Number (dimension of the space) of conformal blocks

Question

In 2d CFTs, we can study correlation functions of holomorphic primaries. For example, in section 8.3.3 of Di Francesco's book, the BPZ differential equation satisfied by $$ \langle \phi_{2,1}(z_0) \phi_1(z_1)\phi_2(z_2)\phi_3(z_3) \rangle $$ is studied. Due to the level-2 null of the primary $\phi_{2,1}$, the above correlation satisfies a 2nd order ODE, which can be solved (up to appropriate factors) by two linear independent hypergeometric functions.

More generally, $\langle \phi_{r,s}(z_0) \phi_1(z_1)\phi_2(z_2)\phi_3(z_3) \rangle$ will satisfy a $rs$-order ODE, giving $rs$ solutions.

However, I wonder what happens if we consider $\langle \phi_{2,1}(z_0) \phi_{r,s}(z_1)\phi_2(z_2)\phi_3(z_3) \rangle$? Does it satisfy the 2nd order or $rs$-order ODE, or both? How many independent solutions will there be?

(I would guess that there will be two independent solutions that simultaneously satisfy a 2nd and an $rs$-order equation, but I'm not entirely sure.)

And what happens if there are more than four operator insertions? What is the general idea to count number of solutions on a Riemann surface $\Sigma$? I have seen some formula/statement on "dimension of conformal blocks/characters", but I'm not sure if this is the same number that the above "BPZ equation approach" is computing.

• Verlinde formula that computes $n$-point function, as mentioned in one answer https://mathoverflow.net/questions/151221/verlindes-formula

• First do a pants decomposition of $\Sigma$, and each pants is treated like a fusion matrix $N$: the final number of blocks is given by suitable product of these $N$'s.

Answer 1

In your example with equations of orders $2$ and $rs$, yes, both equations should hold. This actually gives you constraints on the dimensions of the fields $\phi_2$ and $\phi_3$: for generic values you get zero solution, and for other values you may get one or at most two solutions.

The BPZ approach should work but it is overly complicated. To efficiently count blocks, your second solution is better: reduce the problem to a computation in terms of the fusion rules $N_{ab}^c$ i.e. the structure constants of the fusion ring.

BPZ equations and their solutions are in some sense a representation of the fusion ring in terms of more complicated objects i.e. functions of the positions $z_i$. Introducing complex variables $z_i$ for computing a few integer numbers is definitely not the most efficient approach.