Associativity of fusion products of $C_1$-cofinite $\mathbb{N}$-gradable modules of vertex operator algebra

Abstract

We prove an associative law of the fusion products $⊠$ of $C_1$-cofinite $\mathbb{N}$-gradable modules for a vertex operator algebra $V$. To be more precise, for $C_1$-cofinite $\mathbb{N}$-gradable $V$-modules $A,B,C$ and their fusion products $(A⊠B,{\mathcal{Y}}^{AB})$, $((A⊠B)⊠C,{\mathcal{Y}}^{(AB)C})$, $(B⊠C,{\mathcal{Y}}^{BC})$, $(A⊠(B⊠C),{\mathcal{Y}}^{A(BC)})$ with logarithmic intertwining operators ${\mathcal{Y}}^{AB},\dots ,{\mathcal{Y}}^{A(BC)}$ satisfying the universal properties for $\mathbb{N}$-gradable modules, we prove that four-point correlation functions $⟨\theta ,{\mathcal{Y}}^{A(BC)}(v,x){\mathcal{Y}}^{BC}(u,y)w⟩$ and $⟨{\theta }^{\prime },{\mathcal{Y}}^{(AB)C}({\mathcal{Y}}^{AB}(v,x-y)u,y)w⟩$ are locally normally convergent over $\{(x,y)\in {\mathbb{C}}^2\mid 0<|x-y|<|y|<|x|\}$. We then take their respective principal branches $\tilde{F}(⟨\theta ,{\mathcal{Y}}^{A(BC)}(v,x){\mathcal{Y}}^{BC}(u,y)w⟩)$ and $\tilde{F}(⟨\theta ,{\mathcal{Y}}^{(AB)C}({\mathcal{Y}}^{AB)}(v,x-y)u,y)w⟩)$ on ${\cal D}^2\!=\!\{(x,y)\in {\mathbb C}^2 \mid 0\!<\!|x\!-\!y|\!<\!|y|\!<\!|x|, \mbox{ and } x,y,x\!-\!y\not\in {\mathbb R}^{\leq 0}\}$ and then show that there is an isomorphism ${\varphi }_{[AB]C}:(A⊠B)⊠C\to A⊠(B⊠C)$ such that $$\tilde{F}(⟨\theta ,{\mathcal{Y}}^{A(BC)}(v,x){\mathcal{Y}}^{BC}(u,y)w⟩)=\tilde{F}(⟨{\varphi }_{[AB]C}^{\ast }(\theta ),{\mathcal{Y}}^{(AB)C}({\mathcal{Y}}^{AB}(v,x-y)u,y)w)⟩$$ on ${\mathcal{D}}^2$ for $\theta \in (A⊠(B⊠C){)}^{\vee }$, $v\in A$, $u\in B$, and $w\in C$, where $W^{\vee }$ denotes the contragredient module of $W$ and ${\varphi }_{[AB]C}^{\ast }$ denotes the dual of ${\varphi }_{[AB]C}$. We also prove the pentagon identity.

26 pages, I corrected typos and simplified the arguments

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