  We analyze the structure of the infinite jet algebra, or arc algebra, associated to level one Feigin-Stoyanovsky's principal subspaces. For $A$-series, we show that their Hilbert series can be computed either using the quantum dilogarithm or as certain generating functions over finite-dimensional representations of $A$-type quivers. In particular, we obtain new fermionic character formulas for level one $A$-type principal subspaces, which implies that they are classically free. We also analyze the infinite jet algebra of the principal subspace of the affine vertex algebra $L_{1}(so(5))$ and of $L_{1}(so(8))$. We give new conjectural expressions for their characters and provide strong evidence that they are also classically free. 
Jet schemes, Quantum dilogarithm and Feigin-Stoyanovsky's principal   subspaces