Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

T-duality, Gerbes and Loop Spaces

Dmitriy M. Belov, Chris M. Hull, Ruben Minasian

TL;DR

This work analyzes T-duality for sigma-models with target spaces that are principal torus bundles by treating the $B$-field as a gerbe connection and constructing the generalized correspondence space $Y$. It shows that duality can be understood as an $O(n,n;\mathbb{Z})$-symmetric operation on the loop-space phase space $T^*LY$, with two Hamiltonian reductions yielding the original and dual backgrounds when the obstructions vanish. A key finding is that the obstruction to geometric T-duality is the non-Hamiltonian action arising from a nontrivial de Rham class $[H_1]_{dR}$; when $H_1$ is exact (and $B_0^{IJ}$ is globally well-defined), $Y$ becomes a principal double torus bundle with a consistent affine connection, restoring Hamiltonian structure and allowing dual reductions. This framework clarifies the role of gerbes and affine double torus bundles in global T-duality, and accommodates obstructed cases by working on the enlarged space $Y$ and its loop-space symplectic geometry. Overall, the paper provides a unifying geometric and topological perspective on T-duality, linking gerbe data, double torus fibrations, and loop-space symplectic reductions, with potential implications for T-fold descriptions of duality.

Abstract

We revisit sigma models on target spaces given by a principal torus fibration $X\to M$, and show how treating the 2-form B as a gerbe connection captures the gauging obstructions and the global constraints on the T-duality. We show that a gerbe connection on X, which is invariant with respect to the torus action, yields an affine double torus fibration Y over the base space M - the generalization of the correspondence space. We construct a symplectic form on the cotangent bundle to the loop space LY and study the relation of its symmetries to T-duality. We find that geometric T-duality is possible if and only if the torus symmetry is generated by Hamiltonian vector fields. Put differently, the obstruction to T-duality is the non-Hamiltonian action of the symmetry group.

T-duality, Gerbes and Loop Spaces

TL;DR

This work analyzes T-duality for sigma-models with target spaces that are principal torus bundles by treating the -field as a gerbe connection and constructing the generalized correspondence space . It shows that duality can be understood as an -symmetric operation on the loop-space phase space , with two Hamiltonian reductions yielding the original and dual backgrounds when the obstructions vanish. A key finding is that the obstruction to geometric T-duality is the non-Hamiltonian action arising from a nontrivial de Rham class ; when is exact (and is globally well-defined), becomes a principal double torus bundle with a consistent affine connection, restoring Hamiltonian structure and allowing dual reductions. This framework clarifies the role of gerbes and affine double torus bundles in global T-duality, and accommodates obstructed cases by working on the enlarged space and its loop-space symplectic geometry. Overall, the paper provides a unifying geometric and topological perspective on T-duality, linking gerbe data, double torus fibrations, and loop-space symplectic reductions, with potential implications for T-fold descriptions of duality.

Abstract

We revisit sigma models on target spaces given by a principal torus fibration , and show how treating the 2-form B as a gerbe connection captures the gauging obstructions and the global constraints on the T-duality. We show that a gerbe connection on X, which is invariant with respect to the torus action, yields an affine double torus fibration Y over the base space M - the generalization of the correspondence space. We construct a symplectic form on the cotangent bundle to the loop space LY and study the relation of its symmetries to T-duality. We find that geometric T-duality is possible if and only if the torus symmetry is generated by Hamiltonian vector fields. Put differently, the obstruction to T-duality is the non-Hamiltonian action of the symmetry group.

Paper Structure

This paper contains 33 sections, 13 theorems, 106 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Obstructions to gauging and $T$-duality.
  3. The generalized correspondence space.
  4. T-duality as a symmetry of a loop space.
  5. Sigma-models on principal torus bundles
  6. Review of the obstructions to T-duality
  7. Principal torus bundle.
  8. Restrictions on the $3$-form $H$.
  9. Double fibration.
  10. Torus actions on the double fibration.
  11. Gerbes
  12. Gerbe.
  13. Gerbe on a principal torus bundle.
  14. T-duality constraints.
  15. T-duality in string sigma models
  16. ...and 18 more sections

Key Result

Theorem 2.1

The contraction of the invariant $3$-form $H$iso-H with the fundamental vector field defines a closed $2$-form $F_{\#}$ with integral periods on $X$. One can think of it as a curvature of a connection $\Theta_{\#}$ on a principal torus bundle $\mathbb{T}^n_{\#}\hookrightarrow Y\stackrel{p}{\longrigh

Figures (5)

  • Figure :
  • Figure :
  • Figure :
  • Figure :
  • Figure :

Theorems & Definitions (17)

  • Theorem 2.1
  • Theorem 2.2
  • Corollary 2.1
  • Corollary 2.2
  • Corollary 2.3
  • proof : Proof of the theorem:
  • Theorem 3.1
  • Corollary 3.1
  • proof
  • Lemma 3.1
  • ...and 7 more