Cofiniteness for Twisted Fusion Products in Vertex Operator Algebra Theory

TL;DR

This paper proves that under certain conditions, the $C_1$-cofiniteness property is preserved in twisted fusion products of vertex operator algebra modules, ensuring finite fusion rules and enabling fusion product construction.

Contribution

It establishes the preservation of $C_1$-cofiniteness in twisted fusion products of vertex operator algebra modules, a novel result in VOA theory.

Findings

$W^3$ is $C_1$-cofinite if $W^1$ and $W^2$ are, under certain conditions.

Finiteness of fusion rules is proven for $C_1$-cofinite modules.

Fusion products can be explicitly constructed under these conditions.

Abstract

Let $V$ be a vertex operator algebra equipped with two commuting finite-order automorphisms $g_1$ and $g_2$, and set $g_3 = g_1 g_2$. For $k = 1, 2, 3$, let $W^k$ be a $g_k$-twisted $V$-module. Assuming that $W^1$ and $W^2$ are $C_1$-cofinite and that there exists a surjective twisted logarithmic intertwining operator of type $\binom{W^3}{W^1 \ W^2}$, we prove that $W^3$ is also $C_1$-cofinite. The cofiniteness follows from the finite-dimensionality of the solution space of an associated complex-coefficient linear differential equation. As an application, under the condition of $C_1$-cofiniteness, we establish the finiteness of the fusion rules and construct the fusion product.