Constructing W-algebras

Question

I am following the algorithm in W-algebras with two and three generators, in order to construct consistent (anti-)commutator relations for a particular W-algebra.

I am considering $W(2,4,4)$ where both dimension four operators are fermionic. I have two questions related to the method introduced in the paper, namely:

1. They use Jacobi identities (e.g. $\{\Psi_m,\{\Psi_n,\Psi_\ell\}\} + \mathrm{permutations} = 0$) to fix some of the arbitrary constants, but this isn't sufficient to constrain all of them. How do they calculate the rest? (There is some step in the example involving computing determinants but it isn't clear how this relates to the notation of the previous section.)
2. For fermionic operators, when computing $\{\Psi_m,\{\Psi_n,\Psi_\ell\}\}$, due to the terms that appear in the inner anti-commutator, you encounter stuff like $\{\Psi_m, L_{n+\ell}\}$ but normally, we use a commutator in this case, as one element is even and one is odd. So when computing the Jacobi identity with fermionic operators, should one actually use $[,\}$ as opposed to $\{,\}$?

Answer 1

As it turns out, even though the authors of the paper I linked claim there are benefits to working with the commutators rather than the OPEs, I found using the OPEs much simpler, and didn't need the machinery provided in the algorithm in the paper.

In my case, the two fermionic generators carry an $\mathfrak{sl}(2)$ charge which we require the OPE to respect, and so I simply wrote down all possible terms that may arise order by order, i.e.

$$\Psi^+(z) \Psi^{-}(z) = \frac{a_1}{z^8}\mathbb{I} + \frac{a_2}{z^6}T + \frac{a_3}{z^5}\partial T + \frac{a_4}{z^4}\partial^2 T + \frac{a_5}{z^4}T^2 + \dots$$

using combinations of derivatives and $T$ with arbitrary constants $a_i$. Imposing the Jacobi results in some trivial constraints already satisfied (e.g. taking $TTT$) but others (e.g. $\Psi^+ \Psi^{-}T$) led to constraints on the $a_i$ up to null states.

A useful paper for carrying this out is An Algorithmic Approach to Operator Product Expansions, W-Algebras and W-Strings.