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Homotopy Quantum Field Theory
EMS Tracts in Mathematics Vol. 10

Vladimir Turaev (Indiana University, Bloomington, USA)

Homotopy Quantum Field Theory

With Appendices by Michael Müger and Alexis Virelizier

ISBN print 978-3-03719-086-9, ISBN online 978-3-03719-586-4
DOI 10.4171/086
June 2010, 290 pages, hardcover, 17 x 24 cm.
58.00 Euro

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.

Keywords: Topological Quantum Field Theory, surfaces, 3-manifolds, Frobenius algebras, monoidal categories, sections of fiber bundles

Further Information

Review in Zentralblatt MATH 1243.81016

Review in MR 2674592

Review in Bull. Amer. Math. Soc. 49 (2012), 337–345