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12.6 Spin Complexes 420 12.7 The Heat Kernel and Generalised z-functions 424 12.8 The Atiyah–Patodi–Singer Index Theorem 429 Problems 12 433 13 Anomalies in Gauge Field Theories 434 13.1 Introduction 434 13.2 Abelian Anomalies 435 13.3 Non-Abelian Anomalies 441 13.4 The Wess–Zumino Consistency Conditions 445 13.5 Abelian Anomalies Versus Non-Abelian Anomalies 451 13.6 The Parity Anomaly in Odd-Dimensional Spaces 456 14 Bosonic String Theory 461 14.1 Differential Geometry on Riemann Surfaces 461 14.2 Quantum Theory of Bosonic Strings 468 14.3 One-Loop Amplitudes 487 References 492 Index 497