Geometric Quantum Field Theories

This graduate-level volume is a coherent and self-contained introduction to Quantum Field Theory, uniquely focused on geometric and non-perturbative aspects. The first part covers quantum fields and Euclidean path integral, Yang-Mills field theories, and Wilsonian renormalization. Wilson's notion of the effective field theory and its heavy implication for the QFT framework itself are given particular attention. Next, geometrical and topological aspects are thoroughly treated, accompanied by a healthy dose of underlying mathematics. Anomalies, or quantum failures of classical symmetries, follow as crucial litmus tests for self-consistency, which are delineated in unprecedented detail, spanning decades of development. In the final part, the book asks how relativistic gravity, known to resist standard quantization schemes, may reconcile with the quantum world. This question is approached by invoking d=2 Weyl anomaly, Hawking effects, black hole partition functions, and the renormalization of fundamental strings, with a view toward quantum gravity and superstring theory.