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T-duality for principal torus bundles and dimensionally reduced Gysin sequences

Peter Bouwknegt, Keith Hannabuss, Varghese Mathai

TL;DR

This work develops a global framework for T-duality of principal torus bundles with NS $H$-flux by constructing a dimensionally reduced Gysin sequence via the Chern-Weil mechanism. It shows that the T-dual is generically a continuous field of noncommutative, nonassociative tori over the base, classified by a 3‑tuple of flux components and dual curvature data, with the $O(n,n;\mathbb{Z})$ duality group acting linearly on the associated invariants. Twisted cohomology and a dimensionally reduced Gysin sequence underpin the duality, providing explicit maps and exact sequences that connect base and total space data. The results extend Buscher rules from circles to higher-rank tori and reveal deep connections to operator algebraic constructions, suggesting a rich geometry of T-duals beyond classical bundle theories. The findings offer a path toward manifestly T-duality-invariant descriptions in string theory that naturally incorporate noncommutative and nonassociative structures.

Abstract

We reexamine the results on the global properties of T-duality for principal circle bundles in the context of a dimensionally reduced Gysin sequence. We will then construct a Gysin sequence for principal torus bundles and examine the consequences. In particular, we will argue that the T-dual of a principal torus bundle with nontrivial H-flux is, in general, a continuous field of noncommutative, nonassociative tori.

T-duality for principal torus bundles and dimensionally reduced Gysin sequences

TL;DR

This work develops a global framework for T-duality of principal torus bundles with NS -flux by constructing a dimensionally reduced Gysin sequence via the Chern-Weil mechanism. It shows that the T-dual is generically a continuous field of noncommutative, nonassociative tori over the base, classified by a 3‑tuple of flux components and dual curvature data, with the duality group acting linearly on the associated invariants. Twisted cohomology and a dimensionally reduced Gysin sequence underpin the duality, providing explicit maps and exact sequences that connect base and total space data. The results extend Buscher rules from circles to higher-rank tori and reveal deep connections to operator algebraic constructions, suggesting a rich geometry of T-duals beyond classical bundle theories. The findings offer a path toward manifestly T-duality-invariant descriptions in string theory that naturally incorporate noncommutative and nonassociative structures.

Abstract

We reexamine the results on the global properties of T-duality for principal circle bundles in the context of a dimensionally reduced Gysin sequence. We will then construct a Gysin sequence for principal torus bundles and examine the consequences. In particular, we will argue that the T-dual of a principal torus bundle with nontrivial H-flux is, in general, a continuous field of noncommutative, nonassociative tori.

Paper Structure

This paper contains 15 sections, 3 theorems, 53 equations.

Table of Contents

  1. Introduction
  2. Principal circle bundles
  3. A bit of notation
  4. The Buscher rules revisited
  5. Dimensionally reduced Gysin sequence
  6. Twisted cohomology
  7. Principal torus bundles
  8. Dimensional reduction -- the Chern-Weil homomorphism
  9. Dimensionally reduced Gysin sequences
  10. Application to T-duality
  11. T-duality for principal torus bundles
  12. The T-duality group
  13. Example
  14. Conclusions and further generalizations
  15. Duality from the operator algebraic perspective

Key Result

Theorem 2.1

The sequence eqBg defines an exact complex.

Theorems & Definitions (7)

  • Theorem 2.1
  • proof
  • Remark
  • Remark
  • Theorem 3.1
  • proof
  • Theorem 3.2