D-branes, actions effectives et sym\'etrie miroir

Abstract

This thesis is devoted to derivative corrections to the effective action of D-branes, and to mirror symmetry with D-branes. Series of derivative corrections first predicted by non-commutative gauge theory are completed by couplings between the metric and the gauge field. The result is interpreted as a deformation of the non-commutative gauge theory, whose structure survives. The derivation is applied to the tachyon field, whose potential is shown to be deformed by the very same corrections. Moreover, a prescription is given for the coupling of p-adic strings to a magnetic field, thus allowing to study p-adic solitons using non-commutative field-theory techniques. The link with topological D-branes is provided by the non-commutative description of D-branes in the B-model. The fibre bundles supported by the D-branes are still holomorphic in this description. Establishing this property involves the realization of D-branes as boundary conditions, within the framework of generalized complex geometry. This geometric framework is then used to describe mirror symmetry with D-branes on a Calabi--Yau manifold admitting a $T^3$-fibration. Two pure spinors, involved in the stability equations for topological D-branes, and modified by gauge fields, are exchanged, thus unifying Lagrangian and non-Lagrangian D-branes of the A-model as mirrors of stable D-branes of the B-model.