Vertex algebraic construction of modules for twisted affine Lie algebras of type $A_{2l}^{(2)}$

TL;DR

This paper constructs modules for twisted affine Lie algebras of type $A_{2l}^{(2)}$ using vertex algebra techniques, providing bases, character formulas, and confirming a conjecture for vacuum modules.

Contribution

It introduces a vertex algebraic construction for modules of type $A_{2l}^{(2)}$, including bases and character formulas, and confirms a prior conjecture.

Findings

Constructed bases for standard modules, principal subspaces, and parafermionic spaces.

Derived explicit character formulas for these modules.

Confirmed the conjecture for vacuum modules from prior work.

Abstract

Let $\tilde{\mathfrak{g}}$ be the affine Lie algebra of type $A_{2l}^{(2)}$. The integrable highest weight $\tilde{\mathfrak{g}}$-module $L(k\Lambda_0)$ called the standard $\tilde{\mathfrak{g}}$-module is realized by a tensor product of the twisted module $V_L^T$ for the lattice vertex operator algebra $V_L$. By using such vertex algebraic construction, we construct bases of the standard module, its principal subspace and the parafermionic space. As a consequence, we obtain their character formulas and settle the conjecture for vacuum modules stated in arXiv:math/0102113.