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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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Cited by 16 (0 self)
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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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Cited by 15 (5 self)
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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.
Orbifolding Frobenius algebras
, 2000
"... Abstract. We study the general theory of Frobenius algebras with group actions. These structures arise when one is studying the algebraic structures associated to a geometry stemming from a physical theory with a global finite gauge group, i.e. orbifold theories. In this context, we introduce and ax ..."
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Cited by 9 (1 self)
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Abstract. We study the general theory of Frobenius algebras with group actions. These structures arise when one is studying the algebraic structures associated to a geometry stemming from a physical theory with a global finite gauge group, i.e. orbifold theories. In this context, we introduce and axiomatize these algebras. Furthermore, we define geometric cobordism categories whose functors to the category of vector spaces are parameterized by these algebras. The theory is also extended to the graded and super–graded cases. As an application, we consider Frobenius algebras having some additional properties making them more tractable. These properties are present in Frobenius algebras arising as quotients of Jacobian ideal, such as those having their origin in quasi–homogeneous singularities and their symmetries.
The stable mapping class group of simply connected 4manifolds
, 2006
"... We consider mapping class groups Γ(M) = π0Diff(M fix ∂M) of smooth compact simply connected oriented 4–manifolds M bounded by a collection of 3–spheres. We show that if M contains CP 2 or CP 2 as a connected summand then Γ(M) is independent of the number of boundary components. By repackaging cla ..."
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Cited by 6 (1 self)
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We consider mapping class groups Γ(M) = π0Diff(M fix ∂M) of smooth compact simply connected oriented 4–manifolds M bounded by a collection of 3–spheres. We show that if M contains CP 2 or CP 2 as a connected summand then Γ(M) is independent of the number of boundary components. By repackaging classical results of Wall, Kreck and Quinn, we show that the natural homomorphism from the mapping class group to the group of automorphisms of the intersection form becomes an isomorphism after stabilization with respect to connected sum with CP 2 #CP 2. We next consider the 3+1 dimensional cobordism 2–category C of 3–spheres, 4–manifolds (as above) and enriched with isotopy classes of diffeomorphisms as 2–morphisms. We identify the homotopy type of the classifying space of this category as the Hermitian algebraic Ktheory of the integers. We also comment on versions of these results for simply connected spin 4–manifolds.
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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Cited by 5 (2 self)
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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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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
Discrete torsion, symmetric products and the Hilbert scheme
, 2006
"... Recently the understanding of the cohomology of the Hilbert scheme of points on K3 surfaces has been greatly improved by Lehn and Sorger [18]. Their approach uses the connection of the Hilbert scheme to the orbifolds given by the symmetric products of these surfaces. We introduced a general theory r ..."
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Cited by 4 (3 self)
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Recently the understanding of the cohomology of the Hilbert scheme of points on K3 surfaces has been greatly improved by Lehn and Sorger [18]. Their approach uses the connection of the Hilbert scheme to the orbifolds given by the symmetric products of these surfaces. We introduced a general theory replacing cohomology
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
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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Cited by 1 (0 self)
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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.
Super, Quantum and NonCommutative Species
, 2009
"... Dedicated to the memory of GianCarlo Rota. We introduce an approach to the categorification of rings, via the notion of distributive categories with negative objects, and use it to lay down categorical foundations for the study of super, quantum and noncommutative combinatorics. Via the usual dual ..."
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Cited by 1 (1 self)
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Dedicated to the memory of GianCarlo Rota. We introduce an approach to the categorification of rings, via the notion of distributive categories with negative objects, and use it to lay down categorical foundations for the study of super, quantum and noncommutative combinatorics. Via the usual duality between algebra and geometry, these constructions provide categorifications for various types of affine spaces, thus our works may be regarded as a starting point towards the construction of a categorical geometry.