WZW Commutants, Lattices, and Level 1 Partition Functions

TL;DR

This paper extends the understanding of modular invariant partition functions by analyzing the commutant of modular matrices for semi-simple Lie algebras, demonstrating that lattice-based methods can generate all such functions, and explicitly classifying level 1 cases.

Contribution

It generalizes the commutant analysis from $SU(N)$ to arbitrary semi-simple Lie algebras and proves the completeness of lattice methods in generating all partition functions.

Findings

The commutant of $S$ and $T$ matrices is spanned by even self-dual lattice partition functions.

All level 1 partition functions for $B_n$, $C_n$, $D_n$, and five exceptional algebras are explicitly classified.

Lattice methods are sufficient to generate all modular invariant partition functions.

Abstract

A natural first step in the classification of all `physical' modular invariant partition functions $\sum N_{LR}\,\c_L\,\C_R$ lies in understanding the commutant of the modular matrices $S$ and $T$. We begin this paper extending the work of Bauer and Itzykson on the commutant from the $SU(N)$ case they consider to the case where the underlying algebra is any semi-simple Lie algebra (and the levels are arbitrary). We then use this analysis to show that the partition functions associated with even self-dual lattices span the commutant. This proves that the lattice method due to Roberts and Terao, and Warner, will succeed in generating all partition functions. We then make some general remarks concerning certain properties of the coefficient matrices $N_{LR}$, and use those to explicitly find all level 1 partition functions corresponding to the algebras $B_n$, $C_n$, $D_n$, and the 5 exceptionals. Previously, only those associated to $A_n$ seemed to be generally known.