Twisted associative algebras and intertwining operators

For a vertex algebra $V$ with a finite-order automorphism $g$ satisfying $g^T = 1$ for some $T \in \mathbb{N}$, we construct an associative algebra $\tilde{\mathbf{A}}^{g,\infty}(V)$ and prove that the category of $\frac{1}{T}\mathbb{N}$-graded $g$-twisted $\phi$-coordinated $V$-modules is isomorphic to the category of graded $\tilde{\mathbf{A}}^{g,\infty}(V)$-modules. Furthermore, when $V$ is a vertex operator algebra, we construct associative algebras $\mathbf{A}^{g,\infty}(V)$ and $A^{g,\infty}(V)$, and establish that the categories of admissible $g$-twisted $V$-modules and ordinary $g$-twisted $V$-modules are isomorphic to the categories of graded $\mathbf{A}^{g,\infty}(V)$-modules and graded $A^{g,\infty}(V)$-modules, respectively. By proving that $\tilde{\mathbf{A}}^{g,\infty}(V)$ is isomorphic to $\mathbf{A}^{g,\infty}(V)$, we obtain the equivalence between the category of $\frac{1}{T}\mathbb{N}$-graded $g$-twisted $\phi$-coordinated $V$-modules and the category of admissible $g$-twisted $V$-modules. Let $g_1, g_2, g_3$ be three commuting automorphisms of $V$ of finite order such that $g_1 g_2 = g_3$ and $g_i^T = 1$ for $i = 1, 2, 3$ and some $T \in \mathbb{N}$. Suppose that $W_i$ is a $g_i$-twisted $V$-module for each $i = 1, 2, 3$. We then construct an $A^{g_3,\infty}(V)$-$A^{g_2,\infty}(V)$-bimodule ${A}^{g_3,g_2,\infty}(W_1)$, and prove that the space of intertwining operators of type $\binom{W_3}{W_1 \; W_2}$ is isomorphic to $
\operatorname{Hom}_{A^{g_3,\infty}(V)}\!\left(
{A}^{g_3,g_2,\infty}(W_1) \otimes_{A^{g_2,\infty}(V)} W_2, \, W_3
\right). $