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This book presents the (to date) most general approach to combinatorial constructions of topological quantum field theories (TQFTs) in three dimensions. The authors describe extended TQFTs as double functors between two naturally defined double categories: one of topological nature, made of 3-manifolds with corners, the other of algebraic nature, made of linear categories, functors, vector spaces and maps. Atiyah's conventional notion of TQFTs as well as the notion of modular functor from axiomatic conformal field theory are unified in this concept. A large class of such extended modular catergory is constructed, assigning a double functor to every abelian modular category, which does not have to be semisimple.
This book reviews recent results on low-dimensional quantum field
theories and their connection with quantum group theory and the
theory of braided, balanced tensor categories. It presents
detailed, mathematically precise introductions to these subjects
and then continues with new results. Among the main results are a
detailed analysis of the representation theory of U (sl ), for q a
primitive root of unity, and a semi-simple quotient thereof, a
classfication of braided tensor categories generated by an object
of q-dimension less than two, and an application of these results
to the theory of sectors in algebraic quantum field theory. This
clarifies the notion of "quantized symmetries" in quantum
fieldtheory. The reader is expected to be familiar with basic
notions and resultsin algebra. The book is intended for research
mathematicians, mathematical physicists and graduate students.
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