On Gaugino Condensation in String Theory

We derive the precise form of the low-energy four-dimensional EFT for type IIB string theory compactified on the complex cone over Kähler-Einstein del Pezzo surfaces, including N spacetime-filling D3-branes and assuming Minkowski externally. We explicitly derive the theory for the K\"ahler modulus in the simplest case of a complex cone over the complex projective plane, with a stack of four D7-branes and one O7-plane wrapped around the base of the cone. An effective scalar potential appears in the theory, due to gaugino condensation taking place at low energies over the D7-branes stack, exhibiting a runaway direction and an unstable de Sitter vacuum. We find an explicit cosmological-like solution for the K\"ahler modulus, showing that the warped volume of the internal complex projective plane inflates with time in a runaway fashion. We conclude that type IIB string theory compactified on the complex cone over the complex projective plane, with four D7-branes and one O7-plane wrapped around the complex projective plane, is unstable. We explicitly derive the ten-dimensional equations of motion for maximally symmetric time-dependent metric perturbations by means of an ad hoc procedure, and we exhibit both stationary and time-dependent solutions, whose boundary conditions are imposed in part by the gaugino condensate stress-energy tensor. We partially fix the free parameters of the time-dependent solution using the results from the four-dimensional low-energy EFT. This thesis contains also an introduction to string compactifications, to the KKLT scenario and to the literature about ten-dimensional effects of gaugino condensation.