Hidden exceptional symmetry in the pure spinor superstring

The pure spinor formulation of superstring theory includes an interacting sector of central charge $c_{lambda}=22$, which can be realized as a curved $betagamma$ system on the cone over the orthogonal Grassmannian $text{OG}^{+}(5,10)$. We find that the spectrum of the $betagamma$ system organizes into representations of the $mathfrak{g}=mathfrak{e}_6$ affine algebra at level $-3$, whose $mathfrak{so}(10)_{-3}oplus {mathfrak u}(1)_{-4}$ subalgebra encodes the rotational and ghost symmetries of the system. As a consequence, the pure spinor partition function decomposes as a sum of affine $mathfrak{e}_6$ characters. We interpret this as an instance of a more general pattern of enhancements in curved $betagamma$ systems, which also includes the cases $mathfrak{g}=mathfrak{so}(8)$ and $mathfrak{e}_7$, corresponding to target spaces that are cones over the complex Grassmannian $text{Gr}(2,4)$ and the complex Cayley plane $mathbb{OP}^2$. We identify these curved $betagamma$ systems with the chiral algebras of certain $2d$ $(0,2)$ CFTs arising from twisted compactification of 4d $mathcal{N}=2$ SCFTs on $S^2$.