Three fairy tales in mathematical physics

This thesis considers three distinct topics related to String Theory:; BPS geodesics in N = 2 supersymmetric Yang-Mills theory. Seiberg-Witten theory gives a prescription how to find the full quantum prepotential of N = 2 supersymmetric Yang-Mills theories in terms of a Riemann surface and a meromorphic differential {dollar}lambdasb{lcub}rm SW{rcub}{dollar}. This Riemann surface can be derived from String Theory, such that the BPS spectrum is given by the geodesics on the surface using the metric {dollar}dssp2 = vertlambdasb{lcub}SW{rcub}vertsp2.{dollar} We consider the BPS states of N = 2 supersymmetric SU(2) gauge theory with a massive adjoint matter multiplet using the geodesic approach. Our introduction of the notion of "geodesic horizons" helps to analyse the BPS spectrum in various phases of the Yang-Mills theory, and especially to test the strings approach in the pure-gauge limit.; Coulomb gas on the half plane. Due to applications in Condensed Matter Physics and in String Theory, it is interesting to consider conformal field theories on surfaces with boundaries. The Feigin-Fuchs integral representation (Coulomb gas) leads to a useful description of the bulk theory. I describe how known results for boundary conformal field theories can be derived in terms of the integral representation, using the Ising model as an example. I develop screening prescriptions by perturbatively expanding the Liouville theory with imaginary coupling and with Neumann boundary condition on the bosonic field. To generate the conformal blocks of more general boundary conditions, I propose the insertion of new boundary operators.; Folded strings falling into a black hole. Gauged Wess-Zumino-Witten models based on non-compact groups form an exact conformal background for string theory in carved spacetime. We find classical solutions of strings in two-dimensional GWZW backgrounds. The solutions. which describe longitudinally oscillating folded strings, are given in lattice-like patches of the world sheet, and a transfer operation determines the future evolution. As an example, the SL(2,R)/R two-dimensional black hole is considered. Several new features are found which are not shared by particles, such as the tunnelling of the string into the bare singularity region.