Strings in Space-Time Cotangent Bundle and T-Duality

Abstract

A simple geometric description of T-duality is given by identifying the cotangent bundles of the original and the dual manifold. Strings propagate naturally in the cotangent bundle and the original and the dual string phase spaces are obtained by different projections. Buscher's transformation follows readily and it is literally projective. As an application of the formalism, we prove that the duality is a symplectomorphism of the string phase spaces.

Mirror Symmetry as a Poisson-Lie T-Duality

“…The equations (37) are equivalent to zero curvature equations for the F +− -component of the super stress tensor F M N [8,24] …”

Untitled

“…As we will see in next section, this defines a Lie bialgebra structure on G. It is shown in [6] that there exists a dual (equivalent) σ-model defined by a matrixẼ ij where the role of groups G and G * is interchanged, that is, there is an action of G * on the target manifold of the dual model such that ifṽ a are the generators of this action then…”

“…In this section we summarize the basic facts about Poisson-Lie T-duality [6,7] We consider a two-dimensional σ-model on a target manifold M described by the action…”

“…Eq. (8) can be explicitely solved in that case [6,11] for the matrix E ij . If G is connected and simply connected, any cocycle on G (the Lie algebra of G) with values in G ⊗ G can be integrated to a cocycle in G with values in G ⊗ G. Given the cocommutator δ K on G, we denote by Π R : G → G ⊗ G the corresponding cocycle in G. It satisfies the relation…”

“…Following [6] one can lift the extremal surface to D = G × G * by l(z,z) = g(z,z)h(z,z). The projection of l(z,z) onto G * by means of the opposite decomposition l(z,z) = h(z,z)g(z,z) defines an extremal surface h(z,z) of the dual model.…”