I'm trying to understand CFT from a purely axiomatic-field-theoretical perspective. That is, there is a vector space $V$ associated to the circle, and an element of $V^{\otimes n}$ associated to every conformal manifold with $n$ boundary circles. This map is subject to the constraint that gluing two boundary circles of a manifold amounts to taking the trace of the two according $V$ components. By "conformal manifold", I mean an equivalence class of 2-manifolds with conformal structure, which is labelled by a finite set of real parameters for a fixed topology. Also, to make gluing unambiguous we need a base point on each boundary component.
What do all the conventional notions of CFT mean in this picture? It seems to me that the "vertex operator algebra", and the "operator product expansion" are very closely related to the "tensor" which the axiomatic CFT assigns to the pair of pants with different conformal parameters. I also understand that the conformal annuli form a $\mathbb R^+\times U(1)$ group under stacking ($\mathbb R^+$ for the with/radius ratio of the annulus, and $U(1)$ for the shift of the base point), and the two according generators are the Hamiltonian $H$ and momentum $p$ of the CFT. Is there any way to see that if $V$ is finite, the axiomatic CFT is actually an axiomatic TQFT? Also is there any way to derive that the spectrum of $H$ consists of these infinite towers? In other words, how do I arrive at the Virasoro algebra starting from the axiomatic CFT? Is there any example for which the axiomatic-CFT tensors assigned to different conformal manifolds have been worked out explicitly? Is there an explicit generators-and-relations presentation of the conformal cobordism category? Is it even true that all the properties of conventional CFT follow from only the axiomatic-CFT axioms? If this is not rigorously known yet, what do people believe is the case?
