Decreasing filtrations, $C_2$-algebra and twisted modules

We investigate a question posed by Gaberdiel and Gannon concerning the relationship between $C_2$-algebras and twisted modules. To each twisted module $W$ of a vertex algebra $V$, we first associate a decreasing sequence of subspaces $\{E_n^T(W){\}}_{n\in \mathbb{Z}}$ and demonstrate that the associated graded vector space ${\mathrm{g}\mathrm{r}}_{\mathcal{E}}^T(W)$ is a twisted module of vertex Poisson algebra ${\mathrm{g}\mathrm{r}}_{\mathcal{E}}^T(V)$. We introduce another decreasing sequence of subspace $\{C_n^T(W){\}}_{n\in {\mathbb{Z}}_{\ge 2}}$ and establish a connection between $\{E_n^T(W){\}}_{n\in \mathbb{Z}}$ and $\{C_n^T(W){\}}_{n\in {\mathbb{Z}}_{\ge 2}}$. By utilizing the twisted module ${\mathrm{g}\mathrm{r}}_{\mathcal{E}}^T(W)$ of vertex Poisson algebra ${\mathrm{g}\mathrm{r}}_{\mathcal{E}}^T(V)$, we prove that for any twisted module $W$ of a vertex algebra $V$, $C_2$-cofiniteness implies $C_n$-cofiniteness for all $n\ge 2$. Furthermore, we employ ${\mathrm{g}\mathrm{r}}_{\mathcal{E}}^T(W)$ to study generating subspaces of $\frac{1}{T}\mathbb{N}$-graded twisted modules of lower truncated $\mathbb{Z}$-graded vertex algebras.