Global analysis of duality maps in quantum field theory

Abstract

Abstract. A global analysis of duality transformations is presented. It is shown that duality between quantum field theories exists only when the geometrical structure of the quantum configuration spaces of the theories comply with certain precise conditions. Applications to S-dual actions and to T duality of string theories and D-branes are briefly discussed. It is shown that a new topological term in the dual open string actions is required. We also study an extension of the procedure to construct duality maps among abelian gauge theories to the non abelian case.Duality transformations were introduced by Dirac and extended later on by Montonen and Olive. More recently were used by Seiberg and Witten [1] to relate the weak and strong coupling regime in the analysis of the low energy effective action of the N=2 SUSY SU(2) Yang-Mills. This approach to non-perturbative QFT was then introduced in string theory with spectacular success. It has been shown that the strong coupling regime of one string theory can be mapped to the weak coupling regime of another perturbatively different string theory, giving rise to a possible unification of all string theories in the context of a hypothetical M-theory. In this lecture, we analyse the duality transformations from a global point of view. This approach requires the introduction of a more general geometrical structure than the associated to line bundles over a general euclidean base manifold. We describe [2] the general structure of higher order line bundles, and define over them dual maps between theories described locally by p-forms.Duality maps for theories with p-forms have been discussed in [3] and appear naturally in the description of D-brane theories. However, the global aspects of the configuration space of these local p-forms was never described. We show that the local analysis used in [3] is not enough to ensure quantum duality equivalence and give the necessary conditions to achieve it. The interesting result related to this global structure is that duality between theories of local 1) Invited talk at I-SILAFAE in a splendid site of Yucatán. 2) Plenary talk at I-SILAFAE, Merida,Mexico.