Towards a Classification of su(2)$\bigoplus\cdots\bigoplus$su(2) Modular Invariant Partition Functions

TL;DR

This paper advances the classification of modular invariant partition functions for high-rank semi-simple affine algebras, providing explicit results for certain cases and identifying new physical invariants.

Contribution

It explicitly classifies automorphism invariants for all levels of $(A_1^{(1)})^{igoplus r}$ and completes the classification for specific low-rank cases, also discovering new physical invariants.

Findings

Complete classification for $A_1^{(1)}igoplus A_1^{(1)}$ at all levels.

Explicit automorphism invariants for all levels of $(A_1^{(1)})^{igoplus r}$.

Discovery of new physical invariants.

Abstract

The complete classification of WZNW modular invariant partition functions is known for very few affine algebras and levels, the most significant being all levels of $A_1$ and $A_2$ and level 1 of all simple algebras. Here, we address the classification problem for the nicest high rank semi-simple affine algebras: $(A_1^{(1)})^{\oplus_r}$. Among other things, we explicitly find all automorphism invariants, for all levels $k=(k_1,\ldots,k_r)$, and complete the classification for $A_1^{(1)}\oplus A_1^{(1)}$, for all levels $k_1,k_2$. We also solve the classification problem for $(A_1^{(1)})^{\oplus_r}$, for any levels $k_i$ with the property that for $i\ne j$ each $gcd(k_i+2,k_j+2)\leq 3$. In addition, we find some physical invariants which seem to be new. Together with some recent work by Stanev, the classification for all $(A^{(1)}_1)^{\oplus_r}_k$ could now be within sight.