Geometric Quantum Field Theories

This graduate-level book is a coherent and self-contained introduction to quantum field theory (QFT), with a distinctive focus on geometric and nonperturbative aspects. The opening part covers quantum fields and the 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, geometric and topological aspects are thoroughly treated, accompanied by a healthy dose of underlying mathematics. Anomalies—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 be reconciled with the quantum world. This question is approached by invoking the d = 2 Weyl anomaly, Hawking effects, black hole partition functions, and the renormalization of fundamental strings, with a view toward superstring theory.