I am trying to understand CFT from the viewpoint of both math(in particular using VOA) and physics. Now, in Math, we use the VOA to make sense of fields corresponding to certain states. We define for each state, $\lvert a \rangle $, in the Fock space, a field say $V(\lvert a \rangle, z )$ and we then expect these fields to be local, i.e.
\begin{equation}
(z-w)^k \ [ \ V(\lvert a \rangle, z), V(\lvert b \rangle, w ) \ ] = 0
\end{equation}
where they are treating $V(\lvert a \rangle, z)$ and $V(\lvert b \rangle, w)$ to be formal series and both the products, $V(\lvert a \rangle, z)V(\lvert b \rangle, w )$ and $V(\lvert b \rangle, w)V(\lvert a \rangle, z )$ are defined.
This is the standard treatment in texts such as
Vertex Algebras and Algebraic Curves by David Ben-Zvi and Edward Frenkel
Christoph A. Keller - Introduction to VOA - https://www.math.arizona.edu/~cakeller/VertexOperatorAlgebras.pdf \
However, in physics we treat $z$ in Riemann sphere and $V(\lvert a \rangle, z)V(\lvert b \rangle, w )$ is only defined when $\lvert z \lvert \ > \ \lvert w \lvert$. Further, by locality we mean that $$\langle c \lvert V(\lvert a \rangle, z)V(\lvert b \rangle, w )\lvert d \rangle = \langle c \lvert V(\lvert b \rangle, w)V(\lvert a \rangle, z ) \lvert d \rangle, \ \forall \ \lvert c \rangle, \lvert d \rangle \in \mathcal{F},$$ after analytic continuation of these functions into regions where they were not defined.
Such definitions can be found for example in https://arxiv.org/abs/hep-th/9410029.
Now, how are these two definitions of locality equivalent. Further, I haven't seen where exactly in Math references they start treating $z$ as complex variable.