Simple Currents, Modular Invariants and Fixed Points

Schellekens, A.N.; Yankielowicz, Shimon

doi:10.1142/s0217751x90001367

Cited by 168 publications

(341 citation statements)

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“…Simple current theory [10,11,6,4] shows that in the decomposition (7.6) isomorphicĀ-modules appear precisely as a consequence of fixed point resolution; therefore the multiplicity ofH Jλ in this decomposition is given by |U * λ |. On the other hand, as seen above, elements of the orbifold group G that are not in the stabilizer S λ relate isomorphicĀ-modules.…”

Section: (76)mentioning

confidence: 89%

“…(For a concrete realization of these maps in WZW theories, see [5].) In general, such an implementation Θ (λ) g maps a given space H λ to some other A-module H g ⋆ λ , thereby organizing the primary fields of the A-theory into orbits, much like the simple current group G ∼ = G * organizes [6] theĀ-primaries into orbits.…”

Section: The Action Of the Orbifold Group On A-modulesmentioning

confidence: 99%

“…As a consequence, they can be chosen independently. 6 Not to be confused with the global object for which the term chiral algebra has also been used in the recent mathematical literature [18], which we prefer to call a block algebra.…”

Section: T-dualitymentioning

confidence: 99%

“…(Examples for such automorphisms are those induced by simple currents, see e.g. [6,22].) Every element of Γ Z gives rise to a modular invariant torus partition function…”

Section: T-dualitymentioning

confidence: 99%

“…HereS is the S-matrix of the A k+1 -model, i.e.Sμ ,ν = 2/(k+2) sin((μ+1)(ν+1)π/(k+2)), and we introduced the number 6) which is nothing but the (one-by-one) matrix S J . In contrast, for the diagonalizing matrix o S of o C(Ā) there will typically not exist any preferred ordering of the labels α ∈ I • (for the rows of o S) and a ∈ I 1/2 (for the columns); amusingly, in the present case it is possible to order rows and columns in such a fashion that o S is symmetric.…”

Section: Sl(2) Wzw Theoriesmentioning

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: (76)mentioning

confidence: 89%

Section: The Action Of the Orbifold Group On A-modulesmentioning

confidence: 99%

Section: T-dualitymentioning

confidence: 99%

“…(Examples for such automorphisms are those induced by simple currents, see e.g. [6,22].) Every element of Γ Z gives rise to a modular invariant torus partition function…”

Section: T-dualitymentioning

confidence: 99%

Section: Sl(2) Wzw Theoriesmentioning

confidence: 99%

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Symmetry breaking boundaries

Fuchs

Schweigert

2000

Nuclear Physics B

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Fusion Rules in Conformal Field Theory

Fuchs

1994

Fortschr. Phys.

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Several aspects of fusion rings and fusion rule algebras, and of their manifestations in twodimensional (conformal) field theory, are described: diagonalization and the connection with modular invariance; the presentation in terms of quotients of polynomial rings; fusion graphs; various strategies that allow for a partial classification; and the role of the fusion rules in the conformal bootstrap programme. J. FUCHS, Fusion Rules -The coupling of primary fields of W-algebras in two-dimensional conformal field theory [9-111. (For a few further realizations see section 7 below.) If the axioms of a fusion rule algebra are slightly relaxed, one can also describe: -The multiplication of (classes of) polynomials in any quotient of a polynomial ring, e.g. the ring of chiral primary fields in N = 2 superconformal field theory [12,13]. -Operator products in topological field theory [14,15]. In the present paper the main interest is in the realization of fusion rules in conformal field theory, but to motivate the concept of fusion rings and fusion rule algebras it seems to me most convenient to start with the first example in the above list, i.e. with the decomposition of tensor products of finite-dimensional representations, respectively modules, of a reductive Lie algebra. Thus let g denote a simple Lie algebra (the generalization to arbitrary reductive Lie algebras will be immediate). Any finitedimensional module of g and any tensor product of such modules is fully reducible, i.e. the direct sum of irreducible modules. The finite-dimensional irreducible modules are highest weight modules labelled by a dominant integral highest weight A of g; I denote them by LA, and their Kronecker tensor product and its decomposition into irreducible modules by LA x LA. = @ NAA."" L&. A"(Here and below I use, for V a vector space and n E Z,,, the short-hand notation nV in place of @a=1 V @ ) with I / ( ' ) Let me recall a few well-known properties of such tensor products: (a) The addition 0 and product x are commutative, associative, and distributive.(b) By definition, the numbers N,,,"" are non-negative integers.(c) The number of highest weight modules LA is infinite; but for fixed A and A', NAAeA" is nonzero only for a finite number of highest weights A". (d) The highest weight module with highest weight A = 0 (the trivial one-dimensional module) acts as the identity, i.e. LA x Lo = LA, or in other words V.)for any dominant integral highest weight A . (e) To any module LA there exists a unique conjugate module L; which is again a finite-dimensional highest weight module (namely Lfi = LA+, with A' being minus the lowest weight of LA), such that (L;)' = LA, that Lo appears in LA * L; precisely once, i.e. and that the tensor product of conjugate modules is conjugate to the tensor product of the modules themselves, in the sense that (f) The trivial module Lo is self-conjugate. J. FUCHS, Fusion Rules Then commutativity means while associativity is expressed by (F2), J u^i j k N k F = 1 J j . , " 4 k m . ksl kei The positivity axiom (F3...

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Introduction to Conformal Field Theory

Schellekens

1996

Fortschr. Phys.

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Simple Currents, Modular Invariants and Fixed Points

Cited by 168 publications

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Symmetry breaking boundaries

Symmetry breaking boundaries

Fusion Rules in Conformal Field Theory

Introduction to Conformal Field Theory

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