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Homotopy Quantum Field Theory
Vladimir Turaev, Indiana University, Bloomington
with Appendices by Michael Müger and Alexis Virelizier
A publication of the European Mathematical Society.
EMS Tracts in Mathematics
2010; 290 pp; hardcover
Volume: 10
ISBN-10: 3-03719-086-8
ISBN-13: 978-3-03719-086-9
List Price: US$78
Member Price: US$62.40
Order Code: EMSTM/10
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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. Müger 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.

A publication of the European Mathematical Society (EMS). Distributed within the Americas by the American Mathematical Society.


Graduate students and research mathematicians interested in topological aspects of quantum field theory.

Table of Contents

  • Generalities on HQFTs
  • Group-algebras
  • Two-dimensional HQFTs
  • Biangular algebras and lattice HQFTs
  • Enumeration problems in dimension two
  • Crossed \(G\)-categories and invariants of links
  • Modular \(G\)-categories and HQFTs
  • Miscellaneous algebra
  • Appendix 1. Relative HQFTs
  • Appendix 2. State sum invariants of 3-dimensional \(G\)-manifolds
  • Appendix 3. Recent work on HQFTs
  • Appendix 4. Open problems
  • Appendix 5. On the structure of braided crossed \(G\)-categories
  • Appendix 6. Algebraic properties of Hopf \(G\)-coalgebras
  • Appendix 7. Invariants of 3-dimensional \(G\)-manifolds from Hopf coalgebras
  • Bibliography
  • Index
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