# Homotopy Quantum Field Theory

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*Vladimir Turaev*

with Appendices by Michael Müger and Alexis Virelizier

A publication of the European Mathematical Society

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.

#### Readership

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