The $N=1$ Triplet Vertex Operator Superalgebras: Twisted Sector

We classify irreducible $\sigma$-twisted modules for the $N=1$ super triplet vertex operator superalgebra $\mathcal{SW}(m)$ introduced recently [Adamović 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 [Adamović D., Milas A., Adv. Math. 217 (2008), 2664–2699, arXiv:0707.1857].