I would like to understand what correlation functions compute in Conformal Field Theory in mathematics. Let me begin with basic definitions. We define a free boson field $\phi(z)$ as a formal power series $$ \phi(z)=q+a_0\log(z)-\sum_{n\ne0}\frac{a_n}{n}z^{-n}, $$ where $q,a_n$ for $n\in \mathbb{Z}$ are operators satisfying relations $$ [a_n,q]=\delta_{n,0}, \ \ \ [a_m,a_n]=m\delta_{m+n,0}. $$ Here these operators acts on the $space$ generated by $vacuum$ vector $|0\rangle$ with properties $$ a_n |0\rangle=0\ (n\ge0), \ \ \langle0|q=\langle0|a_n=0\ (n<0), $$ where $\langle0|$ is the dual of the vacuum, i.e. $\langle0|0\rangle=1$. We then define the current $J(z)$ of $\phi(z)$ as a formal derivative $$ J(z)=\sum_{n\in \mathbb{Z}}a_nz^{-n-1}. $$

Here is my question. In CFT one is interested in $correlation$ $functions$ such as $$ \langle0|J(z_1)J(z_2)|0\rangle, \ \ \langle0|J(z_1)J(z_2)J(z_3)J(z_4)|0\rangle $$ What do they compute in this context?

In physics, correlation function like $\langle0|\phi(x)\phi(y)|0\rangle$ computes the possibility of the field $\phi(x)$ to become $\phi(y)$ etc where $x,y$ are coordinate in for example Minkowski space. How should one interprete the correlation function in our case?