Non Semisimple Extended Topological Quantum Field Theories
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Non Semisimple Extended Topological Quantum Field Theories
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Author : Marco De Renzi
language : en
Publisher:
Release Date : 2022
Non Semisimple Extended Topological Quantum Field Theories written by Marco De Renzi and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2022 with Electronic books categories.
"We develop the general theory for the construction of Extended Topological Quantum Field Theories (ETQFTs) associated with the Costantino-Geer-Patureau quantum invariants of closed 3-manifolds. In order to do so, we introduce relative modular categories, a class of ribbon categories which are modeled on representations of unrolled quantum groups, and which can be thought of as a non-semisimple analogue to modular categories. Our approach exploits a 2-categorical version of the universal construction introduced by Blanchet, Habegger, Masbaum, and Vogel. The 1+1+1-EQFTs thus obtained are realized by symmetric monoidal 2-functors which are defined over non-rigid 2-categories of admissible cobordisms decorated with colored ribbon graphs and cohomology classes, and which take values in 2- categories of complete graded linear categories. In particular, our construction extends the family of graded 2+1-TQFTs defined for the unrolled version of quantum sl2 by Blanchet, Costantino, Geer, and Patureau to a new family of graded ETQFTs. The non-semisimplicity of the theory is witnessed by the presence of non-semisimple graded linear categories associated with critical 1-manifolds"--
Non Semisimple Topological Quantum Field Theories For 3 Manifolds With Corners
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Author : Thomas Kerler
language : en
Publisher: Springer
Release Date : 2003-07-01
Non Semisimple Topological Quantum Field Theories For 3 Manifolds With Corners written by Thomas Kerler and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2003-07-01 with Mathematics categories.
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.
Construction Of Extended Topological Quantum Field Theories
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Author : Marco De Renzi
language : en
Publisher:
Release Date : 2017
Construction Of Extended Topological Quantum Field Theories written by Marco De Renzi and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2017 with categories.
The central position held by Topological Quantum Field Theories (TQFTs) in the study of low dimensional topology is due to their extraordinarily rich structure, which allows for various interactions with and applications to questions of geometric nature. Ever since their first appearance, a great effort has been put into extending quantum invariants of 3-dimensional manifolds to TQFTs and Extended TQFTs (ETQFTs). This thesis tackles this problem in two different general frameworks. The first one is the study of the semisimple quantum invariants of Witten, Reshetikhin and Turaev issued from modular categories. Although the corresponding ETQFTs were known to exist for a while, an explicit realization based on the universal construction of Blanchet, Habegger, Masbaum and Vogel appears here for the first time. The aim is to set a golden standard for the second part of the thesis, where the same procedure is applied to a new family of non-semisimple quantum invariants due to Costantino, Geer and Patureau. These invariants had been previously extended to graded TQFTs by Blanchet, Costantino, Geer an Patureau, but only for an explicit family of examples. We lay the first stone by introducing the definition of relative modular category, a non-semisimple analogue to modular categories. Then, we refine the universal construction to obtain graded ETQFTs extending both the quantum invariants of Costantino, Geer and Patureau and the graded TQFTs of Blanchet, Costantino, Geer and Patureau in this general setting.
Non Semisimple Topological Quantum Field Theories For 3 Manifolds With Corners
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Author : Thomas Kerler
language : en
Publisher:
Release Date : 2014-09-01
Non Semisimple Topological Quantum Field Theories For 3 Manifolds With Corners written by Thomas Kerler and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2014-09-01 with categories.
Non Semisimple Extended Topological Quantum Field Theories
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Author : Marco De Renzi
language : en
Publisher: American Mathematical Society
Release Date : 2022-05-24
Non Semisimple Extended Topological Quantum Field Theories written by Marco De Renzi and has been published by American Mathematical Society this book supported file pdf, txt, epub, kindle and other format this book has been release on 2022-05-24 with Mathematics categories.
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Topological Quantum Field Theories From Subfactors
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Author : Vijay Kodiyalam
language : en
Publisher: CRC Press
Release Date : 2019-05-20
Topological Quantum Field Theories From Subfactors written by Vijay Kodiyalam and has been published by CRC Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2019-05-20 with Mathematics categories.
Pure mathematicians have only recently begun a rigorous study of topological quantum field theories (TQFTs). Ocneanu, in particular, showed that subfactors yield TQFTs that complement the Turaev-Viro construction. Until now, however, it has been difficult to find an account of this work that is both detailed and accessible. Topological Quant
Quantum Invariants Of Knots And 3 Manifolds
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Author : Vladimir G. Turaev
language : en
Publisher: Walter de Gruyter GmbH & Co KG
Release Date : 2020-03-23
Quantum Invariants Of Knots And 3 Manifolds written by Vladimir G. Turaev and has been published by Walter de Gruyter GmbH & Co KG this book supported file pdf, txt, epub, kindle and other format this book has been release on 2020-03-23 with Mathematics categories.
This monograph provides a systematic treatment of topological quantum field theories (TQFT's) in three dimensions, inspired by the discovery of the Jones polynomial of knots, the Witten-Chern-Simons field theory, and the theory of quantum groups. The author, one of the leading experts in the subject, gives a rigorous and self-contained exposition of new fundamental algebraic and topological concepts that emerged in this theory. The book is divided into three parts. Part I presents a construction of 3-dimensional TQFT's and 2-dimensional modular functors from so-called modular categories. This gives new knot and 3-manifold invariants as well as linear representations of the mapping class groups of surfaces. In Part II the machinery of 6j-symbols is used to define state sum invariants of 3-manifolds. Their relation to the TQFT's constructed in Part I is established via the theory of shadows. Part III provides constructions of modular categories, based on quantum groups and Kauffman's skein modules. This book is accessible to graduate students in mathematics and physics with a knowledge of basic algebra and topology. It will be an indispensable source for everyone who wishes to enter the forefront of this rapidly growing and fascinating area at the borderline of mathematics and physics. Most of the results and techniques presented here appear in book form for the first time.
Advances In Topological Quantum Field Theory
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Author : John M. Bryden
language : en
Publisher: Springer Science & Business Media
Release Date : 2005-03-02
Advances In Topological Quantum Field Theory written by John M. Bryden and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2005-03-02 with Mathematics categories.
Affine Hecke Algebras And Quantum Symmetric Pairs
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Author : Zhaobing Fan
language : en
Publisher: American Mathematical Society
Release Date : 2023-01-18
Affine Hecke Algebras And Quantum Symmetric Pairs written by Zhaobing Fan and has been published by American Mathematical Society this book supported file pdf, txt, epub, kindle and other format this book has been release on 2023-01-18 with Mathematics categories.
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Categories And Representation Theory
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Author : Hideto Asashiba
language : en
Publisher: American Mathematical Society
Release Date : 2022-09-29
Categories And Representation Theory written by Hideto Asashiba and has been published by American Mathematical Society this book supported file pdf, txt, epub, kindle and other format this book has been release on 2022-09-29 with Mathematics categories.
This book gives a self-contained account of applications of category theory to the theory of representations of algebras. Its main focus is on 2-categorical techniques, including 2-categorical covering theory. The book has few prerequisites beyond linear algebra and elementary ring theory, but familiarity with the basics of representations of quivers and of category theory will be helpful. In addition to providing an introduction to category theory, the book develops useful tools such as quivers, adjoints, string diagrams, and tensor products over a small category; gives an exposition of new advances such as a 2-categorical generalization of Cohen-Montgomery duality in pseudo-actions of a group; and develops the moderation level of categories, first proposed by Levy, to avoid the set theoretic paradox in category theory. The book is accessible to advanced undergraduate and graduate students who would like to study the representation theory of algebras, and it contains many exercises. It can be used as the textbook for an introductory course on the category theoretic approach with an emphasis on 2-categories, and as a reference for researchers in algebra interested in derived equivalences and covering theory.