Results 1  10
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33
From subfactors to categories and topology I. Frobenius algebras in and Morita equivalence of tensor categories
 J. Pure Appl. Alg
, 2003
"... We consider certain categorical structures that are implicit in subfactor theory. Making the connection between subfactor theory (at finite index) and category theory explicit sheds light on both subjects. Furthermore, it allows various generalizations of these structures, e.g. to arbitrary ground f ..."
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Cited by 52 (6 self)
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We consider certain categorical structures that are implicit in subfactor theory. Making the connection between subfactor theory (at finite index) and category theory explicit sheds light on both subjects. Furthermore, it allows various generalizations of these structures, e.g. to arbitrary ground fields, and the proof of new results about topological invariants in three dimensions. The central notion is that of a Frobenius algebra in a tensor category A, which reduces to the classical notion if A = FVect, where F is a field. An object X ∈ A with twosided dual X gives rise to a Frobenius algebra in A, and under weak additional conditions we prove a converse: There exists a bicategory E with ObjE = {A, B} such that EndE(A) ⊗ ≃ A and such that there are J, J: B ⇋ A producing the given Frobenius algebra. Many properties (additivity, sphericity, semisimplicity,...) of A carry over to the bicategory E. We define weak monoidal Morita equivalence of tensor categories, denoted A ≈ B, and establish a correspondence between Frobenius algebras in A and tensor categories B ≈ A. While considerably weaker than equivalence of tensor categories, weak monoidal Morita equivalence A ≈ B has remarkable consequences: A and B have equivalent (as braided tensor categories) quantum doubles (‘centers’) and (if A, B are semisimple spherical or ∗categories) have equal dimensions and give rise the same state sum invariant of closed oriented 3manifolds as recently defined by Barrett and Westbury. An instructive example is provided by finite dimensional semisimple and cosemisimple Hopf algebras, for which we prove H − mod ≈ ˆH − mod. The present formalism permits a fairly complete analysis of the center of a semisimple spherical category, which is the subject of the companion paper math.CT/0111205. 1
Finite groups, spherical 2categories, and 4manifold invariants. arXiv:math.QA/9903003
"... In this paper we define a class of statesum invariants of compact closed oriented piecewise linear 4manifolds using finite groups. The definition of these statesums follows from the general abstract construction of 4manifold invariants using spherical 2categories, as we defined in [32], althou ..."
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Cited by 16 (5 self)
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In this paper we define a class of statesum invariants of compact closed oriented piecewise linear 4manifolds using finite groups. The definition of these statesums follows from the general abstract construction of 4manifold invariants using spherical 2categories, as we defined in [32], although it requires a slight generalization of that construction. We show that the statesum invariants of Birmingham and Rakowski [11, 12, 13], who studied DijkgraafWitten type invariants in dimension 4, are special examples of the general construction that we present in this paper. They showed that their invariants are nontrivial by some explicit computations, so our construction includes interesting examples already. Finally, we indicate how our construction is related to homotopy 3types. This connection suggests that there are many more interesting examples of our construction to be found in the work on homotopy 3types, such as [15], for example. 1 1
On the WittenReshetikhinTuraev representations of the mapping class groups
 Proc. Amer. Math. Soc
, 1999
"... We consider a central extension of the mapping class group of a surface with a collection of framed colored points. The WittenReshetikhinTuraev TQFTs associated to SU(2) and SO(3) induce linear representations of this group. We show that the denominators of matrices which describe these representa ..."
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Cited by 14 (5 self)
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We consider a central extension of the mapping class group of a surface with a collection of framed colored points. The WittenReshetikhinTuraev TQFTs associated to SU(2) and SO(3) induce linear representations of this group. We show that the denominators of matrices which describe these representations over a cyclotomic field can be restricted in many cases. In this way, we give a proof of the known result that if the surface is a torus with no colored points, the representations have finite image. Recall that an object in a cobordism category of dimension 2+1 is a closed oriented surfaces Σ perhaps with some specified further structure. A morphism M from Σ to Σ ′ is (loosely speaking) a compact oriented 3manifold perhaps with some specified further structure, called a cobordism, whose boundary is the disjoint union of −Σ and Σ ′. A morphism M ′ from Σ ′ to Σ ′ ′ is composed with a morphism from Σ to Σ ′ by gluing along Σ ′ , inducing any required extra structure from the structures on M and M ′. Also the extra structure on a 3manifold must induce the extra structure on the boundary. A TQFT in dimension 2+1 is then a functor from
DBranes And KTheory In 2D Topological Field Theory,” hepth/0609042; see also lectures by G. Moore, at http://online.itp.ucsb.edu/online/mp01
"... This expository paper describes sewing conditions in twodimensional open/closed topological field theory. We include a description of the Gequivariant case, where G is a finite group. We determine the category of boundary conditions in the case that the closed string algebra is semisimple. In this ..."
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Cited by 13 (0 self)
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This expository paper describes sewing conditions in twodimensional open/closed topological field theory. We include a description of the Gequivariant case, where G is a finite group. We determine the category of boundary conditions in the case that the closed string algebra is semisimple. In this case we find that sewing constraints – the most primitive form of worldsheet locality – already imply that Dbranes are (Gtwisted) vector bundles on spacetime. We comment on extensions to cochainvalued theories and various applications. Finally, we give uniform proofs of all relevant sewing theorems using Morse theory. August
Direct sum decompositions and indecomposable TQFTs
 J. Math. Phys
, 1995
"... Abstract. The decomposition of an arbitrary axiomatic topological quantum field theory or TQFT into indecomposable theories is given. In particular, unitary TQFT’s in arbitrary dimensions are shown to decompose into a sum of theories in which the Hilbert space of the sphere is onedimensional, and i ..."
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Abstract. The decomposition of an arbitrary axiomatic topological quantum field theory or TQFT into indecomposable theories is given. In particular, unitary TQFT’s in arbitrary dimensions are shown to decompose into a sum of theories in which the Hilbert space of the sphere is onedimensional, and indecomposable twodimensional theories are classified. 1.
The Logic of Linear Functors
 Math. Structures Comput. Sci
, 2002
"... This paper describes a family of logics whose categorical semantics is based on functors with structure rather than on categories with structure. This allows the consideration of logics which contain possibly distinct logical subsystems whose interactions are mediated by functorial mappings. For exa ..."
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Cited by 8 (2 self)
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This paper describes a family of logics whose categorical semantics is based on functors with structure rather than on categories with structure. This allows the consideration of logics which contain possibly distinct logical subsystems whose interactions are mediated by functorial mappings. For example, within one unified framework, we shall be able to handle logics as diverse as modal logic, ordinary linear logic, and the "noncommutative logic" of Abrusci and Ruet, a variant of linear logic which has both commutative and noncommutative connectives. Although this paper will not consider in depth the categorical basis of this approach to logic, preferring instead to emphasize the syntactic novelties that it generates in the logic, we shall focus on the particular case when the logics are based on a linear functor, to give a definite presentation of these ideas. However, it will be clear that this approach to logic has considerable generality. There have been several individu...
Setting the quantum integrand of Mtheory
"... Abstract. In anomalyfree quantum field theories the integrand in the bosonic functional integral— the exponential of the effective action after integrating out fermions—is often defined only up to a phase without an additional choice. We term this choice “setting the quantum integrand”. In the low ..."
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Cited by 7 (2 self)
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Abstract. In anomalyfree quantum field theories the integrand in the bosonic functional integral— the exponential of the effective action after integrating out fermions—is often defined only up to a phase without an additional choice. We term this choice “setting the quantum integrand”. In the lowenergy approximation to Mtheory the E8model for the Cfield allows us to set the quantum integrand using geometric index theory. We derive mathematical results of independent interest about pfaffians of Dirac operators in 8k+3 dimensions, both on closed manifolds and manifolds with boundary. These theorems are used to set the quantum integrand of Mtheory for closed manifolds and for compact manifolds with either temporal (global) or spatial (local) boundary conditions. In particular, we show that Mtheory makes sense on arbitrary 11manifolds with spatial boundary, generalizing the construction of heterotic Mtheory on cylinders. The lowenergy approximation to Mtheory is a refinement of classical 11dimensional supergravity. It has a simple field content: a metric g, a 3form gauge potential C, and a gravitino. The Mtheory action contains rather subtle “ChernSimons ” terms which, on a topologically nontrivial manifold Y, raise delicate issues in the definition of the (exponentiated) action. Some aspects of the problem were resolved by Witten [W1]. The key ingredients are: a quantization law for C
Categorical linear algebra — a setting for questions from physics and lowdimensional topology, Kansas State U. preprint
"... of finite dimensional vector spaces over a fixed field, leftexact functors, and natural transformations has structures closely mimicking those found in ordinary linear algebra. We examine these structures, the relation of this category to the 2categories and weak 2categories (née bicategories) st ..."
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Cited by 6 (0 self)
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of finite dimensional vector spaces over a fixed field, leftexact functors, and natural transformations has structures closely mimicking those found in ordinary linear algebra. We examine these structures, the relation of this category to the 2categories and weak 2categories (née bicategories) studied by Kapranov and Voevodsky [KV]. In a final section, we give a notion of “matrix theory ” applicable to the 2categories considered, and extending to give a computationally convenient ncategorical setting for studying questions related to QFT’s (most especially topological QFT’s). This work was inspired by early drafts of Kapranov and Voevodsky’s manuscript, which has subsequently been entitled Braided Monoidal 2Categories, 2Vector Spaces and Zamolodchikov Tetrahedral Equations [KV]. Analogues of some of the results in it are now found in section V.3 of the most recent version of [KV] available to the author. The most important results however are new, specifically, those dealing with dual categories and homcategories and the reduction of theory of Vmodule functors and Vmodular transformations to exact functors and arbitrary natural transformations, as is the cleaner proof of the coherence properties of the braided monoidal 2category of Vmodules which is afforded by the use of a universal property satisfied by the Vtensor product. As with Kapranov and Voevodsky [KV] and Lawrence [Law], our purpose is to provide an algebraic footing for the extension to higher dimensions of the successful interaction between 3manifold topology, quantum field theory and monoidal category theory (cf. [A, C, FY, MS, RT1, RT2, TV, T, Y3]). Specifically, the algebraic structures describe herein provide the algebraic prerequisites for the study of quantum invariants of 2knots and the formulation of factorization at a corner for (3+1)dimensional topological quantum field theories. In a final section, we present a “matrix theory ” for categorical linear algebra, and note how it can be extended to ncategories, ntuple categories, and the nalgebras of Lawrence [Law].
Les Houches Lectures on Fields, Strings and Duality
 in A. Connes, K. Gawedzki and J. ZinnJustin (eds), Quantum symmetries, Elsevier, Amsterdam. hepth/9703136, Proceedings, NATO Advanced Study Institute, 64th Session, Les Houches
, 1998
"... Notes of my 14 ‘lectures on everything ’ given at the 1995 Les Houches school. An introductory course in topological and conformal field theory, strings, gauge fields, supersymmetry and more. The presentation is more mathematical then usual and takes a modern point of view stressing moduli spaces, d ..."
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Cited by 5 (0 self)
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Notes of my 14 ‘lectures on everything ’ given at the 1995 Les Houches school. An introductory course in topological and conformal field theory, strings, gauge fields, supersymmetry and more. The presentation is more mathematical then usual and takes a modern point of view stressing moduli spaces, duality and the interconnectedness of the subject. An apocryphal lecture on BPS states and Dbranes is added. Lectures given at the Les Houches Summer School on Theoretical Physics, Session LXIV: Quantum