##
**Two-dimensional conformal geometry and vertex operator algebras.**
*(English)*
Zbl 0884.17021

Progress in Mathematics (Boston, Mass.). 148. Boston, MA: Birkhäuser. xii, 280 p. (1997).

This book is an extension of the author’s Ph.D. thesis in 1990, whose main results were announced in his article “Geometric interpretation of vertex operator algebras” [Proc. Natl. Acad. Sci. USA 88, 9964-9968 (1991; Zbl 0810.17019)]. The notion of vertex algebra was introduced by R. E. Borcherds [Proc. Natl. Acad. Sci. USA 83, 3068-3071 (1986; Zbl 0613.17012)]. Vertex operator algebras are vertex algebras with a certain Virasoro algebra structure, which were formulated by I. Frenkel, J. Lepowsky and A. Meurman [Vertex operator algebras and the Monster, Pure Appl. Math. 134 (Academic Press, New York, 1988; Zbl 0674.17001)]. Vertex operator algebras are fundamental algebraic structures in conformal field theory. Based on work of G. B. Segal [Proc. IXth international congress on mathematical physics, Swansea, 1988 (Hilger, Bristol), 22-37 (1989)], C. Vafa [Phys. Lett. B 199, 195-202 (1987)] and Frenkel (see Appendices in the above FLM’s book), the author gives a geometric interpretation of the axioms of vertex operator algebra in terms of so-called “partial operads” of complex powers of the determinant line bundles over certain moduli spaces of spheres with punctures and local analytic coordinates.

The first axiom restricts vertex operator algebras to be positive energy modules for what turns out eventually to be the Virasoro algebra. The second axiom corresponds to the eigenspace decomposition with respect to the operator of rescaling local coordinates. The third axiom states that the correlation functions of vertex operators are meromorphic as functions of any one of the punctures in a sphere and with the other punctures as the only possible poles. The fourth axiom requires that the ordering of the punctures of spheres with tubes does not enter the structure of a geometric vertex operator algebra. The fifth axiom requires that the image of an element sew from two elements in the moduli space is equal to the contraction (a generalization of the composition of maps) of the images of the two elements, multiplied by a projective factor.

The book is a good reference for mathematicians who are interested in the interplay of geometry and algebra.

The first axiom restricts vertex operator algebras to be positive energy modules for what turns out eventually to be the Virasoro algebra. The second axiom corresponds to the eigenspace decomposition with respect to the operator of rescaling local coordinates. The third axiom states that the correlation functions of vertex operators are meromorphic as functions of any one of the punctures in a sphere and with the other punctures as the only possible poles. The fourth axiom requires that the ordering of the punctures of spheres with tubes does not enter the structure of a geometric vertex operator algebra. The fifth axiom requires that the image of an element sew from two elements in the moduli space is equal to the contraction (a generalization of the composition of maps) of the images of the two elements, multiplied by a projective factor.

The book is a good reference for mathematicians who are interested in the interplay of geometry and algebra.

Reviewer: Xu Xiaoping (Kowloon)

### MSC:

17B69 | Vertex operators; vertex operator algebras and related structures |

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

81T40 | Two-dimensional field theories, conformal field theories, etc. in quantum mechanics |

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 |

30F10 | Compact Riemann surfaces and uniformization |

32G15 | Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables) |