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T-duality for torus bundles with H-fluxes via noncommutative topology, II: the high-dimensional case and the T-duality group

Varghese Mathai, Jonathan Rosenberg

TL;DR

This work extends T-duality for principal torus bundles with H-flux to higher dimensions using noncommutative topology. It characterizes when a classical dual exists via the kernel of the fiber-inclusion map on $H^3$ and the fiberwise flux $p_!\delta$, otherwise yielding a non-classical dual as a continuous field of noncommutative tori; twisted K-theory is preserved under duality, with a dimension shift, and twisted cyclic homology substitutes twisted cohomology in the non-classical case. The paper develops the classifying space framework, identifying a T-duality group $GO(n,n;\mathbb{Z})$ acting on the data, and reveals a natural splitting of the classifying space when $n=2$. It provides explicit examples illustrating both classical and non-classical duals, analyzes the two-stage Postnikov structure in Bunke–Schick’s setup, and discusses potential generalizations to discrete groups via Baum–Connes. Overall, it links geometric T-duality with operator-algebraic invariants, offering concrete tools for understanding dualities in string theory and noncommutative geometry.

Abstract

We use noncommutative topology to study T-duality for principal torus bundles with H-flux. We characterize precisely when there is a "classical" T-dual, i.e., a dual bundle with dual H-flux, and when the T-dual must be "non-classical," that is, a continuous field of noncommutative tori. The duality comes with an isomorphism of twisted $K$-theories, required for matching of D-brane charges, just as in the classical case. The isomorphism of twisted cohomology which one gets in the classical case is replaced in the non-classical case by an isomorphism of twisted cyclic homology. An important part of the paper contains a detailed analysis of the classifying space for topological T-duality, as well as the T-duality group and its action. The issue of possible non-uniqueness of T-duals can be studied via the action of the T-duality group.

T-duality for torus bundles with H-fluxes via noncommutative topology, II: the high-dimensional case and the T-duality group

TL;DR

This work extends T-duality for principal torus bundles with H-flux to higher dimensions using noncommutative topology. It characterizes when a classical dual exists via the kernel of the fiber-inclusion map on and the fiberwise flux , otherwise yielding a non-classical dual as a continuous field of noncommutative tori; twisted K-theory is preserved under duality, with a dimension shift, and twisted cyclic homology substitutes twisted cohomology in the non-classical case. The paper develops the classifying space framework, identifying a T-duality group acting on the data, and reveals a natural splitting of the classifying space when . It provides explicit examples illustrating both classical and non-classical duals, analyzes the two-stage Postnikov structure in Bunke–Schick’s setup, and discusses potential generalizations to discrete groups via Baum–Connes. Overall, it links geometric T-duality with operator-algebraic invariants, offering concrete tools for understanding dualities in string theory and noncommutative geometry.

Abstract

We use noncommutative topology to study T-duality for principal torus bundles with H-flux. We characterize precisely when there is a "classical" T-dual, i.e., a dual bundle with dual H-flux, and when the T-dual must be "non-classical," that is, a continuous field of noncommutative tori. The duality comes with an isomorphism of twisted -theories, required for matching of D-brane charges, just as in the classical case. The isomorphism of twisted cohomology which one gets in the classical case is replaced in the non-classical case by an isomorphism of twisted cyclic homology. An important part of the paper contains a detailed analysis of the classifying space for topological T-duality, as well as the T-duality group and its action. The issue of possible non-uniqueness of T-duals can be studied via the action of the T-duality group.

Paper Structure

This paper contains 12 sections, 18 theorems, 52 equations, 2 figures.

Table of Contents

  1. Notation and review of results from MR
  2. Main mathematical results
  3. Applications to T-duality
  4. Examples
  5. The classifying space for topological T-duality, and action of the T-duality group
  6. Review of the Bunke-Schick construction for $n=1$
  7. The higher-rank case and the T-duality group
  8. A splitting of the classifying space when $n=2$
  9. The T-duality group in action
  10. Some classical T-dual examples
  11. Possible generalizations
  12. Open questions

Key Result

Theorem 1.1

Suppose $G\cong \mathbb R^n$ is a vector group and $X$ is a locally compact $G$-space (satisfying our finiteness assumptions). Then there is an exact sequence: \xymatrix@C-1.2pc{ H^2(X,\mathbb Z)\ar[r]^(.4){d_2''} & H^2_M(G, C(X,{\mathbb T})) \ar[r]^(.65)\xi & \operatorname{Br}_G(X) \ar[r]^F & H^3(

Figures (2)

  • 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 $f(z)$.
  • Figure 2: The Serre spectral sequence

Theorems & Definitions (37)

  • Theorem 1.1: MR
  • Lemma 1.2: MR
  • Theorem 1.3: Gu, quoted in MR
  • Lemma 2.1
  • proof
  • Theorem 2.2
  • proof
  • Theorem 2.3
  • proof
  • Theorem 3.1
  • ...and 27 more