From the name "conformal blocks", one would guess that "The space of states are representations of X, which decompose into simple modules, and conformal blocks are those modules...". So my problem is what is X.
In the "Conformal Field Theory" by DiFrancesco et al's, Section 6.6.4, they claim that the correlation function on four points $0,1,\infty,x$, are in the form of $\sum_p \mathcal{F}_p(x) \bar{\mathcal{F}}_p(\bar x)$(e.g. formula 6.187), here $p$ is index for conformal families, which are from representation of Virasoro algebra.
And in other material (say Gui Bin's book https://binguimath.github.io/Files/2022_VOA_Lectures.pdf, Formula 1.14) the conformal blocks are decomposition of representation of vertex operator algebra (and its conjugation).
More interestingly, in A pedestrian explanation of conformal blocks the top two answers make use of these two algebras. The Virasoro algebra is a subalgebra of VOA, so Virasoro algebra would induce a finer decomposition. So why there are these two definitions, and are those equivalent in some sence?
