Let $(V,Y)$ be a vertex operator algebra, and $V'$ be the graded dual of its underlying vector space. The contragredient module structure on $V'$ is given by $Y'$ defined by the formula:


$$\langle Y'(v,x)w', w\rangle = \langle w', Y(e^{xL(1)}(-x^{-2})^{L(0)}v,x^{-1})w\rangle.$$


Now, I think of the LHS as putting $v$ at $x^{-1}$ on $\mathbb{CP}^1$ using the local coordinate system $z\mapsto 1/(z+x)$ with $w$ at $0$ and $w'$ at $\infty$. Further, I think of $$\langle w', Y(v,x^{-1})w\rangle$$ as putting $v$ at $x^{-1}$ using the local coordinate system $z\mapsto z-x^{-1}$. So something needs to be done in order to get the above equality. Can someone explain to me conceptually why this has to be putting $e^{xL(1)}(-x^{-2})^{L(0)}$ in front of $v$?



Intuition behind contragredient module of a VOA
 For the contragredient module vertex operator $\langle Y'(v,x)w',w\rangle$, I believe the local coordinate at the puncture $x^{-1}$ on $\mathbb{C}\mathbb{P}^1$ should actually be $z\rightarrow z^{-1}-x$. We can rewrite this coordinate as follows: $$ z\rightarrow z^{-1}-x = -x^2\cdot\frac{z-x^{-1}}{1+x(z-x^{-1})} = e^{-xw^2\frac{d}{dw}} (-x^2)^{w\frac{d}{dw}}\cdot w, $$ where $w=z-x^{-1}$. Now for vertex operator algebras, the Virasoro algebra is supposed to implement local coordinate changes, with the operator $L(n)$ corresponding to the infinitesimal local coordinate change $-w^{n+1}\frac{d}{dw}$. So if we want to change from the trivial coordinate $w$ to something like $\exp(-xw^2\frac{d}{dw}) (-x^2)^{w\frac{d}{dw}}\cdot w$, we should include the operator $e^{xL(1)}(-x^{-2})^{L(0)}$ in the vertex operator.


