Twisted associative algebras and intertwining operators

Abstract

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 $\ddot{I}•$-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 $\ddot{I}•$-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 $\left(\genfrac{}{}{0ex}{}{W_3}{W_1{W}_2}\right)$ is isomorphic to ${\mathrm{Hom}}_{A^{g_3,\infty }(V)}\left(A^{g_3,g_2,\infty }(W_1){\otimes }_{A^{g_2,\infty }(V)}{W}_2,W_3\right).$

51 pages