Solitons in Affine and Permutation Orbifolds

Kač, Victor G.; Longo, Roberto; Xu, Feng

doi:10.1007/s00220-004-1160-1

Cited by 38 publications

(72 citation statements)

References 48 publications

(135 reference statements)

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“…Clearly the above is a somewhat abstract result, and in concrete models hard work is required to determine the category G − Loc f A of twisted representations. (For a beautiful analysis of orbifolds of affine models in the present axiomatic setting see the series of papers [64,37,27].) However, Theorem 3.12 can be used to clarify the structure of Loc f A G quite completely in the holomorphic case, cf.…”

Section: E(ρ) Acts Trivially) This Expression Defines An Element Of Hommentioning

confidence: 99%

“…(As to the labelling of the irreducible sectors of the permutation orbifold stated in [1] without even a hint of proof, such a proof has recently been provided in [27].) ✷…”

Section: Remarkmentioning

confidence: 99%

“…In the operator algebraic setting, conformal orbifold models were considered in particular in [64,37,27]. In [64] it is shown that A G is completely rational if A is completely rational and G is finite, a result that we will use.…”

Section: Introductionmentioning

confidence: 99%

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Conformal Orbifold Theories and Braided Crossed G-Categories

Mueger

2005

Commun. Math. Phys.

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The aim of the paper is twofold. First, we show that a quantum field theory A living on the line and having a group G of inner symmetries gives rise to a category G − LocA of twisted representations. This category is a braided crossed G-category in the sense of Turaev [60]. Its degree zero subcategory is braided and equivalent to the usual representation category Rep A. Combining this with [29], where Rep A was proven to be modular for a nice class of rational conformal models, and with the construction of invariants of G-manifolds in [60], we obtain an equivariant version of the following chain of constructions: Rational CFT ❀ modular category ❀ 3-manifold invariant.Secondly, we study the relation between G−LocA and the braided (in the usual sense) representation category Rep A G of the orbifold theory A G . We prove the equivalence Rep A G ≃ (G−LocA) G , which is a rigorous implementation of the insight that one needs to take the twisted representations of A into account in order to determine Rep A G . In the opposite direction we have G− LocA ≃ Rep A G ⋊ S, where S ⊂ Rep A G is the full subcategory of representations of A G contained in the vacuum representation of A, and ⋊ refers to the Galois extensions of braided tensor categories of [44,48].Under the assumptions that A is completely rational and G is finite we prove that A has g-twisted representations for every g ∈ G and that the sum over the squared dimensions of the simple g-twisted representations for fixed g equals dim Rep A. In the holomorphic case (where Rep A ≃ Vect C ) this allows to classify the possible categories G−LocA and to clarify the rôle of the twisted quantum doubles D ω (G) in this context, as will be done in a sequel. We conclude with some remarks on non-holomorphic orbifolds and surprising counterexamples concerning permutation orbifolds. * Supported by NWO.

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Section: E(ρ) Acts Trivially) This Expression Defines An Element Of Hommentioning

confidence: 99%

“…(As to the labelling of the irreducible sectors of the permutation orbifold stated in [1] without even a hint of proof, such a proof has recently been provided in [27].) ✷…”

Section: Remarkmentioning

confidence: 99%

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Conformal Orbifold Theories and Braided Crossed G-Categories

Mueger

2005

Commun. Math. Phys.

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“…We have put our mathematical results in Appendix C, in order not to interrupt the main theme of the paper. A quick introduction to Operator Algebras and Conformal Field Theory can be found in [29].…”

Section: Introductionmentioning

confidence: 99%

Noncommutative Spectral Invariants and Black Hole Entropy

Kawahigashi

Longo

2005

Commun. Math. Phys.

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We consider an intrinsic entropy associated with a local conformal net A by the coefficients in the expansion of the logarithm of the trace of the "heat kernel" semigroup. In analogy with Weyl theorem on the asymptotic density distribution of the Laplacian eigenvalues, passing to a quantum system with infinitely many degrees of freedom, we regard these coefficients as noncommutative geometric invariants. Under a natural modularity assumption, the leading term of the entropy (noncommutative area) is proportional to the central charge c, the first order correction (noncommutative Euler characteristic) is proportional to log µ A , where µ A is the global index of A, and the second spectral invariant is again proportional to c.We give a further general method to define a mean entropy by considering conformal symmetries that preserve a discretization of S 1 and we get the same value proportional to c.We then make the corresponding analysis with the proper Hamiltonian associated to an interval. We find here, in complete generality, a proper mean entropy proportional to log µ A with a first order correction defined by means of the relative entropy associated with canonical states. * Supported in part by JSPS. † Supported in part by GNAMPA and MIUR. 1By considering a class of black holes with an associated conformal quantum field theory on the horizon, and relying on arguments in the literature, we indicate a possible way to link the noncommutative area with the BekensteinHawking classical area description of entropy.

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“…By using formulas for conformal dimensions in §3.1.2 and Lemma 2.4 it is easy to check that the following set (27) of [38]):…”

mentioning

confidence: 99%