Modular Categories and Orbifold Models

By Jr Alexander Kirillov. 2001

91 citations recorded by the source corpus.

Abstract: In this paper, we try to answer the following question: given a modular tensor category ? with an action of a compact group G, is it possible to describe in a suitable sense the “quotient” category ?/G? We give a full answer in the case when ?=?ℯ? is the category of vector spaces; in this case, ?ℯ?/G turns out to be the category of representation of Drinfeld's double D(G). This should be considered as the category theory analog of the topological identity {pt}/G=BG.This implies a conjecture of Dijkgraaf, Vafa, E. Verlinde and H. Verlinde regarding so-called orbifold conformal field theories: if ? is a vertex operator algebra which has a unique irreducible module, ? itself, and G is a compact group of automorphisms of ?, and some not too restrictive technical conditions are satisfied, then G is finite, and the category of representations of the algebra of invariants, ?G, is equivalent as a tensor category to the category of representations of Drinfeld's double D(G). We also get some partial results in the non-holomorphic case, i.e. when ? has more than one simple module.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/282ee617626e9e2a1b2f1f8bb08e1baaa8a0d833

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Citations in the corpus

WEAKLY GROUP-THEORETICAL AND SOLVABLE FUSION CATEGORIES (2008, background): “tor on C, hence a group-theoretical Hopf algebra H= H(G,K,L,ω,ψ,φ) with WEAKLY GROUP-THEORETICAL AND SOLVABLE FUSION CATEGORIES 9 Rep(H) = C. This Hopf algebra is an abelian extension C→ Fun(L) → H→ C[K] → C, and is sometimes called a Kac algebra. 3. ProofofTheorem 1.3 We start with the following ch”
Classification of irreducible modules for the vertex operator algebra VL+: general case (2002, methodology): “L/K such that χ(ι(κ)) = −1, and Tχ the irreducible ˆ L/ K -module with central character χ. Then V +”
Modular invariants for group-theoretical modular data. I (2009, background): “ons of) so-called quantum doubles of ﬁnite groups [12], also known as (untwisted) group-theoretical modular categories. Appearing in conformal ﬁeld theory as the modular data of holomorphic orbifolds [13, 29], the grouptheoretical modular data and corresponding modular invariants were studied extens”
Module categories over the Drinfeld double of a finite group (2002, unknown): “In this note we consider another class of examples, known in physics as holomorphic orbifold models, see e.g. [7, 12 ].”
On braided fusion categories I (2009, methodology): “tary points of view: one based on analysis of Tannakian subcategories and another based on group actions on braided fusion categories, see §4.2. In §4.4 we present a description (due to Kirillov. Jr. [Ki] and M¨uger [Mu3]) of the structure of braided fusion categories containing a subcategory Rep(G)”
Centers of graded fusion categories (2009, background): “One can check that the two constructions above (from braided fusion categories containing Rep(G) to braided G-crossed categories and vice versa) are inverses of each other; see [Kirillov 2002; Müger 2004; Drinfeld et al. 2009] for details.”