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15
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
Homological Quantum Field Theory
 in M. Levy (Ed.), Mathematical Physics Research Developments, Nova Publishers
"... We show that the space of chains of smooth maps from spheres into a fixed compact oriented manifold has a natural structure of a transversal dalgebra. We construct a structure of transversal 1category on the space of chains of maps from a suspension space S(Y), with certain boundary restrictions, ..."
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Cited by 3 (3 self)
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We show that the space of chains of smooth maps from spheres into a fixed compact oriented manifold has a natural structure of a transversal dalgebra. We construct a structure of transversal 1category on the space of chains of maps from a suspension space S(Y), with certain boundary restrictions, into a fixed compact oriented manifold. We define homological quantum field theories HLQFT and construct several examples of such structures. Our definition is based on the notions of string topology of Chas and Sullivan, and homotopy quantum field theories of Turaev. 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
2Representations and Equivariant 2D Topological Field Theories Contents
, 2008
"... 2 Frobenius algebras with twisted Gaction 4 ..."
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.
ATG TQFT’s and gerbes
, 2004
"... Abstract We generalize the notion of parallel transport along paths for abelian bundles to parallel transport along surfaces for abelian gerbes using an embedded Topological Quantum Field Theory (TQFT) approach. We show both for bundles and gerbes with connection that there is a onetoone correspon ..."
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Abstract We generalize the notion of parallel transport along paths for abelian bundles to parallel transport along surfaces for abelian gerbes using an embedded Topological Quantum Field Theory (TQFT) approach. We show both for bundles and gerbes with connection that there is a onetoone correspondence between their local description in terms of locallydefined functions and forms and their nonlocal description in terms of a suitable class of embedded TQFT’s. AMS Classification 55R65; 53C29
Contemporary Mathematics Formal Homotopy Quantum Field Theories, II: Simplicial Formal Maps
, 2005
"... Abstract. Simplicial formal maps were introduced in the first paper of this series as a tool for studying Homotopy Quantum Field Theories with background a general homotopy 2type. Here we continue their study, showing how a natural generalisation can handle much more general backgrounds. The questi ..."
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Abstract. Simplicial formal maps were introduced in the first paper of this series as a tool for studying Homotopy Quantum Field Theories with background a general homotopy 2type. Here we continue their study, showing how a natural generalisation can handle much more general backgrounds. The question of the geometric interpretation of these formal maps is partially answered in terms of combinatorial bundles. This suggests new interpretations of HQFTs.
Homotopy Quantum Field Theories
, 2002
"... Abstract. In this short note we provide a review of some developments in the area of homotopy quantum field theories, loosely based on a talk ..."
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Abstract. In this short note we provide a review of some developments in the area of homotopy quantum field theories, loosely based on a talk