UC Santa Cruz

Let $V$ be a vertex operator superalgebra and $\sigma$ the order $2$ automorphism \linebreak associated with the superstructure of $V$. For a finite order automorphism $g$ with $o(g\sigma)=T'$, we follow~\cite{DONG-LI-MASON--AGNV} to construct a sequence of associative algebras $\AGNV$ for $n\in \frac{1}{T'}\Z_+$ such that $A_{g,n-\frac{1}{T'}}(V)$ is a quotient of $\AGNV$. There is a bijection between the irreducible $\AGNV$-modules which cannot factor through $A_{g,n-\frac{1}{T'}}(V)$ and the irreducible admissible $g$-twisted $V$-modules. These results are then\newline applied to $g$-rational vertex operator superalgebras. In this case it is shown that $V$ is $g$-rational if and only if all the $\AGNV$ are finite-dimensional. Taking $n=0$ we obtain the associative algebra $\AGV$ constructed in~\cite{DONG-ZHAO--AGV}. With $g=1$ we recover $A_n(V)$ as in~\cite{JIANG-JIANG}.