Refining twisted bimodules associated to VOAs

Let $V$ be a vertex operator algebra and $g$ an automorphism of $V$ of finite order $T$. For any $m,n\in (1\mathrm{/}T)\mathbb{N}$, an $A_{g,n}(V)\text{\hspace{0.17em}⁣}-\text{\hspace{0.17em}⁣}{A}_{g,m}(V)$ bimodule $A_{g,n,m}(V)=V\mathrm{/}{O}_{g,n,m}(V)$ was defined by Dong and Jiang, where $O_{g,n,m}(V)$ is the sum of three certain subspaces $O_{g,n,m}^{\mathrm{\prime }}(V),O_{g,n,m}^{\mathrm{\prime }\mathrm{\prime }}(V)$ and $O_{g,n,m}^{\mathrm{\prime }\mathrm{\prime }\mathrm{\prime }}(V)$. In this paper, we show that $O_{g,n,m}(V)=O_{g,n,m}^{\mathrm{\prime }}(V)$.