*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 ...
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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.

*Giuseppe Mussardo*

- Published in print:
- 2020
- Published Online:
- May 2020
- ISBN:
- 9780198788102
- eISBN:
- 9780191830082
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198788102.003.0013
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics

The conformal transformations may be part of a larger group of symmetry. Chapter 13 discusses several of the extensions of conformal field theory, including supersymmetry, ZN transformations and ...
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The conformal transformations may be part of a larger group of symmetry. Chapter 13 discusses several of the extensions of conformal field theory, including supersymmetry, ZN transformations and current algebras. It also covers superconformal models, the Neveu–Schwarz and Ramond sectors, irreducible representations and minimal models, additional symmetry, the supersymmetric Landau–Ginzburg theory, parafermion models, the relation to lattice models, Kac–Moody algebras, Virasoro operators, the Sugawara Formula, maximal weights and conformal models as cosets. The appendix provides for the interested reader a self-contained discussion on the Lie algebras, include the dual Coxeter numbers, properties of weight vectors and roots/simple roots.Less

The conformal transformations may be part of a larger group of symmetry. Chapter 13 discusses several of the extensions of conformal field theory, including supersymmetry, **Z**_{N} transformations and current algebras. It also covers superconformal models, the Neveu–Schwarz and Ramond sectors, irreducible representations and minimal models, additional symmetry, the supersymmetric Landau–Ginzburg theory, parafermion models, the relation to lattice models, Kac–Moody algebras, Virasoro operators, the Sugawara Formula, maximal weights and conformal models as cosets. The appendix provides for the interested reader a self-contained discussion on the Lie algebras, include the dual Coxeter numbers, properties of weight vectors and roots/simple roots.

*Adam M. Bincer*

- Published in print:
- 2012
- Published Online:
- January 2013
- ISBN:
- 9780199662920
- eISBN:
- 9780191745492
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199662920.003.0020
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics

The Poincaré and Liouville groups are defined. The Poincaré group is the universal cover of the group of transformations that leave invariant the distance between two points in space-time. This means ...
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The Poincaré and Liouville groups are defined. The Poincaré group is the universal cover of the group of transformations that leave invariant the distance between two points in space-time. This means that it is the semidirect product of the group of translations and the universal cover of the proper orthochronous Lorentz transformations. There are two Poincaré Casimirs: the square of the momentum four-vector and the square of the Pauli–Lubanski four-vector. The irreducible representations of the Poincaré group come in three classes corresponding to the momentum four-vector being, time-like, light-like and space-like. This can be rephrased in terms of the little group. All known massive elementary particles belong to class one, all known massless particles belong to class two. The Poincaré group is non-compact so the unitary representations are infinite-dimensional except for a subset of representations in class 2, which are one-dimensional. This feat is accomplished by representing several generators by zero. Extending the Poincaré group by dilations results in the Weyl group, and extending it further by special conformal transformations results in the Liouville group. In conclusion, the Virasoro and Kac–Moody algebras are briefly mentioned. Biographical notes on Poincaré, Pauli, Lubanski, Kac and Moody are given.Less

The Poincaré and Liouville groups are defined. The Poincaré group is the universal cover of the group of transformations that leave invariant the distance between two points in space-time. This means that it is the semidirect product of the group of translations and the universal cover of the proper orthochronous Lorentz transformations. There are two Poincaré Casimirs: the square of the momentum four-vector and the square of the Pauli–Lubanski four-vector. The irreducible representations of the Poincaré group come in three classes corresponding to the momentum four-vector being, time-like, light-like and space-like. This can be rephrased in terms of the little group. All known massive elementary particles belong to class one, all known massless particles belong to class two. The Poincaré group is non-compact so the unitary representations are infinite-dimensional except for a subset of representations in class 2, which are one-dimensional. This feat is accomplished by representing several generators by zero. Extending the Poincaré group by dilations results in the Weyl group, and extending it further by special conformal transformations results in the Liouville group. In conclusion, the Virasoro and Kac–Moody algebras are briefly mentioned. Biographical notes on Poincaré, Pauli, Lubanski, Kac and Moody are given.