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Duality and defects in rational conformal field theory
, 2006
"... We study topological defect lines in twodimensional rational conformal field theory. Continuous variation of the location of such a defect does not change the value of a correlator. Defects separating different phases of local CFTs with the same chiral symmetry are included in our discussion. We sh ..."
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We study topological defect lines in twodimensional rational conformal field theory. Continuous variation of the location of such a defect does not change the value of a correlator. Defects separating different phases of local CFTs with the same chiral symmetry are included in our discussion. We show how the resulting onedimensional phase boundaries can be used to extract symmetries and orderdisorder dualities of the CFT. The case of central charge c = 4/5, for which there are two inequivalent world sheet phases corresponding to the tetracritical Ising model and the critical threestates
A 2categories companion
"... Abstract. This paper is a rather informal guide to some of the basic theory of 2categories and bicategories, including notions of limit and colimit, 2dimensional universal algebra, formal category theory, and nerves of bicategories. 1. Overview and basic examples This paper is a rather informal gu ..."
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Cited by 4 (0 self)
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Abstract. This paper is a rather informal guide to some of the basic theory of 2categories and bicategories, including notions of limit and colimit, 2dimensional universal algebra, formal category theory, and nerves of bicategories. 1. Overview and basic examples This paper is a rather informal guide to some of the basic theory of 2categories and bicategories, including notions of limit and colimit, 2dimensional universal algebra, formal category theory, and nerves of bicategories. As is the way of these things, the choice of topics is somewhat personal. No attempt is made at either rigour or completeness. Nor is it completely introductory: you will not find a definition of bicategory; but then nor will you really need one to read it. In keeping with the philosophy of category theory, the morphisms between bicategories play more of a role than the bicategories themselves. 1.1. The key players. There are bicategories, 2categories, and Catcategories. The latter two are exactly the same (except that strictly speaking a Catcategory should have small homcategories, but that need not concern us here). The first two are nominally different — the 2categories are the strict bicategories, and not every bicategory is strict — but every bicategory is biequivalent to a strict one, and biequivalence is the right general notion of equivalence for bicategories and for 2categories. Nonetheless, the theories of bicategories, 2categories, and Catcategories have rather different flavours.
AQFT from nfunctorial QFT
"... There are essentially two different approaches to the axiomatization of quantum field theory (QFT): algebraic QFT, going back to Haag and Kastler, and functorial QFT, going back to Atiyah and Segal. More recently, based on ideas by Baez and Dolan, the latter is being refined to “extended ” functoria ..."
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Cited by 3 (1 self)
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There are essentially two different approaches to the axiomatization of quantum field theory (QFT): algebraic QFT, going back to Haag and Kastler, and functorial QFT, going back to Atiyah and Segal. More recently, based on ideas by Baez and Dolan, the latter is being refined to “extended ” functorial QFT by Freed, Hopkins, Lurie and others. The first approach uses local nets of operator algebras which assign to each patch an algebra “of observables”, the latter uses nfunctors which assign to each patch a “propagator of states”. In this note we present an observation about how these two axiom systems are naturally related: we demonstrate under mild assumptions that every 2dimensional extended Minkowskian QFT 2functor (”parallel surface transport”) naturally yields a local net. This is obtained by postcomposing the propagation 2functor with an operation that mimics the passage from the Schrödinger picture to the Heisenberg picture in quantum mechanics. The argument has a straightforward generalization to general pseudo
Rational CFT is parallel transport
"... From the data of any semisimple modular tensor category C the prescription [2] constructs a 3dimensional TFT by encoding 3manifolds in terms of string diagrams in C. From the additional data of a certain Frobenius algebra object internal to C, the presciption [18, 4] obtains (the combinatorial asp ..."
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Cited by 3 (2 self)
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From the data of any semisimple modular tensor category C the prescription [2] constructs a 3dimensional TFT by encoding 3manifolds in terms of string diagrams in C. From the additional data of a certain Frobenius algebra object internal to C, the presciption [18, 4] obtains (the combinatorial aspect of) the corresponding full boundary CFT by decorating triangulations of surfaces with objects and morphisms in C. We show that these decoration prescriptions are “quantum differential cocycles” on the worldvolume for a 3functorial extended QFT. The boundary CFT arises from a morphism between two chiral copies of the (locally trivialized) TFT 3functor. The crucial observation is that all 3dimensional string diagrams in [18] are Poincarédual to cylinders in BBimod(C) which arise as components of a laxnatural transformation between two 3functors that factor through BBC ↩ → BBimod(C). This exhibits the “holographic ” relation between 3d TFT and 2d CFT as the homadjunction in 3Cat, which says that a transformation between two 3functors is itself, in components, a 2functor.
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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Cited by 2 (1 self)
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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
Contents
, 2006
"... Fuchs, Runkel, Schweigert et al. (“FRS”) have developed a detailed formalism for studying CFT in terms of (Wilson)graphs decorated in modular tensor categories. Here we discuss aspects of how the Poincaré dual of their prescription can be understood in terms of locally trivialized 2functors from s ..."
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Fuchs, Runkel, Schweigert et al. (“FRS”) have developed a detailed formalism for studying CFT in terms of (Wilson)graphs decorated in modular tensor categories. Here we discuss aspects of how the Poincaré dual of their prescription can be understood in terms of locally trivialized 2functors from surface elements to bimodules, similar to how Stolz and Teichner describe enriched elliptic objects. Even though this might begin to look like a paper, the following are
2VECTOR SPACES AND GROUPOIDS
, 810
"... Abstract. This paper describes a relationship between essentially finite groupoids and 2vector spaces. In particular, we show to construct 2vector spaces of Vectvalued presheaves on such groupoids. We define 2linear maps corresponding to functors between groupoids in both a covariant and contrav ..."
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Abstract. This paper describes a relationship between essentially finite groupoids and 2vector spaces. In particular, we show to construct 2vector spaces of Vectvalued presheaves on such groupoids. We define 2linear maps corresponding to functors between groupoids in both a covariant and contravariant way, which are ambidextrous adjoints. This is used to construct a representation— a weak functor—from Span(Gpd) (the bicategory of groupoids and spans of groupoids) into 2Vect. In this paper we prove this and give the construction in detail. It has applications in constructing quantum field theories, among others. 1.
unknown title
, 2006
"... We study topological defect lines in twodimensional rational conformal field theory. Continuous variation of the location of such a defect does not change the value of a correlator. Defects separating different phases of local CFTs with the same chiral symmetry are included in our discussion. We sh ..."
Abstract
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We study topological defect lines in twodimensional rational conformal field theory. Continuous variation of the location of such a defect does not change the value of a correlator. Defects separating different phases of local CFTs with the same chiral symmetry are included in our discussion. We show how the resulting onedimensional phase boundaries can be used to extract symmetries and orderdisorder dualities of the CFT. The case of central charge c = 4/5, for which there are two inequivalent world sheet phases corresponding to the tetracritical Ising model and the critical threestates Potts model, is treated as an illustrative example. Contents