their lifts
  9. Divisors
  10. Lie algebroids and residue maps
  11. Generalized complex structures
  12. Lifting generalized complex structures
  13. Severa class
  14. Lifting Dirac structures
  15. Classification of torus actions and connections
  16. ...and 20 more sections

Key Result

Proposition 1

Let $M, M'$ be manifolds endowed with standard $T^k$-actions and common orbits space $B$ for which the torus actions are locally equivalent. Let $\Theta,\Theta'$ be elliptic connection one-forms on $M$ respectively $M'$. Then the class $[d\Theta-d\Theta'] \in H^2(A_{\partial B};\mathfrak{t}^k)$ (th

Theorems & Definitions (112)

  • Proposition
  • Theorem
  • definition 2.1
  • definition 2.2
  • definition 2.3
  • definition 2.4
  • definition 2.5
  • definition 2.6
  • remark 1
  • definition 2.7
  • ...and 102 more