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Homotopy quantum field theories and the homotopy cobordism category in dimension 1+1
"... Abstract. We define Homotopy quantum field theories (HQFT) as Topological quantum field theories (TQFT) for manifolds endowed with extra structure in the form of a map into some background space X. We also build the category of homotopy cobordisms HCobord(n, X) such that an HQFT is a functor ..."
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Abstract. We define Homotopy quantum field theories (HQFT) as Topological quantum field theories (TQFT) for manifolds endowed with extra structure in the form of a map into some background space X. We also build the category of homotopy cobordisms HCobord(n, X) such that an HQFT is a functor
Gerbes and homotopy quantum field theories
, 2002
"... For manifolds with freely generated first homology, we characterize gerbes with connection as functors on a certain surface cobordism category. This allows us to relate gerbes with connection to Turaev’s 1+1dimensional homotopy quantum field theories, and we show that flat gerbes on such spaces a ..."
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For manifolds with freely generated first homology, we characterize gerbes with connection as functors on a certain surface cobordism category. This allows us to relate gerbes with connection to Turaev’s 1+1dimensional homotopy quantum field theories, and we show that flat gerbes on such spaces as above are the same as a specific class of rank one homotopy quantum field theories.
TQFT’s and gerbes
 Algebr. Geom. Topol
"... We generalise the notions of holonomy and parallel transport for abelian bundles and gerbes using an embedded TQFT construction, and obtain statesumlike formulae for the parallel transport along paths and surfaces. This approach has been greatly influenced by the work of Graeme Segal, in particula ..."
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Cited by 5 (2 self)
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We generalise the notions of holonomy and parallel transport for abelian bundles and gerbes using an embedded TQFT construction, and obtain statesumlike formulae for the parallel transport along paths and surfaces. This approach has been greatly influenced by the work of Graeme Segal, in particular his axiomatic approach to Conformal Field Theory. 1
Formal Homotopy Quantum Field Theories
 II : Simplicial Formal Maps
"... Homotopy Quantum Field Theories (HQFTs) were introduced by the second author to extend the ideas and methods of Topological Quantum Field Theories to closed dmanifolds endowed with extra structure in the form of homotopy classes of maps into a given ‘target ’ space. For d = 1, classifications of HQ ..."
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Homotopy Quantum Field Theories (HQFTs) were introduced by the second author to extend the ideas and methods of Topological Quantum Field Theories to closed dmanifolds endowed with extra structure in the form of homotopy classes of maps into a given ‘target ’ space. For d = 1, classifications of HQFTs in terms of algebraic structures are known when B is a K(G,1) and also when it is simply connected. Here we study general HQFTs with d = 1 and target a general 2type, giving a common generalisation of the classifying algebraic structures for the two cases previously known. The algebraic models for 2types that we use are crossed modules, C, and we introduce a notion of formal Cmap, which extends the usual latticetype constructions to this setting. This leads to a classification of ‘formal ’ 2dimensional HQFTs with target C,
ENRICHED HOMOTOPY QUANTUM FIELD THEORIES AND TORTILE STRUCTURES
, 2001
"... The motivation for this paper was to construct approximations to a conformal version of homotopy quantum field theory using 2categories. A homotopy quantum field theory, as defined by Turaev in [9], is a variant of a topological quantum field theory in which manifolds come equipped with a map to so ..."
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The motivation for this paper was to construct approximations to a conformal version of homotopy quantum field theory using 2categories. A homotopy quantum field theory, as defined by Turaev in [9], is a variant of a topological quantum field theory in which manifolds come equipped with a map to some auxiliary space X.
2Representations and Equivariant 2D Topological Field Theories Contents
, 2008
"... 2 Frobenius algebras with twisted Gaction 4 ..."
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Gerbes and homotopy quantum field theories
, 2004
"... For smooth finite dimensional manifolds, we characterise gerbes with connection as functors on a certain surface cobordism category. This allows us to relate gerbes with connection to Turaev’s 1+1dimensional homotopy quantum field theories, and we show that flat gerbes are related to a specific c ..."
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For smooth finite dimensional manifolds, we characterise gerbes with connection as functors on a certain surface cobordism category. This allows us to relate gerbes with connection to Turaev’s 1+1dimensional homotopy quantum field theories, and we show that flat gerbes are related to a specific class of rank one homotopy quantum field theories.