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Anomalies and Noncommutative Index Theory

While this narrative was probably one of the last actual accounts of the site before it was backfilled, it is reminiscent of the dramatic accounts given by those elite- minded tourists on the Grand Tour. The truth of the matter is that an excavation was conducted by Gennaro Matrone under the supervision of archaeologists who published the results of the excavation in 1901 and 1902. Both of the excavations yielded numer- ous objects that were dispersed to museums throughout the United States and Naples, Italy. Today, the site is known as the Fondo Bottaro Villa or the Contrada Bottaro Villa—names that it. | Anomalies and Noncommutative Index Theory Denis PERROT Institut Camille Jordan Université Claude Bernard Lyon 1 21 av. Claude Bernard 69622 Villeurbanne cedex France perrot@math.univ-lyon1.fr March 10 2010 Abstract These lectures are devoted to a description of anomalies in quantum field theory from the point of view of noncommutative geometry and topology. We will in particular introduce the basic methods of cyclic cohomology and explain the noncommutative counterparts of the Atiyah-Singer index theorem. Contents 1 Introduction 1 2 Quantum Field Theory and Anomalies 2 2.1 Classical gauge theory. 2 2.2 Quantum gauge theory. 7 2.3 Chiral anomalies. 9 3 Noncommutative Geometry 12 3.1 Noncommutative spaces. 12 3.2 K-theory and index theory . 15 3.3 Cyclic cohomology. 18 4 Index Theorems 23 4.1 The Chern-Connes character . 23 4.2 Local formulas and residues. 28 4.3 Anomalies revisited. 30 References 37 1 Introduction The aim of these lectures is to provide a modest insight into the interplay between Quantum Field Theory and Noncommutative Geometry 10 . We choose to focus on the very specialized problem of chiral anomalies in gauge theories 1 4 23 from the viewpoint of noncommutative index theorems. In fact both subjects can be considered as equivalent the link is essentially given by Bott periodicity in K-theory 7 . On one side chiral anomalies arise as the lack of 1 gauge invariance for a quantum field theory after renormalization. This confers a fundamental local nature to anomalies in a geometric sense. One the other hand a particular attention has been drawn in the last years to local index formulas in noncommutative geometry as formulated by Connes and Moscovici 15 . We propose to explain in these notes how local index formulas can be extracted from quantum anomalies. This is achieved by putting together cyclic cohomology and regularized traces 2 24 . These lectures are organized as follows. In the first section we recall some basic material on quantum field .