Homotopy Quantum Field Theory

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European Mathematical Society, 2010 - Science - 276 pages
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Homotopy Quantum Field Theory (HQFT) is a branch of Topological Quantum Field Theory founded by E. Witten and M. Atiyah. It applies ideas from theoretical physics to study principal bundles over manifolds and, more generally, homotopy classes of maps from manifolds to a fixed target space. This book is the first systematic exposition of Homotopy Quantum Field Theory. It starts with a formal definition of an HQFT and provides examples of HQFTs in all dimensions. The main body of the text is focused on $2$-dimensional and $3$-dimensional HQFTs. A study of these HQFTs leads to new algebraic objects: crossed Frobenius group-algebras, crossed ribbon group-categories, and Hopf group-coalgebras. These notions and their connections with HQFTs are discussed in detail. The text ends with several appendices including an outline of recent developments and a list of open problems. Three appendices by M. Muger and A. Virelizier summarize their work in this area. The book is addressed to mathematicians, theoretical physicists, and graduate students interested in topological aspects of quantum field theory. The exposition is self-contained and well suited for a one-semester graduate course. Prerequisites include only basics of algebra and topology.
 

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Contents

Groupalgebras
23
Twodimensional HQFTs
40
Biangular algebras and lattice HQFTs
66
Enumeration problems in dimension two
88
Crossed G categories and invariants of links
130
Modular Gcategories and HQFTs
158
Miscellaneous algebra
186
Relative HQFTs
205
State sum invariants of 3dimensionaI G manifolds
213
Open problems
219
Algebraic properties of Hopf G coalgebras
236
Invariants of 3dimensional G manifolds from Hopf coalgebras
253
Bibliography
263
Index
273
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