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256
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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Cited by 61 (18 self)
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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
Interacting quantum observables
 of Lecture Notes in Computer Science
, 2008
"... Abstract. We formalise the constructive content of an essential feature of quantum mechanics: the interaction of complementary quantum observables, and information flow mediated by them. Using a general categorical formulation, we show that pairs of mutually unbiased quantum account on the quantum d ..."
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Cited by 56 (27 self)
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Abstract. We formalise the constructive content of an essential feature of quantum mechanics: the interaction of complementary quantum observables, and information flow mediated by them. Using a general categorical formulation, we show that pairs of mutually unbiased quantum account on the quantum data encoded in complex phases, and prove a normal form theorem for it. Together these enable us to describe all observables of finite dimensional Hilbert space quantum mechanics. The resulting equations suffice to perform computations with elementary quantum gates, translate between distinct quantum computational models, establish the equivalence of entangled quantum states, and simulate quantum algorithms such as the quantum Fourier transform. All these computations moreover happen within an intuitive diagrammatic calculus. 1
TFT CONSTRUCTION OF RCFT CORRELATORS V: PROOF OF MODULAR INVARIANCE AND FACTORISATION
, 2005
"... The correlators of twodimensional rational conformal field theories that are obtained in the TFT construction of [I, II, IV] are shown to be invariant under the action of the relative modular group and to obey bulk and boundary factorisation constraints. We present results both for conformal field ..."
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Cited by 55 (28 self)
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The correlators of twodimensional rational conformal field theories that are obtained in the TFT construction of [I, II, IV] are shown to be invariant under the action of the relative modular group and to obey bulk and boundary factorisation constraints. We present results both for conformal field theories defined on oriented surfaces and for theories defined on unoriented surfaces. In the latter case, in particular the socalled cross cap constraint is included.
Recursion from Cyclic Sharing: Traced Monoidal Categories and Models of Cyclic Lambda Calculi
, 1997
"... . Cyclic sharing (cyclic graph rewriting) has been used as a practical technique for implementing recursive computation efficiently. To capture its semantic nature, we introduce categorical models for lambda calculi with cyclic sharing (cyclic lambda graphs), using notions of computation by Moggi / ..."
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Cited by 52 (5 self)
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. Cyclic sharing (cyclic graph rewriting) has been used as a practical technique for implementing recursive computation efficiently. To capture its semantic nature, we introduce categorical models for lambda calculi with cyclic sharing (cyclic lambda graphs), using notions of computation by Moggi / Power and Robinson and traced monoidal categories by Joyal, Street and Verity. The former is used for representing the notion of sharing, whereas the latter for cyclic data structures. Our new models provide a semantic framework for understanding recursion created from cyclic sharing, which includes traditional models for recursion created from fixed points as special cases. Our cyclic lambda calculus serves as a uniform language for this wider range of models of recursive computation. 1 Introduction One of the traditional methods of interpreting a recursive program in a semantic domain is to use the least fixedpoint of continuous functions. However, in the real implementations of program...
Monads on Tensor Categories
 J. Pure Appl. Algebra
, 2002
"... this paper we will discuss the combination of two classical notions of category theory, both treated extensively in [CWM]. One of these is the notion of a monad or triple on a category, which goes back to Godement [G] and was rst developed by Eilenberg, Moore, Beck and others. The other is that of a ..."
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Cited by 51 (1 self)
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this paper we will discuss the combination of two classical notions of category theory, both treated extensively in [CWM]. One of these is the notion of a monad or triple on a category, which goes back to Godement [G] and was rst developed by Eilenberg, Moore, Beck and others. The other is that of a monoidal category or tensor category, which originates with Benabou [Be] and with Mac Lane's famous coherence theorem [MacL], and which pervades much of present day mathematics. For a monad S on a tensor category, there is a natural additional structure that one can impose, namely that of a comparison map S(X
Higher dimensional algebra V: 2groups
 Theory Appl. Categ
"... A 2group is a ‘categorified ’ version of a group, in which the underlying set G has been replaced by a category and the multiplication map m: G×G → G has been replaced by a functor. Various versions of this notion have already been explored; our goal here is to provide a detailed introduction to tw ..."
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Cited by 50 (3 self)
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A 2group is a ‘categorified ’ version of a group, in which the underlying set G has been replaced by a category and the multiplication map m: G×G → G has been replaced by a functor. Various versions of this notion have already been explored; our goal here is to provide a detailed introduction to two, which we call ‘weak ’ and ‘coherent ’ 2groups. A weak 2group is a weak monoidal category in which every morphism has an inverse and every object x has a ‘weak inverse’: an object y such that x ⊗ y ∼ = 1 ∼ = y ⊗ x. A coherent 2group is a weak 2group in which every object x is equipped with a specified weak inverse ¯x and isomorphisms ix: 1 → x ⊗ ¯x, ex: ¯x ⊗ x → 1 forming an adjunction. We describe 2categories of weak and coherent 2groups and an ‘improvement ’ 2functor that turns weak 2groups into coherent ones, and prove that this 2functor is a 2equivalence of 2categories. We internalize the concept of coherent 2group, which gives a quick way to define Lie 2groups. We give a tour of examples, including the ‘fundamental 2group ’ of a space and various Lie 2groups. We also explain how coherent 2groups can be classified in terms of 3rd cohomology classes in group cohomology. Finally, using this classification, we construct for any connected and simplyconnected compact simple Lie group G a family of 2groups G � ( � ∈ Z) having G as its group of objects and U(1) as the group of automorphisms of its identity object. These 2groups are built using Chern–Simons theory, and are closely related to the Lie 2algebras g � ( � ∈ R) described in a companion paper. 1 1
Spherical categories
 Adv. Math
, 1999
"... Abstract. This paper is a study of monoidal categories with duals where the tensor product need not be commutative. The motivating examples are categories of representations of Hopf algebras. We introduce the new notion of a spherical category. In the first section we prove a coherence theorem for a ..."
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Cited by 44 (6 self)
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Abstract. This paper is a study of monoidal categories with duals where the tensor product need not be commutative. The motivating examples are categories of representations of Hopf algebras. We introduce the new notion of a spherical category. In the first section we prove a coherence theorem for a monoidal category with duals following [MacLane 1963]. In the second section we give the definition of a spherical category, and construct a natural quotient which is also spherical. In the
Constructing the extended Haagerup planar algebra
, 2009
"... We construct a subfactor planar algebra, and as a corollary a subfactor, with the ‘extended Haagerup´ principal graph pair. This is the last open case from Haagerup’s 1993 list of potential principal graphs of subfactors with index in the range (4, 3 + √ 3). We prove that the subfactor planar algeb ..."
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Cited by 43 (17 self)
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We construct a subfactor planar algebra, and as a corollary a subfactor, with the ‘extended Haagerup´ principal graph pair. This is the last open case from Haagerup’s 1993 list of potential principal graphs of subfactors with index in the range (4, 3 + √ 3). We prove that the subfactor planar algebra with these principal graphs is unique. We give a skein theoretic description, and a description as a subalgebra generated by a certain element in the graph planar algebra of its principal graph. We give an explicit algorithm for evaluating closed diagrams using the skein theoretic description. This evaluation algorithm is unusual because intermediate steps may increase the number of generators in a diagram.
A new description of orthogonal bases
 Math. Structures in Comp. Sci
"... We show that an orthogonal basis for a finitedimensional Hilbert space can be equivalently characterised as a commutative †Frobenius monoid in the category FdHilb, which has finitedimensional Hilbert spaces as objects and continuous linear maps as morphisms, and tensor product for the monoidal st ..."
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Cited by 40 (23 self)
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We show that an orthogonal basis for a finitedimensional Hilbert space can be equivalently characterised as a commutative †Frobenius monoid in the category FdHilb, which has finitedimensional Hilbert spaces as objects and continuous linear maps as morphisms, and tensor product for the monoidal structure. The basis is normalised exactly when the corresponding commutative †Frobenius monoid is special. Hence orthogonal and orthonormal bases can be axiomatised in terms of composition of operations and tensor product only, without any explicit reference to the underlying vector spaces. This axiomatisation moreover admits an operational interpretation, as the comultiplication copies the basis vectors and the counit uniformly deletes them. That is, we rely on the distinct ability to clone and delete classical data as compared to quantum data to capture basis vectors. For this reason our result has important implications for categorical quantum mechanics. 1