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Loop Hori Formulae for T-duality and Twisted Bismut-Chern Character

Fei Han, Varghese Mathai

TL;DR

This work lifts T-duality with $H$-flux to the loop space, introducing the loop Hori map $LT_*$ and its dual as kernel-based, quasi-isomorphic correspondences between exotic twisted cohomology theories on loop spaces. It develops a rigorous loop-space integration along the fiber and uses the horizontal twisted Bismut–Chern character of the Poincaré gerbe module as the loop-space analogue of the classical Hori kernel, thereby establishing loop-space T-duality with $H$-flux and a graded extension. Restricting to constant loops recovers the familiar spacetime Hori transform and clarifies how loop-space duality refines base-space results from earlier work. The paper also provides a graded loop Hori formalism and an explicit example to illustrate how loop-space duality manifests as a refinement of T-duality at the level of loop spaces and gerbe modules.

Abstract

The main purpose of this paper is to establish the loop space formulation of T-duality in the presence of background flux. In particular, we construct a loop space analogue of the Hori formula, termed \textbf{the loop Hori map}, and demonstrate that it induces a quasi-isomorphism between the exotic twisted equivariant cohomologies on the free loop spaces of the T-dual sides. Spacetime, when viewed as the constant loops, is a submanifold of loop space. The duality that we prove on loop space restricts to the T-duality with $H$-flux on spacetime. This significantly refines the earlier work of the authors in 2015 where T-duality was established after localisation to the base space. The construction of the loop Hori map is an application of our generalization of the Bismut--Chern character in 2015, originally introduced in the loop space interpretation of the Atiyah--Singer index theorem by Atiyah--Witten and Bismut.

Loop Hori Formulae for T-duality and Twisted Bismut-Chern Character

TL;DR

This work lifts T-duality with -flux to the loop space, introducing the loop Hori map and its dual as kernel-based, quasi-isomorphic correspondences between exotic twisted cohomology theories on loop spaces. It develops a rigorous loop-space integration along the fiber and uses the horizontal twisted Bismut–Chern character of the Poincaré gerbe module as the loop-space analogue of the classical Hori kernel, thereby establishing loop-space T-duality with -flux and a graded extension. Restricting to constant loops recovers the familiar spacetime Hori transform and clarifies how loop-space duality refines base-space results from earlier work. The paper also provides a graded loop Hori formalism and an explicit example to illustrate how loop-space duality manifests as a refinement of T-duality at the level of loop spaces and gerbe modules.

Abstract

The main purpose of this paper is to establish the loop space formulation of T-duality in the presence of background flux. In particular, we construct a loop space analogue of the Hori formula, termed \textbf{the loop Hori map}, and demonstrate that it induces a quasi-isomorphism between the exotic twisted equivariant cohomologies on the free loop spaces of the T-dual sides. Spacetime, when viewed as the constant loops, is a submanifold of loop space. The duality that we prove on loop space restricts to the T-duality with -flux on spacetime. This significantly refines the earlier work of the authors in 2015 where T-duality was established after localisation to the base space. The construction of the loop Hori map is an application of our generalization of the Bismut--Chern character in 2015, originally introduced in the loop space interpretation of the Atiyah--Singer index theorem by Atiyah--Witten and Bismut.
Paper Structure (21 sections, 10 theorems, 221 equations)

This paper contains 21 sections, 10 theorems, 221 equations.

Table of Contents

  1. Brief Background on T-Duality with $H$-Flux and the Hori Map
  2. Preliminary Study of the Loop Space Perspective of T-Duality with $H$-Flux
  3. The Loop Hori Map and T-Duality on Loop Spaces: Main Results
  4. Integration Along the Fiber in Loop Spaces
  5. Family Localization in Finite Dimension
  6. Integration Along the Fiber in Loop Spaces
  7. The Case of Principal Circle Bundles
  8. The Twisted Bismut-Chern Character of the Poincaré Gerbe Module on the Correspondence Space
  9. Review of Exotic Twisted Equivariant Cohomology of Loop Space and Twisted Bismut-Chern Character
  10. Exotic Twisted Equivariant Cohomology of Loop Space
  11. Twisted Bismut-Chern Character of Gerbe Modules
  12. T-duality with H-flux
  13. The Poincaré Gerbe Module on the Correspondence Space and its Twisted Bismut-Chern Character
  14. The correspondence gerbe on the correspondence space and its horizontal holonomy line bundle on the free loop space of the correspondence space
  15. The Poincaré gerbe module over the correspondence gerbe
  16. ...and 6 more sections

Key Result

Theorem 1.3

For $\omega \in \Omega^*(LM)^{L\mathbb{T}}$, one has:

Theorems & Definitions (19)

  • Definition 1.1
  • Remark 1.2
  • Theorem 1.3
  • Remark 1.4
  • proof
  • Proposition 2.1: Brylinski, Section 6.1 in Bry
  • Theorem 2.2: HM15
  • Theorem 2.3: HM15
  • Theorem 2.4
  • Theorem 2.5: T-duality isomorphism BEM04aBEM04b
  • ...and 9 more