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 $\beta \gamma$ system on the cone over the orthogonal Grassmannian ${\text{OG}}^{+}(5,10)$. We find that the spectrum of the $\beta \gamma$ 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 $\beta \gamma$ systems, which also includes the cases $\mathfrak{g}=\mathfrak{s}\mathfrak{o}(8)$ and ${\mathfrak{e}}_7$, corresponding to target spaces that are cones over the complex Grassmannian $\text{Gr}(2,4)$ and the complex Cayley plane ${\mathbb{O}\mathbb{P}}^2$. We identify these curved $\beta \gamma$ systems with the chiral algebras of certain $2d$ $(0,2)$ CFTs arising from twisted compactification of 4d $\mathcal{N}=2$ SCFTs on $S^2$.