# Moduli Spaces of Curves, Mapping Class Groups and Field Theory

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*Xavier Buff; Jérôme Fehrenbach; Pierre Lochak; Leila Schneps; Pierre Vogel*

A co-publication of the AMS and Société Mathématique de France

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.

#### Readership

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

#### Reviews & Endorsements

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