Loop spaces, characteristic classes and geometric quantization.

*(English)*Zbl 0823.55002
Progress in Mathematics (Boston, Mass.). 107. Boston, MA: Birkhäuser. xvi, 300 p. (1993).

After an introductory section, the present book contains seven chapters, a rich bibliography, a list of notations and an index of notions.

Chapter 1 is an introduction to sheaves, complexes of sheaves, and their cohomology, covering standard aspects, and also a few less standard ones which are needed in the rest of the book. There are discussed de Rham cohomology, Deligne cohomology and the Cheeger-Simons differential characters, as well as the Leray spectral sequence for a mapping and a complex of sheaves.

Chapter 2 introduces line bundles, their relation with \(H^ 2\), connections and curvature. The theory of Kostant and Deligne relating line bundles with connection to Deligne cohomology is detailed, both in the smooth and in the holomorphic case. It is explained the Kostant- Souriau central extension of the Lie algebra of Hamiltonian vector fields and the Kostant central extension of the group of symplectomorphisms.

Chapter 3 presents a number of geometric results on the space \(\widetilde{Y}\) of singular knots in a smooth 3-manifold \(M\). The striking fact is that \(\widetilde {Y}\) looks exactly like a Kähler manifold. It has a complex structure, which, by a theorem of Lempert, is integrable in a weak sense. It has a symplectic structure that is familiar from fluid mechanics. The symplectic form of \(\widetilde {Y}\) has integral cohomology class. Finally, \(\widetilde {Y}\) has a Riemannian structure, which is compatible with both the symplectic and complex structures.

Chapter 4 exposes the first level of degree 3 cohomology theory. The theory of Dixmier and Douady, based on infinite-dimensional algebra bundles is exposed and the general obstruction of lifting the structure group of a principal bundle to a central extension of the group is formulated. The relevant geometric concepts leading to the notion of 3- curvature are developed and examples of projective Hilbert space bundles are presented.

Chapter 5, really the heart of the book, exposes the second level of this theory using sheaves of groupoids. The author begins with a discussion of the glueing properties of sheaves with respect to surjective local homeomorphisms, and introduces the notion of torsors, which is a generalization of principal bundles. Next, he defines sheaves of groupoids and gerbes using glueing axioms that mimic the glueing properties of sheaves. The gerbe associated to a \(G\)-bundle over \(M\) and a group extension \(\widetilde {G}\) of \(G\) is discussed. Given a gerbe \(C\), there is an associated sheaf of groups, called the band of \(C\). Gerbes with band equal to the sheaf of smooth \(C^*\)-valued functions are called Dixmier-Douady sheaves of groupoids. The differential of such sheaves of groupoids is developed, for which the exotic names “connective structure” and “curving” were invented. A differential form of degree 3 is defined and called the 3-curvature of the sheaf of groupoids. The group of equivalence classes of sheaves of groupoids of Dixmier-Douady type, with connective structure and curving, is identified with the Deligne cohomology group \(H^ 3(M,Z(2)_ D)\). Many examples and applications are given.

Chapter 6 gives the construction of a line bundle on the space \(LM\) of loops of \(M\) associated to a sheaf of groupoids on \(M\) with connective structure. The construction is in some sense very explicit, in a spirit similar to the construction of a tautological line bundle. The curvature of the line bundle on \(LM\) is the 2-form obtained by transgressing the 3- curvature of the sheaf of groupoids over \(M\). This construction corresponds to a transgression map \(H^ 3 (M,Z(2)_ D) \to H^ 2(LM, Z(1)_ D)\) in Deligne cohomology. Finally, the notion of parallel transport in the context of sheaves of groupoids is discussed.

Chapter 7 treats the Dirac magnetic monopole, and the resulting quantization condition for a closed 3-form on the 3-sphere. First, the author explains Dirac’s construction and its classical mathematical interpretation using a 2-sphere centered at the monopole. Next, he gives his purely 3-dimensional approach, in which the monopole is viewed as a sheaf of groupoids on the 3-sphere. The quantization condition follows from the theory of Chapter 5. Finally, the obstruction to SU(2)- equivalence is studied and it is shown that the obstruction to make the monopole sheaf of groupoids \(C\) over \(S^ 3\) equivariant under SU(2) coincides with the Chern-Cheeger-Simons class \(\delta_ 2 \in H^ 3 (\text{SU}(2), T)\).

Chapter 1 is an introduction to sheaves, complexes of sheaves, and their cohomology, covering standard aspects, and also a few less standard ones which are needed in the rest of the book. There are discussed de Rham cohomology, Deligne cohomology and the Cheeger-Simons differential characters, as well as the Leray spectral sequence for a mapping and a complex of sheaves.

Chapter 2 introduces line bundles, their relation with \(H^ 2\), connections and curvature. The theory of Kostant and Deligne relating line bundles with connection to Deligne cohomology is detailed, both in the smooth and in the holomorphic case. It is explained the Kostant- Souriau central extension of the Lie algebra of Hamiltonian vector fields and the Kostant central extension of the group of symplectomorphisms.

Chapter 3 presents a number of geometric results on the space \(\widetilde{Y}\) of singular knots in a smooth 3-manifold \(M\). The striking fact is that \(\widetilde {Y}\) looks exactly like a Kähler manifold. It has a complex structure, which, by a theorem of Lempert, is integrable in a weak sense. It has a symplectic structure that is familiar from fluid mechanics. The symplectic form of \(\widetilde {Y}\) has integral cohomology class. Finally, \(\widetilde {Y}\) has a Riemannian structure, which is compatible with both the symplectic and complex structures.

Chapter 4 exposes the first level of degree 3 cohomology theory. The theory of Dixmier and Douady, based on infinite-dimensional algebra bundles is exposed and the general obstruction of lifting the structure group of a principal bundle to a central extension of the group is formulated. The relevant geometric concepts leading to the notion of 3- curvature are developed and examples of projective Hilbert space bundles are presented.

Chapter 5, really the heart of the book, exposes the second level of this theory using sheaves of groupoids. The author begins with a discussion of the glueing properties of sheaves with respect to surjective local homeomorphisms, and introduces the notion of torsors, which is a generalization of principal bundles. Next, he defines sheaves of groupoids and gerbes using glueing axioms that mimic the glueing properties of sheaves. The gerbe associated to a \(G\)-bundle over \(M\) and a group extension \(\widetilde {G}\) of \(G\) is discussed. Given a gerbe \(C\), there is an associated sheaf of groups, called the band of \(C\). Gerbes with band equal to the sheaf of smooth \(C^*\)-valued functions are called Dixmier-Douady sheaves of groupoids. The differential of such sheaves of groupoids is developed, for which the exotic names “connective structure” and “curving” were invented. A differential form of degree 3 is defined and called the 3-curvature of the sheaf of groupoids. The group of equivalence classes of sheaves of groupoids of Dixmier-Douady type, with connective structure and curving, is identified with the Deligne cohomology group \(H^ 3(M,Z(2)_ D)\). Many examples and applications are given.

Chapter 6 gives the construction of a line bundle on the space \(LM\) of loops of \(M\) associated to a sheaf of groupoids on \(M\) with connective structure. The construction is in some sense very explicit, in a spirit similar to the construction of a tautological line bundle. The curvature of the line bundle on \(LM\) is the 2-form obtained by transgressing the 3- curvature of the sheaf of groupoids over \(M\). This construction corresponds to a transgression map \(H^ 3 (M,Z(2)_ D) \to H^ 2(LM, Z(1)_ D)\) in Deligne cohomology. Finally, the notion of parallel transport in the context of sheaves of groupoids is discussed.

Chapter 7 treats the Dirac magnetic monopole, and the resulting quantization condition for a closed 3-form on the 3-sphere. First, the author explains Dirac’s construction and its classical mathematical interpretation using a 2-sphere centered at the monopole. Next, he gives his purely 3-dimensional approach, in which the monopole is viewed as a sheaf of groupoids on the 3-sphere. The quantization condition follows from the theory of Chapter 5. Finally, the obstruction to SU(2)- equivalence is studied and it is shown that the obstruction to make the monopole sheaf of groupoids \(C\) over \(S^ 3\) equivariant under SU(2) coincides with the Chern-Cheeger-Simons class \(\delta_ 2 \in H^ 3 (\text{SU}(2), T)\).

Reviewer: N.Papaghiuc (Iaşi)

##### MSC:

55-02 | Research exposition (monographs, survey articles) pertaining to algebraic topology |

58-02 | Research exposition (monographs, survey articles) pertaining to global analysis |

55N35 | Other homology theories in algebraic topology |

58D10 | Spaces of embeddings and immersions |

57M25 | Knots and links in the \(3\)-sphere (MSC2010) |

53C15 | General geometric structures on manifolds (almost complex, almost product structures, etc.) |

53C55 | Global differential geometry of Hermitian and Kählerian manifolds |

57N20 | Topology of infinite-dimensional manifolds |