On q-analog of McKay correspondence and ADE classification of sl^(2) conformal field theories

TL;DR

The paper develops a q-analogue of the McKay correspondence by studying commutative associative algebras in a modular tensor category $\mathcal{C}$ coming from $U_q(\mathfrak{sl}_2)$ at a root of unity. It shows that such algebras correspond to module categories and are classified by finite Dynkin diagrams of types $A_n$, $D_{2n}$, $E_6$, $E_8$ with Coxeter number equal to $l$, linking the fusion with the fundamental $2$-dimensional representation to the Cartan data via $2-A$. The authors connect this framework to vertex operator algebras and conformal field theory, demonstrating that the resulting categories $\mathrm{Rep}^0 A$ are modular and yield modular invariants matching the ADE classification, with explicit constructions for $A_n$, $D_{2n}$, $E_6$, and $E_8$ through conformal embeddings. The work provides a self-contained, category-theoretic route to the ADE structure, aligning with, yet distinct from, subfactor and VOA approaches, and it culminates in explicit fusion rules for the $D_{2n}$ case. This q-analogue clarifies the relationship between quantum group representations, modular tensor categories, and conformal field theory through a concrete ADE correspondence.

Abstract

The goal of this paper is to classify ``finite subgroups in U_q sl(2)'' where $q=e^{πı/l}$ is a root of unity. We propose a definition of such a subgroup in terms of the category of representations of U_q sl(2); we show that this definition is a natural generalization of the notion of a subgroup in a reductive group, and that it is also related with extensions of the chiral (vertex operator) algebra corresponding to sl^(2) at level k=l-2. We show that ``finite subgroups in U_q sl(2)'' are classified by Dynkin diagrams of types A_n, D_{2n}, E_6, E_8 with Coxeter number equal to $l$, give a description of this correspondence similar to the classical McKay correspondence, and discuss relation with modular invariants in (sl(2))_k conformal field theory.

Figures (19)

• Figure 1: Morphisms $\mu$ and $\iota_A$
• Figure 2: Axioms of a commutative associative algebra.
• Figure 3: Definition of morphisms in $\mathop{\mathrm{Rep}}\nolimits A$
• Figure 4: Identifications $\mathop{\mathrm{Hom}}\nolimits_A(F(V), X)=\mathop{\mathrm{Hom}}\nolimits_\mathcal{C}(V, G(X))$. Here $\varphi\in\mathop{\mathrm{Hom}}\nolimits_A(F(V), X), \ \Phi\in\mathop{\mathrm{Hom}}\nolimits_\mathcal{C}(V, G(X))$.
• Figure 5:

Theorems & Definitions (84)

• Theorem
• Definition 1.1
• Definition 1.2
• Remark 1.3
• Lemma 1.4
• proof
• Theorem 1.5
• proof
• Theorem 1.6
• proof