Aspects of Riemannian Geometry in Quantum Field

In this thesis we study in detail several situations where the areas of Riemannian geometry and quantum field theory come together. This study is carried out in three distinct situations. In the first part we show how to introduce new local gauge invariant variables for V = 1 supersymmetric Yang-Mills theory, explicitly parameterizing the physical Hilbert space of the theory. We show that these gauge invariant variables have a geometrical interpretation, and that they can be constructed such that the emergent geometry is that of AV = 1 supergravity: a Riemannian geometry with vector-spinor generated torsion. In the second part we study bosonic and supersymmetric sigma models, investigating to what extent their geometrical target space properties are encoded in the T-duality symmetry they possess. Starting from the consistency requirement between T-duality symmetry and renormalization group flows, we find the two-loop metric beta function for a d = 2 bosonic sigma model on a generic, torsionless background. We then consider target space duality transformations for heterotic sigma models and strings away from renormalization group fixed points. By imposing the consistency requirements between the T-duality symmetry and renormalization group flows, the one loop gauge beta function is uniquely determined. The issue of heterotic anomalies and their cancelation is addressed from this duality constraining viewpoint, providing new insight and mechanisms of anomaly cancelation. In the third part we compute a radiative contribution to an anomalous correlation function of one axial current and two energy-momentum tensors, (A,,(z)T,,,(y)Tp,(x)), corresponding to a contribution to the gravitational axial anomaly in the massless Abelian Higgs model. In all three situations there is a rich interplay between geometry and field theory. Thesis Supervisor: Jeffrey Goldstone Title: Cecil & Ida Green Professor of Physics