Decreasing filtrations, $C_{2}$-algebra and twisted modules
Shijie Cao, Jiancai Sun
TL;DR
This work develops a twisted Li filtration framework to relate $C_{2}$-cofiniteness and twisted modules for vertex algebras. By constructing decreasing sequences $E_{n}^{T}(V)$ and $E_{n}^{T}(W)$ and their associated graded structures, the authors show $\mathrm{gr}_{\mathcal{E}}^{T}(V)$ is a vertex Poisson algebra and $\mathrm{gr}_{\mathcal{E}}^{T}(W)$ is a twisted module over it; they also introduce $C_{n}^{T}(W)$ and connect it to $E_{n}^{T}(W)$. A key result is that $C_{2}$-cofiniteness of a twisted module $W$ implies $C_{n}$-cofiniteness for all $n\ge2$, extending finiteness properties from the untwisted to the twisted setting. Additionally, the filtration framework yields generating-subspace descriptions for $\tfrac{1}{T}\mathbb{N}$-graded twisted modules of lower truncated $\mathbb{Z}$-graded vertex algebras, clarifying how such modules are built from subspaces of $V$ and of the twisted module. Overall, the paper advances understanding of how $C_{2}$-theory interacts with twisted modules and provides tools for constructing and analyzing generating subspaces in orbifold-like contexts.
Abstract
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 $\mathrm{gr}_{\mathcal{E}}^{T}(W)$ is a twisted module of vertex Poisson algebra $\mathrm{gr}_{\mathcal{E}}^{T}(V)$. We introduce another decreasing sequence of subspace $\{C_{n}^{T}(W)\}_{n\in\mathbb{Z}_{\geq2}}$ and establish a connection between $\{E_{n}^{T}(W)\}_{n\in\mathbb{Z}}$ and $\{C_{n}^{T}(W)\}_{n\in\mathbb{Z}_{\geq2}}$. By utilizing the twisted module $\mathrm{gr}_{\mathcal{E}}^{T}(W)$ of vertex Poisson algebra $\mathrm{gr}_{\mathcal{E}}^{T}(V)$, we prove that for any twisted module $W$ of a vertex algebra $V$, $C_{2}$-cofiniteness implies $C_{n}$-cofiniteness for all $n\geq 2$. Furthermore, we employ $\mathrm{gr}_{\mathcal{E}}^{T}(W)$ to study generating subspaces of $\frac{1}{T}\mathbb{N}$-graded twisted modules of lower truncated $\mathbb{Z}$-graded vertex algebras.