# coset of affine Lie algebra

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)

## 1 Answer

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".