The Low Level Modular Invariant Partition Functions of Rank-Two Algebras

TL;DR

This paper systematically finds all modular invariant partition functions for certain rank-two affine algebras using a complete lattice method, filling gaps in previous classifications and computing associated algebraic dimensions.

Contribution

It introduces a complete lattice-based method to identify all physical invariants for specific affine algebras, extending previous partial classifications.

Findings

All invariants for A2 at levels ≤ 32 identified

All invariants for G2 at levels ≤ 31 identified

Dimensions of Weyl-folded commutants computed

Abstract

Using the self-dual lattice method, we make a systematic search for modular invariant partition functions of the affine algebras $g\*{(1)}$ of $g=A_2$, $A_1+A_1$, $G_2$, and $C_2$. Unlike previous computer searches, this method is necessarily complete. We succeed in finding all physical invariants for $A_2$ at levels $\le 32$, for $G_2$ at levels $\le 31$, for $C_2$ at levels $\le 26$, and for $A_1+A_1$ at levels $k_1=k_2\le 21$. This work thus completes a recent $A_2$ classification proof, where the levels $k=3,5,6,9,12,15,21$ had been left out. We also compute the dimension of the (Weyl-folded) commutant for these algebras and levels.