## Homotopy Quantum Field TheoryHomotopy 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 |

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### Common terms and phrases

2-cocycle a e G A'-cobordism action of G automorphisms Axiom base point biangular G-algebra bijection canonical class of maps closed connected oriented closed oriented coalgebras cobordisms cocycle cohomology class colored G-graph components compute Consider Corollary crossed Frobenius G-algebra crossed G-algebra crossed Hopf G-coalgebra CW-decomposition defined definition denote diagram direct sum disjoint union edge element epimorphism equality equivalence extended G-surface finite type follows formula functor fusion category G-link G-set G-system GLn(K gluing rule group G group homomorphism Ha}aeG Hermitian homomorphism g homotopy class Hopf algebra HQFTs implies induced inner product invariant invertible isomorphism classes isotopy Lemma loop manifold map g matrix modular crossed modules monoidal category morphisms multiplication obtain oriented surface pair principal G-bundle Proof quantum field theory representations resp ribbon Hopf Section semisimple semisimple crossed simple objects tensor product Theorem topological TQFTs trivial twist vector X-cobordism X-homeomorphism X-HQFT