[Paper Review] The Classification of SU(m)_{k} Automorphism Invariants

Terry Gannon|ArXiv.org|Aug 22, 1994

Algebraic structures and combinatorial models3 references3 citations

TL;DR

This paper classifies all automorphism invariants for the affine Lie algebra $\widehat{su}(m)_k$ by identifying permutations of level $k$ weights that commute with the modular $S$ and $T$ matrices. It proves that all such automorphisms are simple current automorphisms and their conjugates, extending previous classifications beyond $r=1,2$ and $k=1$, and provides a foundational step toward the full classification of $A_{r,k}^{(1)}$ modular invariants using a streamlined, conceptually simple method.

ABSTRACT

In this paper we find all permutations of the level k weights of the affine algebra A_r^{(1)} which commute with both its S and T modular matrices. We find that all of these are simple current automorphisms and their conjugations. Previously, the A_{r,k}^{(1)} automorphism invariants were known only for r=1,2 \forall k, and k=1 \forall r. This is a major step toward the full classification of all A_{r,k}^{(1)} modular invariants; the simplicity of this proof strongly suggests that the full classification should be accomplishable. In an appendix we collect some new results concerning the A_{r,k}^{(1)} fusion ring.

Motivation & Objective

• To extend the classification of $A_{r,k}^{(1)}$ modular invariants beyond previously known cases for $r=1,2$ and $k=1$.
• To identify all permutations of level $k$ weights of $\widehat{su}(m)_k$ that commute with the modular $S$ and $T$ matrices.
• To determine whether such automorphisms are systematically simple current automorphisms or their conjugates.
• To provide a conceptual and technically simple framework that suggests the full classification of $A_{r,k}^{(1)}$ modular invariants is attainable.

Proposed method

• The method involves analyzing the action of weight permutations on the level $k$ weights of the affine algebra $A_r^{(1)}$.
• It uses the condition that such permutations must commute with the modular transformation matrices $S$ and $T$ of the modular group $SL(2,\mathbb{Z})$.
• The analysis relies on the representation theory of affine Lie algebras and the structure of the modular $S$ and $T$ matrices.
• The proof leverages properties of the fusion ring of $A_{r,k}^{(1)}$, particularly focusing on the role of simple currents.
• It employs group-theoretic techniques to classify automorphisms that preserve modular invariance.
• The argument is supported by results in an appendix on the $A_{r,k}^{(1)}$ fusion ring, which underpins the classification.

Experimental results

Research questions

• RQ1 Which permutations of the level $k$ weights of $\widehat{su}(m)_k$ commute with both the $S$ and $T$ modular matrices?
• RQ2 Are all such automorphisms of $A_{r,k}^{(1)}$ modular invariants necessarily simple current automorphisms or their conjugates?
• RQ3 Can the classification of $A_{r,k}^{(1)}$ modular invariants be systematically extended beyond the known cases of $r=1,2$ and $k=1$?
• RQ4 What structural properties of the fusion ring of $A_{r,k}^{(1)}$ support the classification of its modular invariants?
• RQ5 Does the simplicity of the proof suggest a generalizable method for the full classification of $A_{r,k}^{(1)}$ modular invariants?

Key findings

• All automorphisms of the level $k$ weights of $\widehat{su}(m)_k$ that commute with the $S$ and $T$ matrices are shown to be simple current automorphisms or their conjugates.
• The classification extends beyond the previously known cases of $r=1,2$ for all $k$, and $k=1$ for all $r$, to general $m$ and $k$.
• The proof is conceptually simple and suggests that the full classification of $A_{r,k}^{(1)}$ modular invariants is achievable.
• The results are supported by new findings on the structure of the $A_{r,k}^{(1)}$ fusion ring, particularly regarding the role of simple currents.
• The method provides a strong foundation for the complete classification of modular invariants in the $A_{r,k}^{(1)}$ series.
• The work establishes a systematic and generalizable approach to classifying automorphism invariants in affine Lie algebras.