These lecture notes describe the method of N = 4 supergravity gaugings used as a four-dimensional effective Lagrangian description of the moduli superpotentials generated by superstring vacua with fluxes.
367of compactifications with fluxes in many classes of superstring backgrounds. 1 This approach of the moduli problem is not the subject of the present notes.The second approach uses four-dimensional effective Lagrangians. The idea underlying the effective field theory method is to translate the known properties, and in particular the symmetry content, of the underlying fundamental theory into constraints on a field theory description of the light (four-dimensional) modes only. This effective field theory is then a tool to investigate various aspects of the expected low-energy physics predicted by the fundamental theory. It is also used to isolate the situations relevant to phenomenology and then the classes of vacua deserving a full-fledged, ten-dimensional study. The effective Lagrangian approach can be viewed as a bottom-up approach, in contrast to the top-down method provided by direct studies of compactifications with fluxes in classes of string vacua.Superstring vacua relevant to phenomenology have sixteen supercharges. This large class of solutions includes in particular heterotic (and type I) strings and type II orientifolds. In four dimensions, sixteen supercharges lead to N = 4 supergravity coupled to the N = 4 super-Yang-Mills system. This theory has a severely restricted structure. The sigma-model defining its scalar sector is for instance unique. In fact, the only freedom to introduce parameters in N = 4 supergravity resides in the choice of gauging applied to its vector fields and multiplets. Hence, a gauged N = 4 supergravity treatment of the massless modes of superstring compactifications, supplemented with a breaking mechanism to N = 1 for potentially realistic compactifications, seems an appropriate starting point for an effective Lagrangian description of string compactifications: it is expected that the gauging parameters of the effective N = 4 supergravity theory encode the data of underlying string vacua, including non-trivial fluxes. This approach of N = 4 supergravity gaugings used for the derivation and study of moduli superpotentials has been developed in [6] 2 , and expanded to the inclusion of non-perturbative gaugino condensates in [8].The purpose of the present notes 3 is to describe the method of supergravity gaugings in relation with the effective description of string moduli physics. They do not however discuss the application of the method to the study of specific physics problems or classes of compactifications with fluxes. The next four sections describe in general terms various aspects of field theory and supergravity gaugings, starting with elementary considerations. After a detailed discussion of the relevant aspects of N = 4 supergravity (Sect. 6), the specific use of N = 4 supergravity gaugings to describe moduli effective supergravities is the subject of sections 7, whic...