coset of affine Lie algebra

Question

In many books about conformal field theory, when we talk about a coset $\mathfrak{g}_k/\mathfrak{h}_{k'}$, we would talk about how the modules of $\mathfrak{g}_k$ are decomposed into those of $\mathfrak{h}_{k'}$ tensoring those of $\mathfrak{g}_k/\mathfrak{h}_{k'}$, for example, for the vacuum module $$ \mathcal{R}_\text{vac}[\mathfrak{g}_k] = \mathcal{R}_\text{vac}[\mathfrak{g}_k/\mathfrak{h}_{k'}] \otimes \mathcal{R}_\text{vac}[\mathfrak{h}_{k'}] \oplus ... $$

I wonder if there is an "extra procedure" where we actually single out the vacuum representation $\mathcal{R}_\text{vac}[\mathfrak{g}_k/\mathfrak{h}_{k'}]$ from the sum, namely, a procedure where we actually get the vertex operator algebra $\mathfrak{g}_k/\mathfrak{h}_{k'}$ alone? (maybe it's as simple as focusing on states annihilated by some $\mathfrak{h}_{k'}$ generators, however I'm not entirely sure)

Answer 1

This is typically given by the commutant, or coset construction. You take the vector subspace of $\mathcal{R}_\text{vac}[\mathfrak{g}_k]$ spanned by vectors $v$ satisfying $Y(u,z)v \in \mathcal{R}_\text{vac}[\mathfrak{g}_k][[z]]$ for all $u \in \mathcal{R}_\text{vac}[\mathfrak{h}_{k'}]$. Equivalently, you take fields that have no singularities when applied to vectors in $\mathcal{R}_\text{vac}[\mathfrak{h}_{k'}]$. This subspace has a vertex operator algebra structure.

One possible reference is section 5.7.2 of Frenkel and Ben-Zvi's "Vertex algebras and algebraic curves".