Results 1  10
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28
Two linear transformations each tridiagonal with respect to an eigenbasis of the other; comments on the split decomposition
, 2003
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Category theory for conformal boundary conditions
 FIELDS INST. COMMUN. AMER. MATH. SOC., PROVIDENCE, RI
, 2003
"... ... inherits various structures from C, provided that A is a Frobenius algebra with certain additional properties. As a byproduct we obtain results about the FrobeniusSchur indicator in sovereign tensor categories. A braiding on C is not needed, nor is semisimplicity. We apply our results to the d ..."
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Cited by 50 (14 self)
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... inherits various structures from C, provided that A is a Frobenius algebra with certain additional properties. As a byproduct we obtain results about the FrobeniusSchur indicator in sovereign tensor categories. A braiding on C is not needed, nor is semisimplicity. We apply our results to the description of boundary conditions in twodimensional conformal field theory and present illustrative examples. We show that when the module category is tensor, then it gives rise to a NIMrep of the fusion rules, and discuss a possible relation with the representation theory of vertex operator algebras.
Classification of local conformal nets. Case c < 1
"... We completely classify diffeomorphism covariant local nets of von Neumann algebras on the circle with central charge c less than 1. The irreducible ones are in bijective correspondence with the pairs of AD2nE6,8 Dynkin diagrams such that the difference of their Coxeter numbers is equal to 1. We f ..."
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Cited by 27 (13 self)
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We completely classify diffeomorphism covariant local nets of von Neumann algebras on the circle with central charge c less than 1. The irreducible ones are in bijective correspondence with the pairs of AD2nE6,8 Dynkin diagrams such that the difference of their Coxeter numbers is equal to 1. We first identify the nets generated by irreducible representations of the Virasoro algebra for c<1 with certain coset nets. Then, by using the classification of modular invariants for the minimal models by CappelliItzyksonZuber and the method of αinduction in subfactor theory, we classify all local irreducible extensions of the Virasoro nets for c<1 and infer our main classification result. As an application, we identify in our classification list certain concrete coset nets studied in the literature.
Boundary conformal field theory and fusion ring representations
"... To an RCFT corresponds two combinatorial structures: the 1loop partition function of a closed string (the amplitude of a torus, sometimes called a modular invariant), and the 1loop partition function of an open string (a representation of the fusion ring called a NIMrep or equivalently a fusion g ..."
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Cited by 26 (2 self)
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To an RCFT corresponds two combinatorial structures: the 1loop partition function of a closed string (the amplitude of a torus, sometimes called a modular invariant), and the 1loop partition function of an open string (a representation of the fusion ring called a NIMrep or equivalently a fusion graph). In this paper we develop some basic theory of NIMreps, obtain several new NIMrep classifications, and compare them with the corresponding modular invariant classifications. Among other things, we make the following fairly disturbing observation: there are infinitely many (WZW) modular invariants which do not correspond to any NIMrep. The resolution could be that those modular invariants are physically sick. Is classifying modular invariants really the right thing to do? For current algebras, the answer seems to be: Usually but not always. For finite groups à la DijkgraafVafaVerlindeVerlinde, the answer seems to be: Rarely. 1.
The charges of a twisted brane
, 2003
"... The charges of the twisted Dbranes of certain WZW models are determined. The twisted Dbranes are labelled by twisted representations of the affine algebra, and their charge is simply the ground state multiplicity of the twisted representation. It is shown that the resulting charge group is isomorp ..."
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Cited by 19 (7 self)
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The charges of the twisted Dbranes of certain WZW models are determined. The twisted Dbranes are labelled by twisted representations of the affine algebra, and their charge is simply the ground state multiplicity of the twisted representation. It is shown that the resulting charge group is isomorphic to the charge group of the untwisted branes, as had been anticipated from a Ktheory calculation. Our arguments rely on a number of nontrivial Lie theoretic identities.
On exotic modular tensor categories
 Commun. Contemp. Math
"... Abstract. We classify all unitary modular tensor categories (UMTCs) of rank ≤ 4. There are a total of 35 UMTCs of rank ≤ 4 up to ribbon tensor equivalence. Since the distinction between the modular Smatrix S and −S has both topological and physical significance, so in our convention there are a tot ..."
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Cited by 13 (7 self)
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Abstract. We classify all unitary modular tensor categories (UMTCs) of rank ≤ 4. There are a total of 35 UMTCs of rank ≤ 4 up to ribbon tensor equivalence. Since the distinction between the modular Smatrix S and −S has both topological and physical significance, so in our convention there are a total of 70 UMTCs of rank ≤ 4. In particular, there are two trivial UMTCs with S = (±1). Each such UMTC can be obtained from 10 nontrivial prime UMTCs by direct product, and some symmetry operations. Explicit data of the 10 nontrivial prime UMTCs are given in Section 5. Relevance of UMTCs to topological quantum computation and various conjectures are given in Section 6. 1.
HIGHERLEVEL APPELL FUNCTIONS, MODULAR TRANSFORMATIONS, AND CHARACTERS
, 2004
"... We study modular transformation properties of a class of indefinite theta series involved in characters of infinitedimensional Lie superalgebras. The levelℓ Appell functions Kℓ satisfy open quasiperiodicity relations with additive thetafunction terms emerging in translating by the “period. ” Ge ..."
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Cited by 9 (0 self)
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We study modular transformation properties of a class of indefinite theta series involved in characters of infinitedimensional Lie superalgebras. The levelℓ Appell functions Kℓ satisfy open quasiperiodicity relations with additive thetafunction terms emerging in translating by the “period. ” Generalizing the wellknown interpretation of theta functions as sections of line bundles, the Kℓ function enters the construction of a section of a rank(ℓ + 1) bundle Vℓ,τ. We evaluate modular transformations of the Kℓ functions and construct the action of an SL(2, Z) subgroup that leaves the section of Vℓ,τ constructed from Kℓ invariant. Modular transformation properties of Kℓ are applied to the affine Lie superalgebra ̂sℓ(21) at rational level k> −1 and to the N = 2 superVirasoro algebra, to derive modular transformations of “admissible ” characters, which are not periodic under the spectral flow and cannot therefore be rationally expressed through theta functions. This gives an example where constructing a modular group action involves extensions among representations in a nonrational conformal model.
Classification of operator algebraic conformal field theories
 in “Advances in Quantum Dynamics”, Contemp. Math. 335
, 2003
"... We give an exposition on the current status of classification of operator algebraic conformal field theories. We explain roles of complete rationality and αinduction for nets of subfactors in such a classification and present the current classification result, a joint work with R. Longo, for the ca ..."
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Cited by 8 (2 self)
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We give an exposition on the current status of classification of operator algebraic conformal field theories. We explain roles of complete rationality and αinduction for nets of subfactors in such a classification and present the current classification result, a joint work with R. Longo, for the case with central charge less than 1, where we have a complete classification list consisting of the Virasoro nets, their simple current extensions of index 2, and four exceptionals. Two of the four exceptionals appear to be new. 1
Commutative association schemes
 European J. Combin
"... Abstract. Association schemes were originally introduced by Bose and his coworkers in the design of statistical experiments. Since that point of inception, the concept has proved useful in the study of group actions, in algebraic graph theory, in algebraic coding theory, and in areas as far afield ..."
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Cited by 6 (3 self)
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Abstract. Association schemes were originally introduced by Bose and his coworkers in the design of statistical experiments. Since that point of inception, the concept has proved useful in the study of group actions, in algebraic graph theory, in algebraic coding theory, and in areas as far afield as knot theory and numerical integration. This branch of the theory, viewed in this collection of surveys as the “commutative case, ” has seen significant activity in the last few decades. The goal of the present survey is to discuss the most important new developments in several directions, including Gelfand pairs, cometric association schemes, Delsarte Theory, spin models and the semidefinite programming technique. The narrative follows a thread through this list of topics, this being the contrast between combinatorial symmetry and grouptheoretic symmetry, culminating in Schrijver’s SDP bound for binary codes (based on group actions) and its connection to the Terwilliger algebra (based on combinatorial symmetry). We propose this new role of the Terwilliger algebra in Delsarte Theory as a central topic for future work. 1.
Algorithms for affine KacMoody algebras
, 1989
"... Abstract. Weyl groups are ubiquitous, and efficient algorithms for them — especially for the exceptional algebras — are clearly desirable. In this letter we provide several of these, addressing practical concerns arising naturally for instance in computational aspects of the study of affine algebras ..."
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Cited by 4 (1 self)
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Abstract. Weyl groups are ubiquitous, and efficient algorithms for them — especially for the exceptional algebras — are clearly desirable. In this letter we provide several of these, addressing practical concerns arising naturally for instance in computational aspects of the study of affine algebras or WessZuminoWitten (WZW) conformal field theories. We also discuss the efficiency and numerical accuracy of these algorithms.