*Reinhold A. Bertlmann*

- Published in print:
- 2000
- Published Online:
- February 2010
- ISBN:
- 9780198507628
- eISBN:
- 9780191706400
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198507628.003.0012
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics

This chapter focuses on gravitation. It first reviews some basic concepts of the Riemannian geometry and establishes notation. It then discusses the gravitational action, specifically the fermionic ...
More

This chapter focuses on gravitation. It first reviews some basic concepts of the Riemannian geometry and establishes notation. It then discusses the gravitational action, specifically the fermionic action. It introduces Einstein-, Lorentz-, and Weyl anomalies by violating the corresponding Einstein-, Lorentz-, and Weyl symmetries, and establishes consistency conditions. The equivalence of the Einstein- and Lorentz anomaly is demonstrated, and the covariant anomaly is discussed. Finally, the chapter treats gravitation on a BRS level, deriving the SZ chain of descent equations. Index theorems are use to carry out explicit anomaly examples.Less

This chapter focuses on gravitation. It first reviews some basic concepts of the Riemannian geometry and establishes notation. It then discusses the gravitational action, specifically the fermionic action. It introduces Einstein-, Lorentz-, and Weyl anomalies by violating the corresponding Einstein-, Lorentz-, and Weyl symmetries, and establishes consistency conditions. The equivalence of the Einstein- and Lorentz anomaly is demonstrated, and the covariant anomaly is discussed. Finally, the chapter treats gravitation on a BRS level, deriving the SZ chain of descent equations. Index theorems are use to carry out explicit anomaly examples.

*Kazuo Fujikawa and Hiroshi Suzuki*

- Published in print:
- 2004
- Published Online:
- September 2007
- ISBN:
- 9780198529132
- eISBN:
- 9780191712821
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198529132.003.0008
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics

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 ...
More

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.Less

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.