The N = 1 Triplet Vertex Operator Superalgebras: Twisted Sector

TL;DR

This paper classifies twisted modules for the N=1 super triplet vertex operator superalgebra, determines their graded dimensions, and explores the structure of the space of characters, revealing a rich modular and algebraic framework.

Contribution

It provides a classification of irreducible twisted modules for the N=1 super triplet vertex operator superalgebra and analyzes their characters and modular properties.

Findings

Irreducible twisted modules are classified.

Graded dimensions of twisted modules are determined.

The space of characters has dimension (9m+3) and relates to the triplet algebra.

Abstract

We classify irreducible $\sigma$-twisted modules for the N=1 super triplet vertex operator superalgebra $\mathcal{SW}(m)$ introduced recently [Adamovic D., Milas A., Comm. Math. Phys., to appear, arXiv:0712.0379]. Irreducible graded dimensions of $\sigma$-twisted modules are also determined. These results, combined with our previous work in the untwisted case, show that the $SL(2,\mathbb{Z})$-closure of the space spanned by irreducible characters, irreducible supercharacters and $\sigma$-twisted irreducible characters is $(9m+3)$-dimensional. We present strong evidence that this is also the (full) space of generalized characters for $\mathcal{SW}(m)$. We are also able to relate irreducible $\mathcal{SW}(m)$ characters to characters for the triplet vertex algebra $\mathcal{W}(2m+1)$, studied in [Adamovic D., Milas A., Adv. Math. 217 (2008), 2664-2699, arXiv:0707.1857].