Permutation orbifolds

Abstract

A general theory of permutation orbifolds is developed for arbitrary twist groups. Explicit expressions for the number of primaries, the partition function, the genus one characters, the matrix elements of modular transformations and for fusion rule coefficients are presented, together with the relevant mathematical concepts, such as Lambda-matrices and twisted dimensions. The arithmetic restrictions implied by the theory for the allowed modular representations in CFT are discussed. The simplest nonabelian example with twist group S_3 is described to illustrate the general theory.Comment: 10 page

SNorbifolds and string interactions

“…This allows one to follow some analogies with the U (N ) group case, and we give a representation evaluating general characters (as polynomials of a basic set). Our representation is seen to agree (in explicit comparison) with the results of Bantay [29] who has given a beautiful mathematical procedure for orbifold characters through representations of the double. The procedure that we present will have some practical advantages, in particular we will use it to study the interacting features of primaries.…”

“…In this case, one may redefine θ by absorbing the conformal dimension of the primary into θ. And, this prescription is related to taking "regular part" of characters of the seed CFT in [29].…”

Solitons in Affine and Permutation Orbifolds

“…Note that the conjecture [τ ] = [1] at the end of previous section is true, then the above proposition is also true for odd n.…”

“…4.5 α δ is reducible, and [σ k δ] = [δ] for some 1 ≤ k ≤ n − 1 . By (1) of Lemma 7 we have S δfµ = S σ k (δ)fµ = G(σ k , f µ ) * S δfµ Since G(σ k , f µ ) = e …”

A geometric construction for permutation equivariant categories from modular functors

“…These categories [FRS03] enter in the construction of boundary conditions for tensor product theories that break permutation symmetries, so-called permutation branes [Rec03]. Moreover, they conveniently encode refined aspects of the family of representations of mapping class groups that is associated to the modular tensor category C. This explains the role of permutation orbifolds in Bantay's approach to the congruence subgroup conjecture ( [Ban02,Ban03]). (For a different proof of the congruence subgroup conjecture that is based on generalized FrobeniusSchur indicators, see [NS07].)…”