Anomalies and Noncommutative Index Theory

Abstract

These lectures will be devoted to a description of anomalies in quantum field theory from the point of view of noncommutative geometry and topology. We will in particular explain the Atiyah-Singer index theorems and their noncommutative counterparts, and introduce the basics of cyclic cohomology.

Sections

1. Quantum Field Theory and Anomalies

   1.1  Classical and quantum field theory

   1.2  Dyson-Schwinger equations

   1.3  Gauge theories

   1.4  Anomalies

   1.5  BRS cohomology

   1.6  Examples

    

2. Classical Index Theorems

   2.1  Topological K-theory

   2.2  Elliptic operators

   2.3  The Atiyah-Singer index theorem

   2.4  Examples

   2.5  The index theorem for families

   2.6  Geometric interpretation of anomalies

    

3. Noncommutative Index Theorems

   3.1  Noncommutative geometry

   3.2  K-theory of Banach algebras

   3.3  Cyclic cohomology

   3.4  The Chern-Connes character

   3.5  Local formulas and residues

   3.6  Anomalies revisited

Specialized Talk

Anomalies and noncommutative geometry

Updated 14/09/2005

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