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Extended TQFT’s and Quantum Gravity
, 2007
"... Abstract. This paper gives a definition of an extended topological quantum field theory (TQFT) as a weak 2functor Z: nCob2→2Vect, by analogy with the description of a TQFT as a functor Z: nCob→Vect. We also show how to obtain such a theory from any finite group G. This theory is related to a topolo ..."
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Abstract. This paper gives a definition of an extended topological quantum field theory (TQFT) as a weak 2functor Z: nCob2→2Vect, by analogy with the description of a TQFT as a functor Z: nCob→Vect. We also show how to obtain such a theory from any finite group G. This theory is related to a topological gauge theory, the DijkgraafWitten model. To give this definition rigorously, we first define a bicategory of cobordisms between cobordisms. We also give some explicit description of a highercategorical version of Vect, denoted 2Vect, a bicategory of 2vector spaces. Along the way, we prove several results showing how to construct 2vector spaces of Vectvalued presheaves on certain kinds of groupoids. In particular, we use the case when these are groupoids whose objects are connections, and whose morphisms are gauge transformations, on the manifolds on which the extended TQFT is to be defined. On cobordisms between these manifolds, we show how a construction of “pullback and pushforward ” of presheaves gives both the morphisms and 2morphisms in 2Vect for the extended TQFT, and that these
Model structures on the category of small double categories, Algebraic and Geometric Topology 8
, 2008
"... Abstract. In this paper we obtain several model structures on DblCat, the category of small double categories. Our model structures have three sources. We first transfer across a categorificationnerve adjunction. Secondly, we view double categories as internal categories in Cat and take as our weak ..."
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Abstract. In this paper we obtain several model structures on DblCat, the category of small double categories. Our model structures have three sources. We first transfer across a categorificationnerve adjunction. Secondly, we view double categories as internal categories in Cat and take as our weak equivalences various internal equivalences defined via Grothendieck topologies. Thirdly, DblCat inherits a model structure as a category of algebras over a 2monad. Some of these model structures coincide and the different points of view give us further results about cofibrant replacements and cofibrant objects. As part of this program we give explicit descriptions and discuss properties of free double categories, quotient double categories, colimits of double categories, and several nerves
Resolutions of pModular TQFT’s and Representations of Symmetric Groups 1
, 2001
"... 2. FrohmanNicas TQFT’s over Z..............................7 3. Lefschetz Decompositions and Specht Modules............. 11 ..."
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2. FrohmanNicas TQFT’s over Z..............................7 3. Lefschetz Decompositions and Specht Modules............. 11
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.
A PREHISTORY OF nCATEGORICAL PHYSICS
, 2008
"... We begin with a chronology tracing the rise of symmetry concepts in physics, starting with groups and their role in relativity, and leading up to more sophisticated concepts from ncategory theory, which manifest themselves in Feynman diagrams and their higherdimensional generalizations: strings, me ..."
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We begin with a chronology tracing the rise of symmetry concepts in physics, starting with groups and their role in relativity, and leading up to more sophisticated concepts from ncategory theory, which manifest themselves in Feynman diagrams and their higherdimensional generalizations: strings, membranes and spin foams.
CATEGORICAL CENTERS AND RESHETIKHINTURAEV INVARIANTS
, 812
"... Abstract. A theorem of Müger asserts that the center Z(C) of a spherical klinear category C is a modular category if k is an algebraically closed field and the dimension of C is invertible. We generalize this result to the case where k is an arbitrary commutative ring, without restriction on the di ..."
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Abstract. A theorem of Müger asserts that the center Z(C) of a spherical klinear category C is a modular category if k is an algebraically closed field and the dimension of C is invertible. We generalize this result to the case where k is an arbitrary commutative ring, without restriction on the dimension of the category. Moreover we construct the analogue of the ReshetikhinTuraev invariant associated to Z(C) and give an algorithm for computing this invariant in terms of certain explicit morphisms in the category C. Our approach is based on (a) Lyubashenko’s construction of the ReshetikhinTuraev invariant in terms of the coend of a ribbon category; (b) an explicit algorithm for computing this invariant via Hopf diagrams; (c) an algebraic interpretation of the center of C as the category of modules over a certain Hopf monad Z on the category C; (d) a generalization of the classical notion of Drinfeld double to Hopf monads, which, applied to the Hopf monad Z, provides an explicit
Relation between quantum invariants of 3manifolds and 2dimensional CWcomplexes
, 2000
"... We show that the ReshetikhinTuraevWalker invariant of 3manifolds can be normalized to obtain an invariant of 4dimensional thickenings of 2complexes. Moreover when the underlying semisimple tortile category comes from the Lie family (“quantum groups”) over the ring Z (p)[v] where v is a primitiv ..."
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We show that the ReshetikhinTuraevWalker invariant of 3manifolds can be normalized to obtain an invariant of 4dimensional thickenings of 2complexes. Moreover when the underlying semisimple tortile category comes from the Lie family (“quantum groups”) over the ring Z (p)[v] where v is a primitive prime root of unity, the 0term in the Ohtsuki expansion of this invariant depends only on the spine and is the Z/pZ invariant of 2complexes defined in [10]. As a consequence it is shown that when the Euler characteristic is greater or equal to 1, the 2complex invariant depends only on homology. The last statement doesn’t hold for the negative Euler characteristic case.
Contents
, 2006
"... Abstract. In this article, we introduce and study Hopf monads on monoidal categories with duals. Hopf monads generalize Hopf algebras to a nonbraided (and nonlinear) setting. We extend many fundamental results of the theory of Hopf algebras (such as the decomposition of Hopf modules, the existence ..."
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Abstract. In this article, we introduce and study Hopf monads on monoidal categories with duals. Hopf monads generalize Hopf algebras to a nonbraided (and nonlinear) setting. We extend many fundamental results of the theory of Hopf algebras (such as the decomposition of Hopf modules, the existence of integrals, Maschke’s criterium of semisimplicity, etc...) to Hopf monads. We also introduce and study quasitriangular and ribbon Hopf monads (again
Contents
, 2001
"... Abstract: We develop an explicit skein theoretical algorithm to compute the Alexander polynomial of a 3manifold from a surgery presentation employing the methods used in the construction of quantum invariants of 3manifolds. As a prerequisite we establish and prove a rather unexpected equivalence b ..."
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Abstract: We develop an explicit skein theoretical algorithm to compute the Alexander polynomial of a 3manifold from a surgery presentation employing the methods used in the construction of quantum invariants of 3manifolds. As a prerequisite we establish and prove a rather unexpected equivalence between the topological quantum field theory constructed by Frohman and Nicas using the intersection homology of U(1)representation varieties on the one side and the combinatorially constructed HenningsTQFT based on the quasitriangular Hopf algebra N = Z/2 ⋉ ∧ ∗ R 2 on the other side. We find that both TQFT’s are SL(2, R)equivariant functors and also as such isomorphic. The SL(2, R)action in the Hennings construction comes from the natural action on N and in the case of the FrohmanNicas theory from the HardLefschetz decomposition of the U(1)moduli spaces given that they are naturally Kähler. The irreducible components of this TQFT, corresponding to simple representations of SL(2, Z) and Sp(2g, Z), thus yield a large family of homological TQFT’s by taking sums and products. We give several examples of TQFT’s and invariants that appear to fit into this family, such as Milnor and Reidemeister Torsion,