  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.
