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A Geometric Realization of Spherical T-Duality via $\star$-Diagrams

Leonardo F. Cavenaghi, Lino Grama, Ludmil Katzarkov

TL;DR

The paper establishes a direct geometric bridge between the differential-geometric framework of $\star$-diagrams and the topological notion of spherical T-duality. It proves that for linear $S^3$-bundles over $S^4$, the existence of a $\star$-diagram between their total spaces is equivalent to the two bundles forming a spherical T-dual pair, thereby providing a concrete geometric realization of spherical T-duality. A key conceptual advance is the introduction of higher-dimensional generalized logarithmic transformations, which relate dual manifolds via diffeomorphisms on the correspondence space and reveal a Morita-equivalence underlying the T-duality isomorphisms. The results unify the Morita-theoretic view of groupoids with the cohomological and K-theoretic consequences of T-duality, and they lay groundwork for extending these dualities to principal and non-principal Milnor bundles through explicit geometric constructions.

Abstract

This paper establishes an equivalence between two distinct frameworks for constructing and relating smooth manifolds: the geometric theory of \emph{$\star$-diagrams} and the string-theory-inspired notion of \emph{spherical T-duality}. We prove that for linear $\mathrm{S}^3$-bundles over the 4-sphere, the existence of a $\star$-diagram connecting two such bundles is equivalent to them forming a spherical T-dual pair. This result provides a concrete geometric realization of spherical T-duality, interpreting its abstract cohomological definitions in the language of differential geometry. To forge this connection, we introduce a higher-dimensional generalization of \emph{logarithmic transformations}. These topological surgeries change the diffeomorphism type of the homology $Σ\times \mathrm{S}^1$, where $Σ$ is a homotopy sphere. Forgetting the $\mathrm{S}^1$-factor, they realize the constructed spherical T-dualities. Furthermore, we show that the known isomorphisms in the equivariant K-theory and cohomology between the T-dual manifolds are a direct consequence of an underlying \emph{Morita equivalence} between the action groupoids naturally associated with the base manifolds in a $\star$-diagram.

A Geometric Realization of Spherical T-Duality via $\star$-Diagrams

TL;DR

The paper establishes a direct geometric bridge between the differential-geometric framework of -diagrams and the topological notion of spherical T-duality. It proves that for linear -bundles over , the existence of a -diagram between their total spaces is equivalent to the two bundles forming a spherical T-dual pair, thereby providing a concrete geometric realization of spherical T-duality. A key conceptual advance is the introduction of higher-dimensional generalized logarithmic transformations, which relate dual manifolds via diffeomorphisms on the correspondence space and reveal a Morita-equivalence underlying the T-duality isomorphisms. The results unify the Morita-theoretic view of groupoids with the cohomological and K-theoretic consequences of T-duality, and they lay groundwork for extending these dualities to principal and non-principal Milnor bundles through explicit geometric constructions.

Abstract

This paper establishes an equivalence between two distinct frameworks for constructing and relating smooth manifolds: the geometric theory of \emph{-diagrams} and the string-theory-inspired notion of \emph{spherical T-duality}. We prove that for linear -bundles over the 4-sphere, the existence of a -diagram connecting two such bundles is equivalent to them forming a spherical T-dual pair. This result provides a concrete geometric realization of spherical T-duality, interpreting its abstract cohomological definitions in the language of differential geometry. To forge this connection, we introduce a higher-dimensional generalization of \emph{logarithmic transformations}. These topological surgeries change the diffeomorphism type of the homology , where is a homotopy sphere. Forgetting the -factor, they realize the constructed spherical T-dualities. Furthermore, we show that the known isomorphisms in the equivariant K-theory and cohomology between the T-dual manifolds are a direct consequence of an underlying \emph{Morita equivalence} between the action groupoids naturally associated with the base manifolds in a -diagram.
Paper Structure (15 sections, 25 theorems, 127 equations)

This paper contains 15 sections, 25 theorems, 127 equations.

Table of Contents

  1. Introduction
  2. A Concise Account and New Results on $\star$-diagrams
  3. Manufacturing exotic manifolds: the recipe for $\star$-diagrams
  4. Riemannian submersions and $\star$-diagrams
  5. $\star$-diagrams and Morita equivalence
  6. Spherical T-duality
  7. Milnor bundles: A recall
  8. Oriented $\mathrm{S}^3$-bundles over 4-manifolds
  9. Principal Spherical T-duality
  10. Non-principal Spherical T-duality
  11. Higher-Dimensional Logarithmic Transformations, Exotic Hopf Manifolds and Spherical T-duality
  12. Topological Logarithmic Transformations on $\mathrm{T}^2$-Fibrations
  13. A generalized logarithmic transformation proposal
  14. Generalized logarithmic transformations and $\star$-diagrams
  15. Spherical T-dualities and $\star$-diagrams

Key Result

Lemma 2.4

Let $U$ be a $G$-manifold and let $\theta, \theta' : U \to G$ be smooth maps satisfying eq:covariance. Then $\widehat{\theta}$ and $\widehat{\theta'}$ are equivariant diffeomorphisms such that where $\theta \theta'(x) := \theta(x) \theta'(x),~ \forall x \in U.$

Theorems & Definitions (60)

  • Example 2.1: The Gromoll--Meyer Exotic Sphere
  • Example 2.2: Pairs of Diffeomorphic Manifolds via $\star$-Diagrams
  • Definition 2.3
  • Lemma 2.4: Lemma 2.3 in SperancaCavenaghiPublished
  • Theorem 2.5: Theorem 2.2 in SperancaCavenaghiPublished
  • Proposition 2.6
  • Proposition 2.7: Proposition 5.3, p.28 in SperancaCavenaghiPublished
  • proof
  • Theorem 2.8
  • proof
  • ...and 50 more