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Moduli Spaces of Curves, Mapping Class Groups and Field Theory
Xavier Buff, Université Paul Sabatier, Toulouse, France, Jérôme Fehrenbach, University of Nice Sophia-Antipolis, Valbonne, France, Pierre Lochak, Centre de Mathématiques de Jussieu, Université Paris VI, Paris, France, Leila Schneps, Université Paris VI, France, and Pierre Vogel, Université Paris VII, France
A co-publication of the AMS and Société Mathématique de France.

SMF/AMS Texts and Monographs
2003; 131 pp; softcover
Volume: 9
ISBN-10: 0-8218-3167-4
ISBN-13: 978-0-8218-3167-0
List Price: US$48
Member Price: US$38.40
Order Code: SMFAMS/9
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This is a collection of articles that grew out of a workshop organized to discuss deep links among various topics that were previously considered unrelated. Rather than a typical workshop, this gathering was unique as it was structured more like a course for advanced graduate students and research mathematicians.

In the book, the authors present applications of moduli spaces of Riemann surfaces in theoretical physics and number theory and on Grothendieck's dessins d'enfants and their generalizations. Chapter 1 gives an introduction to Teichmüller space that is more concise than the popular textbooks, yet contains full proofs of many useful results which are often difficult to find in the literature. This chapter also contains an introduction to moduli spaces of curves, with a detailed description of the genus zero case, and in particular of the part at infinity. Chapter 2 takes up the subject of the genus zero moduli spaces and gives a complete description of their fundamental groupoids, based at tangential base points neighboring the part at infinity; the description relies on an identification of the structure of these groupoids with that of certain canonical subgroupoids of a free braided tensor category. It concludes with a study of the canonical Galois action on the fundamental groupoids, computed using Grothendieck-Teichmüller theory. Finally, Chapter 3 studies strict ribbon categories, which are closely related to braided tensor categories: Here they are used to construct invariants of 3-manifolds which in turn give rise to quantum field theories. The material is suitable for advanced graduate students and researchers interested in algebra, algebraic geometry, number theory, and geometry and topology.

Titles in this series are co-published with Société Mathématique de France. SMF members are entitled to AMS member discounts.


Advanced graduate students and researchers interested in algebra, algebraic geometry, number theory, and geometry and topology.


From a review of the French edition:

"A collective monograph dedicated to the new and profound relations between various theories previously considered as unrelated ... A specific feature of the book, which distinguishes it from many other monographs and textbooks on the same subjects, is its nature of a `guide for the non-specialist' ... it also contains full proofs of some results difficult to find elsewhere ... Examples are studied in great detail ... Recommended as a first reading for a non-specialist who wants to get acquainted with the subject but who does not want to get lost in its many intricacies and ramifications."

-- Mathematical Reviews

Table of Contents

  • X. Buff, J. Fehrenbach, and P. Lochak -- Elements of the geometry of moduli spaces of curves
  • L. Schneps -- Fundamental groupoids of genus zero moduli spaces and braided tensor categories
  • P. Vogel -- Witten-Reshetikhin-Turaev invariants and quantum field theories
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