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Lectures on tensor categories and modular functors. (English) Zbl 0965.18002
University Lecture Series. 21. Providence, RI: American Mathematical Society (AMS). x, 221 p. (2001).
During the past two decades, the study of quantum field theory in mathematical physics has undergone a tremendous development. The various attempts by physicists to explore possible, physically relevant models in quantum field theory, together with suitable mathematical frameworks for them, have transpired a wealth of new mathematical ideas, concepts, methods, constructions, interrelations, striking results, and unexpected mathematical conjectures. The impact of this development on, and the challenge for mathematics has been enormous. Many of the most spectacular achievements and results in mathematics, during this period, have been inspired by the related mathematical intuition, mathematically experimental constructions, and conjectural mathematical predictions offered by physicists working on quantum field theory, and this tendency of interdisciplinary fructification in the respective developments of mathematics and (quantum field) physics has become increasingly rapid, ubiquitous and almost dominant, in the meantime.
One of the milestones, in this regard, was provided by the pionieering works of E. Witten on topological quantum field theory in the late 1980’s [cf. “Topological quantum field theory”, Commun. Math. Phys. 117, No. 3, 353-386 (1988; Zbl 0656.53078)], followed by the related work of G. Moore and N. Seiberg [“Classical and quantum conformal field theory”, Commun. Math. Phys. 123, No. 2, 177-254 (1989; Zbl 0694.53074)]. Their new fundamental ideas, mathematical constructions, guiding proposals, and conjectures have triggered a real avalanche of papers, both mathematical and physical, investigating various aspects of the conjectured interrelation between three superficially different recent topics in mathematics, namely
(a) tensor categories (arising in the representation theory of quantum groups);
(b) three-dimensional topological quantum field theory (in connection with invariants of three-manifolds).
(c) two-dimensional modular functors (in connection with moduli spaces of algebraic curves, Teichmüller theory, and two-dimensional conformal quantum field theory).
Only very recently, in the late 1990’s, and after a decade of utmost intensive explorations in this direction by an immense number of researchers, it had become a common perception that these three topics are really closely related. However, the crucial statements and results establishing this knowledge are widely scattered through the current research literatur in the field, difficult to localize precisely, hard to systematize within the entire context, incoherently related, and full of gaps to be filled.
The book under review represents the first attempt to provide a comprehensive exposition of the established relations among the three topics (a), (b) and (c). The text grew out of lectures which the second author taught on these topics at Massachusetts Institute of Technology (MIT) in the Spring of 1997, using also several unpublished expository text and very recent research articles. The outcome, after further elaboration of the lecture notes, is this monograph which provides a finely succinct, compact and wholly expository introduction to these topics and their interplay.
The authors have gathered numerous (partly unpublished) results, put them systematically into a coherent context, arranged them according to their interplay, filled existing gaps, and simplified some of the very involved proofs.
Moreover, the various objects of study (e.g., tensor categories, diverse modular functors, distinct topological quantum field theories, complex-analytic moduli spaces of Riemann surfaces, etc.), which have not yet become a commonplace even among the experts in the field, have been systematically compiled and explained.
As to the contents, the text is subdivided into seven chapters. Those are entitled as follows:
Chapter 1: braided tensor categories (including monoidal tensor categories, quantum groups, and Drinfeld’s category associated with a simple Lie algebra);
Chapter 2: ribbon categories (including rigid monoidal categories, the graphical calculus for morphisms in ribbon categories, and semisimple categories);
Chapter 3: modular tensor categories (including the example of the quantum double of a finite group as well as the discussion of quantum groups at roots of unity);
Chapter 4: three-dimensional topological quantum field theory (including invariants of three-manifolds and examples of three-dimensional TQFT’s);
Chapter 5: modular functors (including topological 2D functors and their relation with the Teichmüller modular group tower, the Lego-Teichmüller game, towers of groupoids, and the link between this analytic set-up and three-dimensional topological quantum field theory);
Chapter 6: moduli spaces and complex modular functors (including an introduction to compactified moduli spaces of punctured complex curves, Deligne’s theory of flat connections with regular singularities, complex-analytic modular functors as local systems with regular singularities on moduli spaces, and Drinfeld’s category as an example of a complex-analytic modular functor in genus zero);
Chapter 7: Wess-Zumino-Witten Model (discussing the most famous example of a modular functor, namely the one coming from the Wess-Zumino-Witten model in conformal quantum field theory, based on the unpublished approach by A. Beilinson, B. Feigin and B. Mazur).
According to these contents, the book under review is, for the time being, unique of its kind. This makes it an important addition to the existing literature on these topics, a suitable course text at the advanced-graduate level and, likewise, a useful reference book for researchers in the field.

##### MSC:
 18-02 Research exposition (monographs, survey articles) pertaining to category theory 57-02 Research exposition (monographs, survey articles) pertaining to manifolds and cell complexes 81-02 Research exposition (monographs, survey articles) pertaining to quantum theory 18D10 Monoidal, symmetric monoidal and braided categories (MSC2010) 81T40 Two-dimensional field theories, conformal field theories, etc. in quantum mechanics 57R56 Topological quantum field theories (aspects of differential topology) 81T45 Topological field theories in quantum mechanics 57M27 Invariants of knots and $$3$$-manifolds (MSC2010) 81R10 Infinite-dimensional groups and algebras motivated by physics, including Virasoro, Kac-Moody, $$W$$-algebras and other current algebras and their representations 81R50 Quantum groups and related algebraic methods applied to problems in quantum theory 17B67 Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras 17B37 Quantum groups (quantized enveloping algebras) and related deformations