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COMPUTATIONS OF TURAEVVIROOCNEANU INVARIANTS OF 3MANIFOLDS FROM SUBFACTORS
, 2008
"... In this paper, we establish a rigorous correspondence between the two tube algebras, that one comes from the TuraevViroOcneanu TQFT introduced by Ocneanu and another comes from the sector theory introduced by Izumi, and construct a canonical isomorphism between the centers of the two tube algebras ..."
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In this paper, we establish a rigorous correspondence between the two tube algebras, that one comes from the TuraevViroOcneanu TQFT introduced by Ocneanu and another comes from the sector theory introduced by Izumi, and construct a canonical isomorphism between the centers of the two tube algebras, which is a conjugate linear isomorphism preserving the products of the two algebras and commuting with the actions of SL(2, Z). Via this correspondence and the Dehn surgery formula, we compute TuraevViroOcneanu invariants from several subfactors for basic 3manifolds including lens spaces and Brieskorn 3manifolds by using Izumi’s data written in terms of sectors. 1.
On Σmodels and nonabelian differential cohomology
, 2008
"... A “Σmodel ” can be thought of as a quantum field theory (QFT) which is determined by pulling back nbundles with connection (aka (n−1)gerbes with connection, aka nonabelian differential cocycles) along all possible maps (the “fields”) from a “parameter space ” to the given base space. If formulate ..."
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A “Σmodel ” can be thought of as a quantum field theory (QFT) which is determined by pulling back nbundles with connection (aka (n−1)gerbes with connection, aka nonabelian differential cocycles) along all possible maps (the “fields”) from a “parameter space ” to the given base space. If formulated suitably, such Σmodels include gauge theories such as notably (higher) ChernSimons theory. If the resulting QFT is considered as an “extended ” QFT, it should itself be a nonabelian differential cocycle on parameter space whose parallel transport along pieces of parameter space encodes the QFT propagation and correlators. We are after a conception of nonabelian differential cocycles and their quantization which captures this. Our main motivation is the quantization of differential ChernSimons cocycles to extended ChernSimons QFT and its boundary conformal QFT, reproducing the cocycle structure implicit in [23]. • Classical – We conceive nonabelian differential cohomology in terms of cohomology with coefficients in ωcategoryvalued presheaves [48] of parallel transport ωfunctors from ωpaths to a given
FOUR DIMENSIONAL TOPOLOGICAL QUANTUM FIELD THEORY, HOPF CATEGORIES, AND THE CANONICAL BASES
, 2004
"... Abstract: We propose a new combinatorial method of constructing 4DTQFTs. The method uses a new type of algebraic structure called a Hopf category. We also outline the construction of a family of Hopf categories related to the quantum groups, using the canonical bases. ..."
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Abstract: We propose a new combinatorial method of constructing 4DTQFTs. The method uses a new type of algebraic structure called a Hopf category. We also outline the construction of a family of Hopf categories related to the quantum groups, using the canonical bases.
FOUR DIMENSIONAL TOPOLOGICAL QUANTUM FIELD THEORY, HOPF CATEGORIES, AND THE CANONICAL BASES
, 1994
"... Abstract: We propose a new combinatorial method of constructing 4DTQFTs. The method uses a new type of algebraic structure called a Hopf category. We also outline the construction of a family of Hopf categories related to the quantum groups, using the canonical bases. ..."
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Abstract: We propose a new combinatorial method of constructing 4DTQFTs. The method uses a new type of algebraic structure called a Hopf category. We also outline the construction of a family of Hopf categories related to the quantum groups, using the canonical bases.
REVISED VERSION ChernSimons Theory with Finite Gauge Group
, 1992
"... A typical course in quantum field theory begins with a thorough examination of a “toy model”, usually the φ 4 theory. Our purpose here is to provide a detailed description of a “toy model ” for topological quantum field theory, suitable for use as a foundation for more sophisticated developments. We ..."
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A typical course in quantum field theory begins with a thorough examination of a “toy model”, usually the φ 4 theory. Our purpose here is to provide a detailed description of a “toy model ” for topological quantum field theory, suitable for use as a foundation for more sophisticated developments. We carry through all the steps of the path integral quantization: start with a lagrangian, construct the classical action, construct a measure, and do the integral. When the gauge group is finite the “path integral ” reduces to a finite sum. This remark makes it clear that the analytical difficulties simplify enormously, and that there should be no essential problem in carrying out the process. Many interesting features remain, however. The algebraic and topological structure are essentially unchanged, and are much clearer when not overshadowed by the analysis. And even the analysis does not entirely disappear: the details of the construction of the state spaces requires a much more precise formulation of the classical theory than is usually given, and reveals some incompleteness in the understanding of the classical theory for continuous Lie groups [F1].
Comment.Math.Univ.Carolin.41,3(2000)437–444 437
"... Abstract. Anewalgebraicstructureonthe orbitsofdressingtransformationsofthe quasitriangularPoissonLiegroupsisprovided.Thisgivesthetopologicalinterpretation ofthelinkinvariantsassociatedwiththeWeinsteinXuclassicalsolutionsofthequantum YangBaxterequation.Someapplicationstothethreedimensionaltopologi ..."
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Abstract. Anewalgebraicstructureonthe orbitsofdressingtransformationsofthe quasitriangularPoissonLiegroupsisprovided.Thisgivesthetopologicalinterpretation ofthelinkinvariantsassociatedwiththeWeinsteinXuclassicalsolutionsofthequantum YangBaxterequation.Someapplicationstothethreedimensionaltopologicalquantum fieldtheoriesarediscussed.