General minimal-model reducibility conjecture for tensor-product Virasoro modules

Let $c=c_{p,q} eq 0$ and let $L(c,h)$ be a minimal model. Let $V_{\alpha,\beta}^{\prime}$ be an irreducible module from the intermediate series, and let an admissible triple mean a triple $((m,n),(m^{\prime},n^{\prime}),(m^{\doubleprime},n^{\doubleprime}))$ satisfying the admissibility conditions for the minimal model. General minimal-model reducibility conjecture. The module $V_{\alpha,\beta}^{\prime}\otimes L(c,h)$ is reducible if and only if there exists an admissible triple such that

$$h=h_{m^{\prime},n^{\prime}},\qquad \alpha=h_{m,n}+h_{m^{\prime},n^{\prime}}-h_{m^{\prime\prime},n^{\prime\prime}},\qquad \beta=1-h_{m,n}.$$

In this case there is an irreducible submodule $U$ such that

$$\left(V_{\alpha,\beta}^{\prime}\otimes L(c,h)\right)/U\cong L(c,h_{m^{\prime\prime},n^{\prime\prime}}).$$

The conjecture extends the explicit reducibility and quotient results obtained for several minimal models, including the cases $c=-22/5$ and $c=1/2$; its validity for all nonzero minimal-model central charges remains open.