# Hidden exceptional symmetry in the pure spinor superstring

## TL;DR

This paper uncovers an unexpected exceptional symmetry, specifically an affine $ ext{E}_6$ algebra, in the pure spinor superstring's curved $eta ext{γ}$ system, revealing deeper algebraic structures and symmetries.

## Contribution

It demonstrates the organization of the pure spinor system into affine $ ext{E}_6$ representations and connects curved $eta ext{γ}$ systems to chiral algebras of 2D CFTs from 4D SCFTs.

## Key findings

- Spectrum organizes into $ ext{E}_6$ affine algebra representations.
- Partition function decomposes into $ ext{E}_6$ characters.
- Identifies a pattern of symmetry enhancement in curved $eta ext{γ}$ systems.

## Abstract

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 $\mathfrak{g}=\mathfrak{so}(8)$ 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 $\beta\gamma$ systems with the chiral algebras of certain $2d$ $(0,2)$ CFTs arising from twisted compactification of 4d $\mathcal{N}=2$ SCFTs on $S^2$.