Decreasing Filtrations, C₂-Algebra and Twisted Modules

We investigate a question posed by Gaberdiel and Gannon (2009) concerning the relationship between C₂-algebras and twisted modules. To each twisted module W of a vertex algebra V, we first associate a decreasing sequence of subspaces ₙT(W)\ₙ and demonstrate that the associated graded vector space grET(W) is a twisted module of vertex Poisson algebra grET(V). We introduce another decreasing sequence of subspaces ₙT(W)_n≥2 and establish a connection between ₙT(W)\ₙ and ₙT(W)_n≥2. By utilizing the twisted module grET(W) of vertex Poisson algebra grET(V), we prove that for any twisted module W of a vertex algebra V, C₂-cofiniteness implies Cₙ-cofiniteness for all n≥ 2. Furthermore, we employ grET(W) to study generating subspaces of 1/TN-graded twisted modules of lower truncated Z-graded vertex algebras.