Algebraic formulation of duality transformations for abelian lattice models

Drühl, Kai J.; Wagner, Hermann‐Josef

doi:10.1016/0003-4916(82)90286-x

Cited by 53 publications

(49 citation statements)

References 15 publications

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“…It is known [26] that the duality symmetry of this model maps the free boundary condition to any of the three fixed boundary conditions, and indeed this is a specific case of the general duality [27] between free and 'configurational' boundary conditions of lattice spin models. Similarly, the new boundary condition discovered in [26] gets mapped to any of the three mixed boundary conditions.…”

Section: T-dualitymentioning

confidence: 91%

“…For everỹ r = 1, 2, ... , 2ℓ+2, the integer E(r) lies in the label set of the A 4ℓ+1 graph; indeed, the exponents correspond precisely to the black nodes of A 4ℓ+1 , with the middle node appearing twice. We can therefore define for every s = 1, 2, ... , 4ℓ+1 a matrix V s through 27) where S (A) and S (D) are the unitary diagonalizing matrices for the graphs A 4ℓ+1 and D 2ℓ+2 . Thus S (A) is nothing but the modular S-matrix of the sl(2) 4ℓ WZW model.…”

Section: Virasoro Minimal Modelsmentioning

confidence: 99%

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Symmetry breaking boundaries

Fuchs

Schweigert

2000

Nuclear Physics B

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Various structural properties of the space of symmetry breaking boundary conditions that preserve an orbifold subalgebra are established. To each such boundary condition we associate its automorphism type. We show that correlation functions in the presence of such boundary conditions are expressible in terms of twisted boundary blocks which obey twisted Ward identities. The subset of boundary conditions that share the same automorphism type is controlled by a classifying algebra, whose structure constants are shown to be traces on spaces of chiral blocks. T-duality on boundary conditions is not a one-to-one map in general. These structures are illustrated in a number of examples. Several applications, including the construction of non-BPS boundary conditions in string theory, are exhibited.

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Section: T-dualitymentioning

confidence: 91%

Section: Virasoro Minimal Modelsmentioning

confidence: 99%

Symmetry breaking boundaries

Fuchs

Schweigert

2000

Nuclear Physics B

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“…This "geometrical" duality of cellular complexes pictorially exchanges k-cells with (D − k)-cells, and the physical state is now a (D − k − 2)-cochain. This is the general strategy, explained in detail, for example, in the review part of [2], or in [3].…”

Section: Introduction Of the Problemmentioning

confidence: 99%

General duality for Abelian-group-valued statistical-mechanics models

Caracciolo

Sportiello

2004

J. Phys. A: Math. Gen.

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Abstract. We introduce a general class of statistical-mechanics models, taking values in an abelian group, which includes examples of both spin and gauge models, both ordered and disordered.The model is described by a set of "variables" and a set of "interactions". Each interaction is associated with a linear combination of variables; these are summarized in a matrix J. A Gibbs factor is associated to each variable (onebody term) and to each interaction.Then we introduce a duality transformation for systems in this class. The duality exchanges the abelian group with its dual, the Gibbs factors with their Fourier transforms, and the interactions with the variables. High (low) couplings in the interaction terms are mapped into low (high) couplings in the one-body terms. If the matrix J is interpreted as a vector representation of a matroid, duality exchanges the matroid with its dual.We discuss some physical examples. The main idea is to generalize known models up to eventually include randomness into the pattern of interaction. We introduce and study a random Gaussian Model, a random Potts-like model, and a random variant of discrete scalar QED.Although the classical procedure 'a la Kramers and Wannier does not extend in a natural way to such a wider class of systems, our weaker procedure applies to these models, too. We shortly describe the consequence of duality for each example.

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“…The second reason for rank loss in connection matrices is that if H has a proper automorphism (a permutation of the nodes that preserves both the nodeweights and edgeweights), then in formula (22), any two terms defined by a mapping φ : [k] → V (H) and φσ (σ ∈ Aut(H)) are equal, so the sum of all such terms is still rank 1. So the rank of M (hom(·, H), k) is at most the number of orbits of the automorphism group of H on ordered k-tuples of its nodes.…”

Section: The Exact Rank Of Connection Matrices For Homomorphismsmentioning

confidence: 99%

“…Using arguments related to duality transformations of models in statistical physics (see e.g. [22] and references therein) one can show [28] that this weighted graph H represents sflo in the sense that sflo(·) = hom(·, H). The condition on S that it is closed under inversion can be dropped if we use homomorphisms of directed graphs (Section 3.5).…”

Section: Edge Colorings and Homomorphismsmentioning

confidence: 99%

Counting Graph Homomorphisms

Borgs

Chayes

Lovász

et al.

Algorithms and Combinatorics

156

201

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Counting homomorphisms between graphs (often with weights) comes up in a wide variety of areas, including extremal graph theory, properties of graph products, partition functions in statistical physics and property testing of large graphs.In this paper we survey recent developments in the study of homomorphism numbers, including the characterization of the homomorphism numbers in terms of the semidefiniteness of "connection matrices", and some applications of this fact in extremal graph theory.We define a distance of two graphs in terms of similarity of their global structure, which also reflects the closeness of (appropriately scaled) homomorphism numbers into the two graphs. We use homomorphism numbers to define convergence of a sequence of graphs, and show that a graph sequence is convergent if and only if it is Cauchy in this distance. Every convergent graph sequence has a limit in the form of a symmetric measurable function in two variables. We use these notions of distance and graph limits to give a general theory for parameter testing.The convergence can also be characterized in terms of mappings of the graphs into fixed small graphs, which is strongly connected to important parameters like ground state energy in statistical physics, and to weighted maximum cut problems in computer science.

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Algebraic formulation of duality transformations for abelian lattice models

Cited by 53 publications

References 15 publications

Symmetry breaking boundaries

Symmetry breaking boundaries

General duality for Abelian-group-valued statistical-mechanics models

Counting Graph Homomorphisms

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