Associative algebras and intertwining operators

Abstract

Let $V$ be a vertex operator algebra and $A^{\infty }(V)$ and $A^N(V)$ for $N\in \mathbb{N}$ the associative algebras introduced by the author in [H5]. For a lower-bounded generalized $V$-module $W$, we give $W$ a structure of graded $A^{\infty }(V)$-module and we introduce an $A^{\infty }(V)$-bimodule $A^{\infty }(W)$ and an $A^N(V)$-bimodule $A^N(W)$. We prove that the space of (logarithmic) intertwining operators of type $\left(\genfrac{}{}{0ex}{}{W_3}{W_1{W}_2}\right)$ for lower-bounded generalized $V$-modules $W_1$, $W_2$ and $W_3$ is isomorphic to the space ${\mathrm{hom}}_{A^{\infty }(V)}(A^{\infty }(W_1){\otimes }_{A^{\infty }(V)}{W}_2,W_3)$. Assuming that $W_2$ and $W_3^{\prime }$ are equivalent to certain universal lower-bounded generalized $V$-modules generated by their $A^N(V)$-submodules consisting of elements of levels less than or equal to $N\in \mathbb{N}$, we also prove that the space of (logarithmic) intertwining operators of type $\left(\genfrac{}{}{0ex}{}{W_3}{W_1{W}_2}\right)$ is isomorphic to the space of ${\mathrm{hom}}_{A^N(V)}(A^N(W_1){\otimes }_{A^N(V)}{\Omega }_N^0(W_2),{\Omega }_N^0(W_3))$.

45 pages. One section reviewing the associative algebras $A^{\infty }(V)$ and $A^N(V)$ and graded $A^{\infty }(V)$- and $A^N(V)$-modules is added. Some misprints and typos are corrected. To appear in Comm. Math. Phys

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