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T-duality for torus bundles via noncommutative topology

Varghese Mathai, Jonathan Rosenberg

TL;DR

The paper addresses the problem of T-duality for torus bundles with NS $H$-flux, showing that every principal $T^2$-bundle with $H$-flux admits a T-dual. The authors develop an equivariant Brauer-group framework and use crossed products to reveal a dichotomy: a classical T-dual exists if the fiberwise flux push-forward $p_!\delta$ vanishes, while a nonclassical dual—bundled noncommutative tori—appears when $p_!\delta\neq0$. They prove isomorphisms in twisted $K$-theory and, in the nonclassical case, twisted cyclic homology, connecting to Connes-Thom and Elliott–Natsume–Nest type results. The work unifies classical and noncommutative T-duality, provides explicit constructions for the noncommutative duals (e.g., fibers $A_{f(z)}$), and offers concrete examples (such as $\mathbb{T}^3$) to illustrate the framework. The results have implications for string theory and noncommutative geometry, highlighting when noncommutative duals are essential and how they preserve topological invariants.

Abstract

It is known that the T-dual of a circle bundle with H-flux (given by a Neveu-Schwarz 3-form) is the T-dual circle bundle with dual H-flux. However, it is also known that torus bundles with H-flux do not necessarily have a T-dual which is a torus bundle. A big puzzle has been to explain these mysterious "missing T-duals.'' Here we show that this problem is resolved using noncommutative topology. It turns out that every principal 2-torus-bundle with H-flux does indeed have a T-dual, but in the missing cases (which we characterize), the T-dual is non-classical and is a bundle of noncommutative tori. The duality comes with an isomorphism of twisted K-theories, just as in the classical case. The isomorphism of twisted cohomology which one gets in the classical case is replaced by an isomorphism of twisted cyclic homology.

T-duality for torus bundles via noncommutative topology

TL;DR

The paper addresses the problem of T-duality for torus bundles with NS -flux, showing that every principal -bundle with -flux admits a T-dual. The authors develop an equivariant Brauer-group framework and use crossed products to reveal a dichotomy: a classical T-dual exists if the fiberwise flux push-forward vanishes, while a nonclassical dual—bundled noncommutative tori—appears when . They prove isomorphisms in twisted -theory and, in the nonclassical case, twisted cyclic homology, connecting to Connes-Thom and Elliott–Natsume–Nest type results. The work unifies classical and noncommutative T-duality, provides explicit constructions for the noncommutative duals (e.g., fibers ), and offers concrete examples (such as ) to illustrate the framework. The results have implications for string theory and noncommutative geometry, highlighting when noncommutative duals are essential and how they preserve topological invariants.

Abstract

It is known that the T-dual of a circle bundle with H-flux (given by a Neveu-Schwarz 3-form) is the T-dual circle bundle with dual H-flux. However, it is also known that torus bundles with H-flux do not necessarily have a T-dual which is a torus bundle. A big puzzle has been to explain these mysterious "missing T-duals.'' Here we show that this problem is resolved using noncommutative topology. It turns out that every principal 2-torus-bundle with H-flux does indeed have a T-dual, but in the missing cases (which we characterize), the T-dual is non-classical and is a bundle of noncommutative tori. The duality comes with an isomorphism of twisted K-theories, just as in the classical case. The isomorphism of twisted cohomology which one gets in the classical case is replaced by an isomorphism of twisted cyclic homology.

Paper Structure

This paper contains 10 sections, 13 theorems, 31 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries on noncommutative tori
  3. Mathematical framework
  4. Structure of the equivariant Brauer group and T-duality
  5. Review of the case $n=1$
  6. Features of the general case
  7. Calculations for the case $n=2$
  8. Applications to T-duality
  9. Examples: torus bundles and noncommutative torus bundles over the circle
  10. Concluding remarks

Key Result

Theorem 4.2.1

Suppose $G$ is a connected Lie group and $X$ is a locally compact $G$-space (satisfying our finiteness assumptions). Then there is an exact sequence where and In addition, there is an exact sequence \xymatrix{ H^2(Z,\mathbb Z) \ar[r]^(.4){d_2''} & H^2_M(G, C(X,{\mathbb T})) \ar[r]^(.65)\xi & \ker F \ar[r]^(.3)\eta & H^1_M(G, H^2(X,\mathbb Z)). }

Figures (1)

  • Figure 1: In the diagram, the fiber over $z\in Z$ is the noncommutative torus $A_{f(z)}$, which is represented by a foliated torus, with foliation angle equal to $f(z)$.

Theorems & Definitions (22)

  • Theorem 4.2.1: CKRW
  • Theorem 4.2.2: Wig
  • Corollary 4.2.3
  • proof
  • Theorem 4.2.4
  • Theorem 4.2.5: Wig
  • Theorem 4.2.6: "generalized van Est" Gu
  • Corollary 4.2.7
  • proof
  • Proposition 4.3.1
  • ...and 12 more