Homotopy Quantum box conception (HQFT) is a department of Topological Quantum box idea based by means of E. Witten and M. Atiyah. It applies rules from theoretical physics to review vital bundles over manifolds and, extra in most cases, homotopy periods of maps from manifolds to a hard and fast aim area. This ebook is the 1st systematic exposition of Homotopy Quantum box concept. It starts off with a proper definition of an HQFT and offers examples of HQFTs in all dimensions. the most physique of the textual content is concentrated on $2$-dimensional and $3$-dimensional HQFTs. A examine of those HQFTs ends up 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 a number of appendices together with an overview of contemporary advancements and an inventory of open difficulties. 3 appendices through M. Müger and A. Virelizier summarize their paintings during this region. The booklet is addressed to mathematicians, theoretical physicists, and graduate scholars attracted to topological points of quantum box idea. The exposition is self-contained and well matched for a one-semester graduate direction. necessities comprise simply fundamentals of algebra and topology. A booklet of the ecu Mathematical Society (EMS). allotted in the Americas by way of the yankee Mathematical Society.

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V and g W V ! U in C , we've got tr. fg/ D tr. gf /I (ii) for any endomorphisms f and g of items of C , now we have tr. f / D tr. f / and tr. f ˝ g/ D tr. f / tr. g/: five. 2 measurement. We outline the size of an item U 2 C through dim. U / D tr. idU / D d. UU / cUU;U . ÂU ˝ idU / bU 2 finish C . 1/: If U 2 C˛ , then dim. U / D F . d. ;˛;U / / bU is the worth of F on a coloured G-knot represented by means of a diagram together with an embedded orientated circle categorised with . ˛; U /. This knot ok S three is an (oriented) unknot endowed with the homomorphism 1 . CK / D Z ! G wearing the meridian to ˛. The houses of F proven in part four. 7 (or in Lemma five. 1. 1) indicate that isomorphic gadgets have equivalent dimensions and for any U 2 C and ˇ 2 G, dim. U / D dim. U / and dim. 'ˇ . U // D 'ˇ . dim. U //: For any U; V 2 C , dim. U ˝ V / D dim. U / dim. V /: For morphisms and items of the impartial part C1 C , the definitions above coincide with the normal definitions of hint and size in a ribbon type. this suggests that for any f 2 finish C . 1/, now we have tr. f / D f . specifically, dim. 1C / D tr. id1 / D id1 . five. three Algebra of colours. We outline the Lalgebra of colours or the Verlinde algebra L D L. C/ of C. examine the K-module U 2C finish C . U /, the place U runs over all items of C. The component of this module represented through f 2 finish C . U / is denoted hU; f i 156 VI Crossed G-categories and invariants of hyperlinks or shorter hf i. We issue this module via the next family members: for any morphisms f W U ! V and g W V ! U in C , hV; fgi D hU; gf i: (5. three. a) Denote the quotient K-module by way of L. we offer L with multiplication by way of the formulation hf i hf zero i D hf ˝ f zero i. essentially, L is an associative K-algebra with unit hid1 i. each item U 2 C determines a component hUL i D hidU i 2 L. The algebra L is G-graded: L D ˛2G L˛ the place L˛ is the submodule of L additively generated by way of the vectors hU; f i with U 2 C˛ . we've got L˛ Lˇ L˛ˇ for all ˛; ˇ 2 G. The formulation '˛ . hf i/ D h'˛ . f /i defines an motion of G on L by means of algebra automorphisms. basically, '˛ . Lˇ / D L˛ˇ ˛ 1 for all ˛; ˇ 2 G. The life of the braiding means that ab D '˛ . b/a for any a 2 L˛ ; b 2 L. The life of the twist means that '˛ jL˛ D identity for any ˛ 2 G. therefore L satisfies the 1st 3 axioms (3. 1. 1)–(3. 1. three) (see p. 25) of a crossed G-algebra. The hint of morphisms defines a K-linear practical L ! finish C . 1/ wearing any generator hf i to tr. f / 2 finish C . 1/. We denote this sensible by way of dim. specifically, dim. hU i/ D tr. idU / D dim. U / for any U 2 C . It follows from the homes of the hint that dim is an algebra homomorphism. Sending a generator hU; f i 2 L to the generator hU ; f W U ! U i we outline a K-linear homomorphism L ! L denoted by means of . It follows from the definitions and Corollary four. 6 that's an involutive anti-automorphism of L wearing each one L˛ onto L˛ 1 and commuting with dim W L ! finish C . 1/. five. four Generalized hues. the weather of the algebra of colours L D L. C / can be utilized to paint G-links. A generalized coloring of a G-link . `; z; g/ is a functionality a which assigns to each direction W Œ0; 1 !