By Vladimir Turaev
Homotopy Quantum box idea (HQFT) is a department of Topological Quantum box conception based through E. Witten and M. Atiyah. It applies rules from theoretical physics to check critical bundles over manifolds and, extra as a rule, homotopy sessions of maps from manifolds to a set goal area. This ebook is the 1st systematic exposition of Homotopy Quantum box thought. It starts off with a proper definition of an HQFT and gives examples of HQFTs in all dimensions. the most physique of the textual content is targeted on $2$-dimensional and $3$-dimensional HQFTs. A learn of those HQFTs results in new algebraic gadgets: crossed Frobenius group-algebras, crossed ribbon group-categories, and Hopf group-coalgebras. those notions and their connections with HQFTs are mentioned intimately. The textual content ends with numerous appendices together with an summary of contemporary advancements and an inventory of open difficulties. 3 appendices by means of M. Müger and A. Virelizier summarize their paintings during this zone. The booklet is addressed to mathematicians, theoretical physicists, and graduate scholars drawn to topological elements of quantum box thought. The exposition is self-contained and compatible for a one-semester graduate direction. necessities comprise purely fundamentals of algebra and topology. A ebook of the ecu Mathematical Society (EMS). dispensed in the Americas by means of the yankee Mathematical Society.