##
**A mathematical introduction to conformal field theory. Based on a series of lectures given at the Mathematisches Institut der Universität Hamburg.
2nd revised ed.**
*(English)*
Zbl 1161.17014

Lecture Notes in Physics 759. Berlin: Springer (ISBN 978-3-540-68625-5/hbk). xv, 249 p. (2008).

The first edition of M. Schottenloher’s lecture notes “A Mathematical Introduction to Conformal Field Theory” appeared in 1997, when the reconvergence of theoretical physics and pure mathematics had reached a first peak, after several decades of distinct developments in the middle of the twentieth century. More than ten years ago, the first edition of these notes M. Schottenloher [Lecture Notes in Physics. New Series Monographs. Berlin: Springer (1997; Zbl 0874.17031)] was to explain, in a rigorous and comprehensive way, the modern mathematical background for the conceptual framework of contemporary two-dimensional conformal field theory, mainly in order to provide a profound basis for the communication (and collaboration) between physicists and (pure) mathematicians.

The book under review is the second, completely rewritten, substantially expanded and appropriately updated edition of the original text, thereby reflecting various recent developments in the field during the past ten years, on the one hand, and making especially the physics part of the notes more detailed, self-contained and tutorial on the other. As a result of the author’s revision of this well-tried book, the volume of the text of the new edition has nearly doubled, where half of this considerable expansion is due to two completely new chapters. In general, the leading concept of presenting a reasonably concise and at the same time rigorous introduction to conformal field theory has been throughout maintained in the present second edition. Also, like in the already thoroughly reviewed first edition, the notes are divided into two major parts of different nature. The first part, titled “Mathematical Preliminaries” and comprising the original Chapters 1–5, is rather elementary and detailed, whereas the second part “First Steps Toward Conformal Field Theory” requires much more advanced mathematical prerequisites from functional analysis, complex analysis, and algebraic geometry. Due to this higher degree of complexity of the second part, not all facts are proven in full detail, but the author has also striven here to increase the degree of self-containedness by providing much more proofs and important examples than in the original edition.

Apart from these expository improvements, the main changes are given by the two new Chapters 8 and 10.

Chapter 8 discusses the system of axioms for relativistic quantum field theory which has been formulated by Arthur Wightman in the early 1950s (cf.: R. F. Streater and A. S. Wightman: PCT, Spin and Statistics, and All That, Princeton University Press, Princeton, NJ (1964; Zbl 0135.44305 ), thereby including another, more classical and analysis-based approach to conformal field theory for the purpose of additional motivation and a better complex understanding of the subject. Also, this new Chapter 8 serves as a useful preparation for the subsequent Chapter 9 on the approach to two-dimensional conformally invariant quantum field theory via the Osterwalder-Schrader axioms (1973–1975), which appeared to be quite isolated in the first edition. Moreover, Chapter 9 has been enriched by adding the conformal Ward identities, the state field correspondence, and some links to the more recent approach via the theory of vertex algebras.

In this vein, the new Chapter 10 is exclusively devoted to a brief introduction to vertex algebras, mainly based on the recent standard books by V. Kac [“Vertex algebras for beginners, University Lecture Series 10, Amer. Math. Soc., Providence, RI, 2nd ed. (1998; Zbl 0924.17023) and D. Ben-Zvi and E. Frenkel [Vertex algebras and algebraic curves, Am. Math. Soc., Providence, RI (2001; Zbl 0981.17022). This chapter, which is presented in a largely self-contained manner, provides the link to the more recent developments in quantum field theory and the respective literature. Finally, the present second edition contains a brief appendix pointing to some further developments with respect to boundary conformal field theory, stochastic Loewner evolution, and modularity properties, together with some hints for further reading.

Throughout the text, the author has included a wealth of additional facts and statements, various helpful cross-references linking the different chapters, a large number of further-leading remarks, and many more illustrating examples. And, of course, the already rich bibliography has been supplemented and updated likewise.

All together, the second, revised and enlarged edition of this highly appreciated mediator between contemporary mathematics and physical quantum field theory has grown into an even more valuable source for students, teachers,and active researchers in this fascinating area of modern science.

The book under review is the second, completely rewritten, substantially expanded and appropriately updated edition of the original text, thereby reflecting various recent developments in the field during the past ten years, on the one hand, and making especially the physics part of the notes more detailed, self-contained and tutorial on the other. As a result of the author’s revision of this well-tried book, the volume of the text of the new edition has nearly doubled, where half of this considerable expansion is due to two completely new chapters. In general, the leading concept of presenting a reasonably concise and at the same time rigorous introduction to conformal field theory has been throughout maintained in the present second edition. Also, like in the already thoroughly reviewed first edition, the notes are divided into two major parts of different nature. The first part, titled “Mathematical Preliminaries” and comprising the original Chapters 1–5, is rather elementary and detailed, whereas the second part “First Steps Toward Conformal Field Theory” requires much more advanced mathematical prerequisites from functional analysis, complex analysis, and algebraic geometry. Due to this higher degree of complexity of the second part, not all facts are proven in full detail, but the author has also striven here to increase the degree of self-containedness by providing much more proofs and important examples than in the original edition.

Apart from these expository improvements, the main changes are given by the two new Chapters 8 and 10.

Chapter 8 discusses the system of axioms for relativistic quantum field theory which has been formulated by Arthur Wightman in the early 1950s (cf.: R. F. Streater and A. S. Wightman: PCT, Spin and Statistics, and All That, Princeton University Press, Princeton, NJ (1964; Zbl 0135.44305 ), thereby including another, more classical and analysis-based approach to conformal field theory for the purpose of additional motivation and a better complex understanding of the subject. Also, this new Chapter 8 serves as a useful preparation for the subsequent Chapter 9 on the approach to two-dimensional conformally invariant quantum field theory via the Osterwalder-Schrader axioms (1973–1975), which appeared to be quite isolated in the first edition. Moreover, Chapter 9 has been enriched by adding the conformal Ward identities, the state field correspondence, and some links to the more recent approach via the theory of vertex algebras.

In this vein, the new Chapter 10 is exclusively devoted to a brief introduction to vertex algebras, mainly based on the recent standard books by V. Kac [“Vertex algebras for beginners, University Lecture Series 10, Amer. Math. Soc., Providence, RI, 2nd ed. (1998; Zbl 0924.17023) and D. Ben-Zvi and E. Frenkel [Vertex algebras and algebraic curves, Am. Math. Soc., Providence, RI (2001; Zbl 0981.17022). This chapter, which is presented in a largely self-contained manner, provides the link to the more recent developments in quantum field theory and the respective literature. Finally, the present second edition contains a brief appendix pointing to some further developments with respect to boundary conformal field theory, stochastic Loewner evolution, and modularity properties, together with some hints for further reading.

Throughout the text, the author has included a wealth of additional facts and statements, various helpful cross-references linking the different chapters, a large number of further-leading remarks, and many more illustrating examples. And, of course, the already rich bibliography has been supplemented and updated likewise.

All together, the second, revised and enlarged edition of this highly appreciated mediator between contemporary mathematics and physical quantum field theory has grown into an even more valuable source for students, teachers,and active researchers in this fascinating area of modern science.

Reviewer: Werner Kleinert (Berlin)

### MSC:

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

81-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to quantum theory |

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 |

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

14H60 | Vector bundles on curves and their moduli |