Modular Invariant Partition Functions and Method of Shift Vector

TL;DR

This paper introduces a shift vector method to construct self-dual lattices and derive modular invariants for affine Lie algebras, providing new insights into Lie group classifications and affine algebra partition functions.

Contribution

The paper presents a novel shift vector approach to generate self-dual lattices and explicitly derive modular invariants for affine Lie algebras, including new proofs and classifications.

Findings

Constructed self-dual lattices of dimension (l,l)

Derived modular invariants for affine Lie algebras

Provided new proof for A-D-E classification of SU(2)_k

Abstract

Using shift vector method we obtain a large class of self-dual lattices of dimension $(l,l)$, which has a one to one correspondence with modular invariants of free bosonic theory compactified on co-root lattice of a rank $l$ Lie group. Then a large number of modular invariants of affine Lie algebras are derived explicitly. As two applications of this method, we give a direct derivation of $D$-series of $SU(N)$ and a new proof for the A-D-E classification of the $SU(2)_k$ partition functions.