Cofiniteness for Twisted Fusion Products in Vertex Operator Algebra Theory

TL;DR

This work extends fusion-product theory to twisted modules for a VOA $V$ with two commuting finite-order automorphisms by proving $C_1$-cofiniteness is preserved: if $W^1$ and $W^2$ are $C_1$-cofinite and there exists a surjective twisted logarithmic intertwining operator $inom{W^3}{W^1oldsymbol{W}^2}$, then $oxed{W^3}$ is $C_1$-cofinite. The key mechanism is a finite-dimensional solution space for a complex differential equation attached to the intertwining operators, yielding a uniform weight bound and hence cofiniteness. The authors then construct the fusion product $W^1oxtimes W^2$ as a concrete subquotient of a product of surjective intertwiners, prove its lower-truncation and universal property, and show that the fusion rules $N(W^1,W^2;W^3)$ are finite. In a special case with finitely many irreducible twisted modules, the fusion product decomposes semisimply as a finite direct sum of irreducibles with multiplicities given by the fusion rules, aligning with the modular-tensor-category framework for twisted VOAs.

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.

Theorems & Definitions (50)

• Theorem A
• Theorem B
• Theorem C
• Lemma 2.1
• Lemma 2.2
• proof
• Definition 3.1
• Remark 3.2
• Definition 3.3
• Definition 3.4