Studies in Quantum Field Theory

We analyze several topics in quantum field theory, mainly motivated by their role in the formulation of string theories. The common theme in what follows is the implementation of symmetries, such as local supersymmetry or BRST symmetry, through an action principle and the analysis of anomalies, the latter describing the breakdown of these symmetries at the quantum level. In the first part of this dissertation, we analyze "chiral bosons", i.e. massless scalar fields in a two -dimensional spacetime propagating in only one of the two light-cone directions. We present a general method for constructing couplings for chiral bosons and give details for the coupling to supergravity. The notion of a two dimensional chiral boson is generalized in d = 4k + 2 spacetime dimensions to that of a self-dual antisymmetric tensor field. We derive the coupling to gravity and compute the gravitational anomalies using the Feynman rules obtained from the action. We find agreement with the important work of Alvarez-Gaume and Witten, who conjectured the relevant Feynman rules. Our result therefore completes and justifies the Alvarez-Gaume-Witten findings. For the case of d = 2 we also show how to use the method of Fujikawa for computing anomalies from the non-invariance of the path integral measure. We obtain the full effective action by integrating the anomaly equation. In the second part we focus on a method for computing the consistent anomalies in the Fujikawa scheme. In a first application, we derive the consistent regulators for the various fields of the quantum action of the spinning string in superspace. These regulators produce the anomalies which satisfy the Wess-Zumino consistency conditions. In a second application, we analyze the anomalous structure of the Green-Schwarz formulation of the heterotic string. We find anomalies which generically do not cancel on an arbitrary world-sheet manifold. This raises questions concerning the possible validity of such a formulation of string theory which are not answered in this thesis. In the third and final part, we concentrate on the BRST symmetry, which is as an essential tool for the interpretation of a gauge theory in an arbitrary gauge. We first discuss the problem of deriving a BRST invariant quantum action for the Casalbuoni-Brink-Schwarz superparticle. Then we introduce a general method for treating the change of variables in the path integral by associating a BRST symmetry with it. This allow us to derive the BRST charges for the so-called gauged WZWN model. Eventually, we use BRST techniques, as embodied in the Batalin-Vilkovisky antibracket formalism, to discuss general Ward identities for systems with non-linear symmetries and involving external fields. An example of such a system, the super-W_3 -gravity model, is presented.