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On Higher Topological T-duality Functors

Ashwin S. Pande

TL;DR

The paper develops a higher-dimensional analogue of Bunke-Schick Topological T-duality by geometrizing P(X) through the moduli space G(X) of String Field Theory backgrounds over X. Objects are SFT backgrounds modeled as pairs (E→X, H) and morphisms are gauge transformations, organized into a category whose category of elements yields a functorial, homotopy-theoretic construction of invariants P0 and P1, with P0 recovering Bunke-Schick’s P(X) and P1 encoding mapping class group actions on background data. The framework shows Pk(X) vanishes for k≥2 in the current setting, but it provides a clear path to extend via richer gauge structures and Thomason cohomology, potentially capturing higher T-duality phenomena and connections to triples, doubled geometries, and T-folds. Physically, the approach links closed bosonic SFT massless-mode reductions to geometric moduli spaces that encode Topological T-duality, offering a new, categorially enriched perspective on dualities and their realizations in geometry and string theory.

Abstract

We use String Field Theory (SFT) to construct a higher analogue of Bunke-Schick's functor $P: \mathbf{Top}^{op} \to \mathbf{Set}$ \cite{BunkeS1} by geometrizing $P.$ We use the projection of SFT onto its massless modes \cite{SFTDiffeo} to construct the category $\C$ whose objects are pairs (which we identify with SFT backgrounds) and whose maps are morphisms of pairs (which are gauge transformations). Using $\C$ and categorical equivalence, for any $CW-$complex $X$ we define the moduli space $G(X)$ of SFT backgrounds which are pairs over $X$ up to gauge equivalence. We use the homotopy theory of the moduli space $G(X)$ to define functors on the category of $CW-$complexes $P_k:\mathbf{CW}^{op} \to \mathbf{Grpd}$ such that $P_0 \simeq P,$ $P_1$ is nontrivial and $P_k(X)$ are always trivial for $k \geq 2.$ Arrows in $P_1(X)$ are shown to be isotopy classes of maps in the mapping class group of $X$ acting on (isomorphism classes of) pairs over $X.$ We discuss applications to Topological T-duality for triples and to modelling doubled geometries and T-folds \cite{HullT}.

On Higher Topological T-duality Functors

TL;DR

The paper develops a higher-dimensional analogue of Bunke-Schick Topological T-duality by geometrizing P(X) through the moduli space G(X) of String Field Theory backgrounds over X. Objects are SFT backgrounds modeled as pairs (E→X, H) and morphisms are gauge transformations, organized into a category whose category of elements yields a functorial, homotopy-theoretic construction of invariants P0 and P1, with P0 recovering Bunke-Schick’s P(X) and P1 encoding mapping class group actions on background data. The framework shows Pk(X) vanishes for k≥2 in the current setting, but it provides a clear path to extend via richer gauge structures and Thomason cohomology, potentially capturing higher T-duality phenomena and connections to triples, doubled geometries, and T-folds. Physically, the approach links closed bosonic SFT massless-mode reductions to geometric moduli spaces that encode Topological T-duality, offering a new, categorially enriched perspective on dualities and their realizations in geometry and string theory.

Abstract

We use String Field Theory (SFT) to construct a higher analogue of Bunke-Schick's functor \cite{BunkeS1} by geometrizing We use the projection of SFT onto its massless modes \cite{SFTDiffeo} to construct the category whose objects are pairs (which we identify with SFT backgrounds) and whose maps are morphisms of pairs (which are gauge transformations). Using and categorical equivalence, for any complex we define the moduli space of SFT backgrounds which are pairs over up to gauge equivalence. We use the homotopy theory of the moduli space to define functors on the category of complexes such that is nontrivial and are always trivial for Arrows in are shown to be isotopy classes of maps in the mapping class group of acting on (isomorphism classes of) pairs over We discuss applications to Topological T-duality for triples and to modelling doubled geometries and T-folds \cite{HullT}.
Paper Structure (42 sections, 28 theorems, 24 equations)

This paper contains 42 sections, 28 theorems, 24 equations.

Table of Contents

  1. Introduction \newlabelSSecTTDP701
  2. T-duality and Topological T-duality
  3. Main Results and Physical Relevance
  4. Outline of Paper
  5. Introduction to SFT
  6. Morphisms in the Category of Pairs \newlabelSecIntro701
  7. Definition of the Category of Pairs ${\mathcal{C}}$
  8. Some Properties of Morphisms of Pairs
  9. Morphisms in ${\mathcal{C}}$
  10. Isomorphisms of Pairs covering the identity
  11. Isomorphisms of pairs covering a nontrivial homeomorphism
  12. Arbitrary Morphisms of Pairs
  13. The Space $G(X)$ and its Homotopy Properties \newlabelSecGXHot701
  14. The Lift of $P$ to ${\mathcal{C}}$ \newlabelSSecPCLift701
  15. The subcategory ${\mathcal{C}}_X$ \newlabelSSecDefCX701
  16. ...and 27 more sections

Key Result

Lemma 2.1

Suppose $f:Y \to X$ is a map and let $(E_0, H_0)$ and $(E_1, H_1)$ be isomorphic pairs over $X.$ Then the pullback pairs $(f^{\ast}E_0, f^{\ast}H_0)$ and $(f^{\ast}E_1, f^{\ast}H_1)$ are isomorphic over $Y.$ \newlabelLemPairIso701

Theorems & Definitions (28)

  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Theorem 3.1
  • Theorem 3.2
  • Theorem 3.3
  • Theorem 3.4
  • Theorem 3.5
  • Theorem 3.6
  • ...and 18 more