Homotopy quantum field theory. With appendices by Michael Müger and Alexis Virelizier.

*(English)*Zbl 1243.81016
EMS Tracts in Mathematics 10. Zürich: European Mathematical Society (EMS) (ISBN 978-3-03719-086-9/hbk). xiv, 276 p. (2010).

Topological quantum field theories (TQFTs) have been used to define invariants of manifolds in low dimensions, and this monograph explores a generalization of TQFT called a homotopy quantum field theory (HQFT). This is a theory that allows for maps to a topological space \(X\), which is called the target space, and one recovers a TQFT from an HQFT by taking the target space \(X\) to be a point. Many of the known results for TQFTs can be generalized to HQFTs, and this is achieved in great part in this book.

The book develops HQFTs in general terms, paying special attention to the case where the target is an Eilenberg-MacLane space \(K(G,1)\) corresponding to a discrete group \(G\). Since maps to \(K(G,1)\) classify principal \(G\)-bundles, the numerical invariants provided by HQFTs can be viewed as “quantum” characteristic numbers. The author focuses mainly on the cases of \((d+1)\)-dimensional HQFTs with \(d=1,2,\) which is justified by the fact that these are the cases of particular interest for applications to low-dimensional topology. The book studies in great detail the algebraic structures underlying such HQFTs. For \(d=1\), this involves \(G\)-graded algebras, and the main results proved here are: (i) a bijective correspondence between isomorphism classes of \(2\)-dimensional HQFTs with target \(K(G,1)\) and isomorphism classes of crossed Frobenius \(G\)-algebras, and (ii) a classification of semisimple crossed Frobenius \(G\)-algebras in terms of 2-dimensional cohomology classes of the finite-index subgroups of \(G\). For \(d=2\), this involves crossed \(G\)-categories, and associated to each modular crossed \(G\)-category is a 3-dimensional HQFT with target \(K(G,1)\). This HQFT provides numerical invariants of closed 3-dimensional \(G\)-manifolds, and setting \(G=1\), one recovers the construction of 3-dimensional TQFTs from modular categories in [V. G. Turaev, Quantum invariants of knots and 3-manifolds. de Gruyter Studies in Mathematics. 18. Berlin: Walter de Gruyter (1994; Zbl 0812.57003); 2nd revised ed. (2010; Zbl 1213.57002)].

This book is based on several previous papers by the author, some of which have appeared only in preprint form, and it also includes seven appendices, one of which is written by Michael Müger and two by Alexis Virelizier. The book is reasonably self-contained; at some points the proofs involve advanced techniques, nevertheless all the definitions and statements of theorems can be understood by readers with a background that includes the basics of algebra, topology and (starting with Chapter VI) category theory.

The book develops HQFTs in general terms, paying special attention to the case where the target is an Eilenberg-MacLane space \(K(G,1)\) corresponding to a discrete group \(G\). Since maps to \(K(G,1)\) classify principal \(G\)-bundles, the numerical invariants provided by HQFTs can be viewed as “quantum” characteristic numbers. The author focuses mainly on the cases of \((d+1)\)-dimensional HQFTs with \(d=1,2,\) which is justified by the fact that these are the cases of particular interest for applications to low-dimensional topology. The book studies in great detail the algebraic structures underlying such HQFTs. For \(d=1\), this involves \(G\)-graded algebras, and the main results proved here are: (i) a bijective correspondence between isomorphism classes of \(2\)-dimensional HQFTs with target \(K(G,1)\) and isomorphism classes of crossed Frobenius \(G\)-algebras, and (ii) a classification of semisimple crossed Frobenius \(G\)-algebras in terms of 2-dimensional cohomology classes of the finite-index subgroups of \(G\). For \(d=2\), this involves crossed \(G\)-categories, and associated to each modular crossed \(G\)-category is a 3-dimensional HQFT with target \(K(G,1)\). This HQFT provides numerical invariants of closed 3-dimensional \(G\)-manifolds, and setting \(G=1\), one recovers the construction of 3-dimensional TQFTs from modular categories in [V. G. Turaev, Quantum invariants of knots and 3-manifolds. de Gruyter Studies in Mathematics. 18. Berlin: Walter de Gruyter (1994; Zbl 0812.57003); 2nd revised ed. (2010; Zbl 1213.57002)].

This book is based on several previous papers by the author, some of which have appeared only in preprint form, and it also includes seven appendices, one of which is written by Michael Müger and two by Alexis Virelizier. The book is reasonably self-contained; at some points the proofs involve advanced techniques, nevertheless all the definitions and statements of theorems can be understood by readers with a background that includes the basics of algebra, topology and (starting with Chapter VI) category theory.

Reviewer: Hans U. Boden (Hamilton/Ontario)

##### MSC:

81-02 | Research exposition (monographs, survey articles) pertaining to quantum theory |

57R56 | Topological quantum field theories (aspects of differential topology) |

81T45 | Topological field theories in quantum mechanics |

57M27 | Invariants of knots and \(3\)-manifolds (MSC2010) |

18D10 | Monoidal, symmetric monoidal and braided categories (MSC2010) |

55P20 | Eilenberg-Mac Lane spaces |

53D45 | Gromov-Witten invariants, quantum cohomology, Frobenius manifolds |

57M60 | Group actions on manifolds and cell complexes in low dimensions |