Lie groups, Lie algebras, cohomology and some applications in physics.

*(English)*Zbl 0836.22027
Cambridge Monographs on Mathematical Physics. Cambridge: Cambridge Univ. Press. xvii, 455 p. (1995).

A self-contained monographic exposition of modern cohomology theory of Lie groups and Lie algebras as well as some of its applications in physics are presented. The following main topics are treated in the book: differential geometry of Lie groups, fibre bundles and connections, characteristic classes, index theorems, monopoles, instantons, extensions of Lie groups and Lie algebras, some applications of supersymmetry, Chevalley-Eilenberg approach to Lie algebra cohomology, symplectic cohomology, jet-bundle approach to variational principles in mechanics, Wess-Zumino-Witten terms, infinite (Virasoro, Kac-Moody) algebras, the cohomology descent in mechanics and in gauge field theories and anomalies. The contents of the book are organized as follows:

Ch. 1. Lie groups, fibre bundles and Cartan calculus: Some topics in the differential geometry of Lie groups. Left- and right-invariant vector fields as differential operators acting on Lie group manifolds and vector fibre bundles. Differential forms and Cartan calculus. De Rham cohomology and Hodge-de Rham theory. The Maurer-Cartan equations and Lie algebras. Left- and bi-invariances. The supertranslation group. Some homotopy groups. The Poincaré polynomials of compact simple Lie groups and on the homotopy groups of spheres.

Ch. 2. Connections and characteristic classes: Description of elements of the theory of connections on principal bundles. Equivalent forms of Lie groups. Characteristic (Chern) classes, Chern characters and Chern-Simons forms. The magnetic monopole. Yang-Mills instantons. Pontryagin classes and the Euler class. Index theorems for manifolds without boundary and for the spin complexes. Twisted complexes.

Ch. 3. A first look at cohomology of groups and related topics: Some specific features of classical and quantum mechanics as a physical motivation for introducing group cohomology and group extensions. Two- cocycles in Galilei covariant classical and quantum mechanics, in Bargmann’s theory of projective (ray) Lie group representations as well as in the Weyl-Heisenberg group and in the extended Galilei group theories. Dynamical groups, symplectic cohomology and geometrical quantization. The three-cocycles and associativity breaking. Group contraction procedure and non-trivial group cohomology.

Ch. 4. An introduction to abstract group extension theory: The general theory of group extensions and the formalism of exact sequences of homomorphisms in terms of fibre bundles. Principal bundle description of an abstract group extension \(\widetilde {G}\) of \(G\) by \(K\) (structure group). Group laws for \(G\) in terms of the factor system which determines and is determined by the extension.

Ch. 5. Cohomology groups of a group \(G\) and extensions by an abelian kernel: Introduction of the concept of cocycles and coboundaries for the group cohomology with values in an abelian group. The second cohomology groups and abelian extensions, including the semidirect and central ones. Superspace as group extension. Cohomology induced by the action of the group \(G\) on a manifold \(M^m\) \(({\mathcal F}(M)\)-valued cochains).

Ch. 6. Cohomology of Lie algebras: Cohomology of Lie algebras with values in a vector space, the Whitehead lemmas and Lie algebra extensions which are related to the second cohomology groups. The relations between Lie groups and Lie algebras with the simple example of central extensions of groups and algebras (governed by two complexes) as well as the higher order case. Explicit formulas for obtaining Lie algebra cocycles from the Lie group ones and vice versa. The Chevalley-Eilenberg formulation of the Lie algebra cohomology and invariant differential forms on a Lie group \(G\). The BRST (Becchi, Rouses, Stora and Tyutin) approach to Lie algebra cohomology. Lie algebra cohomology versus Lie group cohomology. \({\mathcal F} (M)\)-valued Lie algebra cohomology.

Ch. 7. Group extensions by non-abelian kernels: Reduction of the theory of group extensions by non-abelian groups \(K\) to an abelian cohomology problem: a \(G\)-kernel is extendible iff a certain three-cocycle with values in the centre \(C_K\) of \(K\) is trivial, and the possible extensions of \(G\) by \(K\) are in one-to-one correspondence with the extensions of \(G\) by the abelian group \(C_K\). Construction of extensions. The right “covering” groups of the complete Lorentz group.

Ch. 8. Cohomology and Wess-Zumino terms. An introduction: The variational principle and the Nöther theorem in Newtonian mechanics. The analysis of the relation between mechanics and cohomology from the Lagrangian point of view introduced in the 1-jet bundle framework. Quasi-invariance of Lagrangians and group (Lie algebra) central extensions. Description of quasi-invariant Lagrangians in terms of one-cocycles as well as two- cocycles and the cohomological descent procedure. The complementary aspect for obtaining physical actions from non-trivial cohomology and the concept of the Wess-Zumino term on a group manifold. The different role of the left and right versions of the symmetry Lie algebra involved. The cohomological descent approach to classical anomalies. The massive superparticle, the Wess-Zumino terms for supersymmetric extended objects and the supersymmetry algebra. The Lagrangian description of the magnetic monopole.

Ch. 9. Infinite-dimensional Lie groups and algebras: Analysis of cohomology properties in the infinite-dimensional case. The group of mappings \(G(M)\) associated with a compact Lie group \(G\) and its Lie algebra \({\mathcal G} (M)\). Current algebras as infinite-dimensional Lie algebras. The Kac-Moody or untwisted affine algebra. The Virasoro algebra. The two-dimensional conformal group and conformal algebra. The Chevalley-Eilenberg cohomology on groups of diffeomorphisms of the circle \(S^1\) and a Wess-Zumino term in the two-dimensional gravity theory (induced by Polyakov).

Ch. 10. Gauge anomalies: The topological and cohomological properties of abelian and non-abelian chiral anomalies in Yang-Mills gauge field theories. The group of gauge transformations and the orbit space of Yang- Mills potentials. The Gribov ambiguity and the appearance of anomalies related to the non-trivial topology of the configuration or Yang-Mills orbit space. The explicit path integral calculation of the abelian chiral anomaly in \(D = 2p\) dimensions and the non-abelian gauge anomalies \((D = 2)\) with help of Fujikawa’s method and their interpretation in terms of suitable index theorems on spaces of \(D = 2p\) and \(D = 2\) dimensions. The consistency conditions for the anomalies and the Schwinger terms and a cohomology in which the cocycles are valued in the space of functionals of the gauge fields. The method for obtaining non-trivial candidates for both the non-abelian anomalies and Schwinger terms based on the use of the cohomological descent procedure starting from the Chern character forms. The ambiguity of the cohomological descent procedure and different (cohomologous) expression for the Schwinger terms, the BRST formulation of the gauge cohomology and the Wess-Zumino-Witten terms. Possible consistency of anomalous gauge field theories.

The bibliographical notes are given at the end of each chapter.

Ch. 1. Lie groups, fibre bundles and Cartan calculus: Some topics in the differential geometry of Lie groups. Left- and right-invariant vector fields as differential operators acting on Lie group manifolds and vector fibre bundles. Differential forms and Cartan calculus. De Rham cohomology and Hodge-de Rham theory. The Maurer-Cartan equations and Lie algebras. Left- and bi-invariances. The supertranslation group. Some homotopy groups. The Poincaré polynomials of compact simple Lie groups and on the homotopy groups of spheres.

Ch. 2. Connections and characteristic classes: Description of elements of the theory of connections on principal bundles. Equivalent forms of Lie groups. Characteristic (Chern) classes, Chern characters and Chern-Simons forms. The magnetic monopole. Yang-Mills instantons. Pontryagin classes and the Euler class. Index theorems for manifolds without boundary and for the spin complexes. Twisted complexes.

Ch. 3. A first look at cohomology of groups and related topics: Some specific features of classical and quantum mechanics as a physical motivation for introducing group cohomology and group extensions. Two- cocycles in Galilei covariant classical and quantum mechanics, in Bargmann’s theory of projective (ray) Lie group representations as well as in the Weyl-Heisenberg group and in the extended Galilei group theories. Dynamical groups, symplectic cohomology and geometrical quantization. The three-cocycles and associativity breaking. Group contraction procedure and non-trivial group cohomology.

Ch. 4. An introduction to abstract group extension theory: The general theory of group extensions and the formalism of exact sequences of homomorphisms in terms of fibre bundles. Principal bundle description of an abstract group extension \(\widetilde {G}\) of \(G\) by \(K\) (structure group). Group laws for \(G\) in terms of the factor system which determines and is determined by the extension.

Ch. 5. Cohomology groups of a group \(G\) and extensions by an abelian kernel: Introduction of the concept of cocycles and coboundaries for the group cohomology with values in an abelian group. The second cohomology groups and abelian extensions, including the semidirect and central ones. Superspace as group extension. Cohomology induced by the action of the group \(G\) on a manifold \(M^m\) \(({\mathcal F}(M)\)-valued cochains).

Ch. 6. Cohomology of Lie algebras: Cohomology of Lie algebras with values in a vector space, the Whitehead lemmas and Lie algebra extensions which are related to the second cohomology groups. The relations between Lie groups and Lie algebras with the simple example of central extensions of groups and algebras (governed by two complexes) as well as the higher order case. Explicit formulas for obtaining Lie algebra cocycles from the Lie group ones and vice versa. The Chevalley-Eilenberg formulation of the Lie algebra cohomology and invariant differential forms on a Lie group \(G\). The BRST (Becchi, Rouses, Stora and Tyutin) approach to Lie algebra cohomology. Lie algebra cohomology versus Lie group cohomology. \({\mathcal F} (M)\)-valued Lie algebra cohomology.

Ch. 7. Group extensions by non-abelian kernels: Reduction of the theory of group extensions by non-abelian groups \(K\) to an abelian cohomology problem: a \(G\)-kernel is extendible iff a certain three-cocycle with values in the centre \(C_K\) of \(K\) is trivial, and the possible extensions of \(G\) by \(K\) are in one-to-one correspondence with the extensions of \(G\) by the abelian group \(C_K\). Construction of extensions. The right “covering” groups of the complete Lorentz group.

Ch. 8. Cohomology and Wess-Zumino terms. An introduction: The variational principle and the Nöther theorem in Newtonian mechanics. The analysis of the relation between mechanics and cohomology from the Lagrangian point of view introduced in the 1-jet bundle framework. Quasi-invariance of Lagrangians and group (Lie algebra) central extensions. Description of quasi-invariant Lagrangians in terms of one-cocycles as well as two- cocycles and the cohomological descent procedure. The complementary aspect for obtaining physical actions from non-trivial cohomology and the concept of the Wess-Zumino term on a group manifold. The different role of the left and right versions of the symmetry Lie algebra involved. The cohomological descent approach to classical anomalies. The massive superparticle, the Wess-Zumino terms for supersymmetric extended objects and the supersymmetry algebra. The Lagrangian description of the magnetic monopole.

Ch. 9. Infinite-dimensional Lie groups and algebras: Analysis of cohomology properties in the infinite-dimensional case. The group of mappings \(G(M)\) associated with a compact Lie group \(G\) and its Lie algebra \({\mathcal G} (M)\). Current algebras as infinite-dimensional Lie algebras. The Kac-Moody or untwisted affine algebra. The Virasoro algebra. The two-dimensional conformal group and conformal algebra. The Chevalley-Eilenberg cohomology on groups of diffeomorphisms of the circle \(S^1\) and a Wess-Zumino term in the two-dimensional gravity theory (induced by Polyakov).

Ch. 10. Gauge anomalies: The topological and cohomological properties of abelian and non-abelian chiral anomalies in Yang-Mills gauge field theories. The group of gauge transformations and the orbit space of Yang- Mills potentials. The Gribov ambiguity and the appearance of anomalies related to the non-trivial topology of the configuration or Yang-Mills orbit space. The explicit path integral calculation of the abelian chiral anomaly in \(D = 2p\) dimensions and the non-abelian gauge anomalies \((D = 2)\) with help of Fujikawa’s method and their interpretation in terms of suitable index theorems on spaces of \(D = 2p\) and \(D = 2\) dimensions. The consistency conditions for the anomalies and the Schwinger terms and a cohomology in which the cocycles are valued in the space of functionals of the gauge fields. The method for obtaining non-trivial candidates for both the non-abelian anomalies and Schwinger terms based on the use of the cohomological descent procedure starting from the Chern character forms. The ambiguity of the cohomological descent procedure and different (cohomologous) expression for the Schwinger terms, the BRST formulation of the gauge cohomology and the Wess-Zumino-Witten terms. Possible consistency of anomalous gauge field theories.

The bibliographical notes are given at the end of each chapter.

Reviewer: A.A.Bogush (Minsk)

##### MSC:

22E60 | Lie algebras of Lie groups |

17B56 | Cohomology of Lie (super)algebras |

17B65 | Infinite-dimensional Lie (super)algebras |

17B67 | Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras |

57T10 | Homology and cohomology of Lie groups |

81R10 | Infinite-dimensional groups and algebras motivated by physics, including Virasoro, Kac-Moody, \(W\)-algebras and other current algebras and their representations |

17B68 | Virasoro and related algebras |

81T13 | Yang-Mills and other gauge theories in quantum field theory |

17-02 | Research exposition (monographs, survey articles) pertaining to nonassociative rings and algebras |

22-02 | Research exposition (monographs, survey articles) pertaining to topological groups |

20J06 | Cohomology of groups |

20J05 | Homological methods in group theory |