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Dualizable Tensor Categories

Author: Christopher L. Douglas
Publisher: American Mathematical Soc.
ISBN: 1470443619
Category : Education
Languages : en
Pages : 88

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Book Description
We investigate the relationship between the algebra of tensor categories and the topology of framed 3-manifolds. On the one hand, tensor categories with cer-tain algebraic properties determine topological invariants. We prove that fusion categories of nonzero global dimension are 3-dualizable, and therefore provide 3-dimensional 3-framed local ﬁeld theories. We also show that all ﬁnite tensor cat-egories are 2-dualizable, and yield categoriﬁed 2-dimensional 3-framed local ﬁeld theories. On the other hand, topological properties of 3-framed manifolds deter-mine algebraic equations among functors of tensor categories. We show that the 1-dimensional loop bordism, which exhibits a single full rotation, acts as the double dual autofunctor of a tensor category. We prove that the 2-dimensional belt-trick bordism, which unravels a double rotation, operates on any ﬁnite tensor category, and therefore supplies a trivialization of the quadruple dual. This approach pro-duces a quadruple-dual theorem for suitably dualizable objects in any symmetric monoidal 3-category. There is furthermore a correspondence between algebraic structures on tensor categories and homotopy ﬁxed point structures, which in turn provide structured ﬁeld theories; we describe the expected connection between piv-otal tensor categories and combed ﬁxed point structures, and between spherical tensor categories and oriented ﬁxed point structures.

Tensor Categories

Author: Pavel Etingof
Publisher: American Mathematical Soc.
ISBN: 1470434415
Category : Mathematics
Languages : en
Pages : 362

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Book Description
Is there a vector space whose dimension is the golden ratio? Of course not—the golden ratio is not an integer! But this can happen for generalizations of vector spaces—objects of a tensor category. The theory of tensor categories is a relatively new field of mathematics that generalizes the theory of group representations. It has deep connections with many other fields, including representation theory, Hopf algebras, operator algebras, low-dimensional topology (in particular, knot theory), homotopy theory, quantum mechanics and field theory, quantum computation, theory of motives, etc. This book gives a systematic introduction to this theory and a review of its applications. While giving a detailed overview of general tensor categories, it focuses especially on the theory of finite tensor categories and fusion categories (in particular, braided and modular ones), and discusses the main results about them with proofs. In particular, it shows how the main properties of finite-dimensional Hopf algebras may be derived from the theory of tensor categories. Many important results are presented as a sequence of exercises, which makes the book valuable for students and suitable for graduate courses. Many applications, connections to other areas, additional results, and references are discussed at the end of each chapter.

On the Asymptotics to all Orders of the Riemann Zeta Function and of a Two-Parameter Generalization of the Riemann Zeta Function

Author: Athanassios S. Fokas
Publisher: American Mathematical Society
ISBN: 1470450984
Category : Mathematics
Languages : en
Pages : 114

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