Let $V$ be a nice vertex algebra, and $M_1, M_2, M_3, M_4, M_5, M_6$ be modules over $V$. Assume that I have intertwining operators $\mathcal{Y}^1\in \begin{pmatrix}M_4 \\M_1 M_5\end{pmatrix}$, $\mathcal{Y}^2\in \begin{pmatrix}M_5 \\M_2 M_3\end{pmatrix}$, $\mathcal{Y}^3\in \begin{pmatrix}M_4 \\M_2 M_6\end{pmatrix}$, $\mathcal{Y}^4\in \begin{pmatrix}M_6 \\M_1 M_3\end{pmatrix}$, such that for every $m_{(1)}\in M_1,m_{(2)}\in M_2,m_{(3)}\in M_3, m'_{(4)}\in M_4'$, and complex numbers $|z_2|> 1 > |z_1|>0$

$$\langle m'_{(4)},\mathcal{Y}^1(m_{(1)},1)\mathcal{Y}^2(m_{(2)},z_1)m_{(3)} \rangle\ \ (1)$$ $$\langle m'_{(4)},\mathcal{Y}^3(m_{(2)},z_2)\mathcal{Y}^4(m_{(1)},1)m_{(3)} \rangle\ \ (2)$$ both converge. Then, the composite $\mathcal{Y}^1\mathcal{Y}^2$ determines a map $$f:M_1\boxtimes_{P(1)}(M_2\boxtimes_{P(z_1)}M_3)\rightarrow M_4$$ in $Rep(V)$, and the composite $\mathcal{Y}^3\mathcal{Y}^4$ determines a map $$g:M_2\boxtimes_{P(z_2)}(M_1\boxtimes_{P(1)}M_3)\rightarrow M_4$$ in $Rep(V)$.

Now, let me fix $z_1 = 1/2$ and $z_2 = 3/2$. Further, in order to avoid having multivalued functions, all the expression I am going to consider next will be interpreted using the regular branch of the logarithm. Now assume that for every $m_{(1)}\in M_1,m_{(2)}\in M_2,m_{(3)}\in M_3, m'_{(4)}\in M_4'$, $(1)$ and $(2)$ (which are now single-valued!) are analytic continuation of one another along a path $\gamma : 1/2\rightarrow 3/2$ in $\mathbb{C}\backslash \{0,1\}$ in the upper half-plane. Can I conclude that $$f = g\circ c,$$

where $c$ is the map in Rep(V) given by the composite

$$\mathcal{A}:M_1\boxtimes_{P(1)}(M_2\boxtimes_{P(1/2)}M_3)\rightarrow (M_1\boxtimes_{P(1/2)}M_2)\boxtimes_{P(1/2)}M_3$$ $$\mathcal{R}\boxtimes Id:(M_1\boxtimes_{P(1/2)}M_2)\boxtimes_{P(1/2)}M_3\rightarrow (M_2\boxtimes_{P(1/2)}M_1)\boxtimes_{P(1/2)}M_3$$ $$\mathcal{A}^{-1}:(M_2\boxtimes_{P(1/2)}M_1)\boxtimes_{P(1/2)}M_3\rightarrow M_2\boxtimes_{P(1)}(M_1\boxtimes_{P(1/2)}M_3)$$

$$\mathcal{T}_{\gamma}\boxtimes Id:M_2\boxtimes_{P(1)}(M_1\boxtimes_{P(1/2)}M_3)\rightarrow M_2\boxtimes_{P(3/2)}(M_1\boxtimes_{P(1)}M_3)\ \ \ \textrm{(parallel transport along }\gamma )\textrm{?}$$

And if so, how?

$\DeclareMathOperator{\Id}{\mathrm{Id}}\DeclareMathOperator{\Rep}{\operatorname{Rep}}$Let $V$ be a nice vertex algebra, and $M_1, M_2, M_3, M_4, M_5, M_6$ be modules over $V$. Assume that I have intertwining operators $\mathcal{Y}^1\in \begin{pmatrix}M_4 \\M_1 M_5\end{pmatrix}$, $\mathcal{Y}^2\in \begin{pmatrix}M_5 \\M_2 M_3\end{pmatrix}$, $\mathcal{Y}^3\in \begin{pmatrix}M_4 \\M_2 M_6\end{pmatrix}$, $\mathcal{Y}^4\in \begin{pmatrix}M_6 \\M_1 M_3\end{pmatrix}$, such that for every $m_{(1)}\in M_1,m_{(2)}\in M_2,m_{(3)}\in M_3, m'_{(4)}\in M_4'$, and complex numbers $|z_2|> 1 > |z_1|>0$

$$\langle m'_{(4)},\mathcal{Y}^1(m_{(1)},1)\mathcal{Y}^2(m_{(2)},z_1)m_{(3)} \rangle\ \ (1)$$ $$\langle m'_{(4)},\mathcal{Y}^3(m_{(2)},z_2)\mathcal{Y}^4(m_{(1)},1)m_{(3)} \rangle\ \ (2)$$ both converge. Then, the composite $\mathcal{Y}^1\mathcal{Y}^2$ determines a map $$f:M_1\boxtimes_{P(1)}(M_2\boxtimes_{P(z_1)}M_3)\rightarrow M_4$$ in $\Rep(V)$, and the composite $\mathcal{Y}^3\mathcal{Y}^4$ determines a map $$g:M_2\boxtimes_{P(z_2)}(M_1\boxtimes_{P(1)}M_3)\rightarrow M_4$$ in $\Rep(V)$.

Now, let me fix $z_1 = 1/2$ and $z_2 = 3/2$. Further, in order to avoid having multivalued functions, all the expression I am going to consider next will be interpreted using the regular branch of the logarithm. Now assume that for every $m_{(1)}\in M_1,m_{(2)}\in M_2,m_{(3)}\in M_3, m'_{(4)}\in M_4'$, $(1)$ and $(2)$ (which are now single-valued!) are analytic continuation of one another along a path $\gamma : 1/2\rightarrow 3/2$ in $\mathbb{C}\backslash \{0,1\}$ in the upper half-plane. Can I conclude that $$f = g\circ c,$$

where $c$ is the map in Rep(V) given by the composite

$$\mathcal{A}:M_1\boxtimes_{P(1)}(M_2\boxtimes_{P(1/2)}M_3)\rightarrow (M_1\boxtimes_{P(1/2)}M_2)\boxtimes_{P(1/2)}M_3$$ $$\mathcal{R}\boxtimes \Id:(M_1\boxtimes_{P(1/2)}M_2)\boxtimes_{P(1/2)}M_3\rightarrow (M_2\boxtimes_{P(1/2)}M_1)\boxtimes_{P(1/2)}M_3$$ $$\mathcal{A}^{-1}:(M_2\boxtimes_{P(1/2)}M_1)\boxtimes_{P(1/2)}M_3\rightarrow M_2\boxtimes_{P(1)}(M_1\boxtimes_{P(1/2)}M_3)$$

$$\mathcal{T}_{\gamma}\boxtimes \Id:M_2\boxtimes_{P(1)}(M_1\boxtimes_{P(1/2)}M_3)\rightarrow M_2\boxtimes_{P(3/2)}(M_1\boxtimes_{P(1)}M_3)\ \ \ \textrm{(parallel transport along }\gamma )\textrm{?}$$

And if so, how?