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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.
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,
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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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
Homotopy Quantum Field Theories and Tortile Structures. arXiv:math.QA/0111084
"... Abstract. We study a variation of Turaev’s homotopy quantum field theories using 2categories of surfaces. We define the homotopy surface 2category of a space X and define an SXstructure to be a monoidal 2functor from this to the 2category of idempotentcomplete additive klinear categories. We ..."
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Abstract. We study a variation of Turaev’s homotopy quantum field theories using 2categories of surfaces. We define the homotopy surface 2category of a space X and define an SXstructure to be a monoidal 2functor from this to the 2category of idempotentcomplete additive klinear categories. We initiate the study of the algebraic structure arising from these functors. In particular we show that, under certain conditions, an SXstructure gives rise to a lax tortile πcategory when the background space is an EilenbergMaclane space X = K(π,1), and to a tortile category with lax π2Xaction when the background space is simplyconnected. 2000 Mathematics Subject Classification 57R56, 18D05
SECTIONS OF FIBER BUNDLES OVER SURFACES
, 904
"... Abstract. We study the existence problem and the enumeration problem for sections of Serre fibrations over compact orientable surfaces. When the fundamental group of the fiber is finite, a complete solution is given in terms of 2dimensional cohomology classes associated with certain irreducible rep ..."
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Abstract. We study the existence problem and the enumeration problem for sections of Serre fibrations over compact orientable surfaces. When the fundamental group of the fiber is finite, a complete solution is given in terms of 2dimensional cohomology classes associated with certain irreducible representations of this group. The proofs are based on Topological Quantum
2Representations and Equivariant 2D Topological Field Theories Contents
, 2008
"... 2 Frobenius algebras with twisted Gaction 4 ..."
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Gerbes and homotopy quantum eld theories
"... Abstract For smooth nite 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 eld theories, and we show that flat gerbes are related to a specic ..."
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Abstract For smooth nite 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 eld theories, and we show that flat gerbes are related to a specic class of rank one homotopy quantum eld theories. AMS Classication 55P48; 57R56, 81T70
ABELIAN HOMOTOPY DIJKGRAAF–WITTEN THEORY
, 2004
"... ABSTRACT. We construct a version of Dijkgraaf–Witten theory based on a compact abelian Lie group within the formalism of Turaev’s homotopy quantum field theory. As an application we show that the 2+1–dimensional theory based on U(1) classifies lens spaces up to homotopy type. 1. ..."
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ABSTRACT. We construct a version of Dijkgraaf–Witten theory based on a compact abelian Lie group within the formalism of Turaev’s homotopy quantum field theory. As an application we show that the 2+1–dimensional theory based on U(1) classifies lens spaces up to homotopy type. 1.