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Path Integrals and Quantum Anomalies$
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Kazuo Fujikawa and Hiroshi Suzuki

Print publication date: 2004

Print ISBN-13: 9780198529132

Published to Oxford Scholarship Online: September 2007

DOI: 10.1093/acprof:oso/9780198529132.001.0001

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TWO-DIMENSIONAL FIELD THEORY AND BOSONIZATION

TWO-DIMENSIONAL FIELD THEORY AND BOSONIZATION

Chapter:
(p.149) 8 TWO-DIMENSIONAL FIELD THEORY AND BOSONIZATION
Source:
Path Integrals and Quantum Anomalies
Author(s):

Kazuo Fujikawa

Hiroshi Suzuki

Publisher:
Oxford University Press
DOI:10.1093/acprof:oso/9780198529132.003.0008

This chapter discusses the quantum anomalies in two-dimensional field theory. Two-dimensional field theory is important in connection with conformal field theory and its applications to string theory and condensed matter theory. The description of fermionic theory in terms of bosonic theory, namely, the bosonization in the path integral formulation is formulated, and an issue related to a local counter-term is clarified. It is explained in this chapter that the central extensions in Kac-Moody and Virasoro algebras are the algebraic representations of chiral and general coordinate anomalies, respectively. The connection of the identities, written in terms of the operator product expansion in conformal field theory with the identities in conventional field theory, is also explained. Finally, the calculational method of Weyl anomalies in string theory and its implications are discussed. The ghost number anomaly in the first quantization of string theory is related to the Riemann-Roch theorem.

Keywords:   conformal field theory, string theory, condensed matter theory, central extension, Kac-Moody algebra, Virasoro algebra, Weyl anomaly, ghost number anomaly, Riemann-Roch theorem

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