Hidden exceptional symmetry in the pure spinor superstring
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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 OG(+) (5,10). We find that the spectrum of the beta gamma system organizes into representations of the g = e(6) affine algebra at level -3, whose so(10)(-3) circle plus 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 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 g = so(8) and e(7), corresponding to target spaces that are cones over the complex Grassmannian Gr(2, 4) and the complex Cayley plane OP2. We identify these curved beta gamma systems with the chiral algebras of certain two-dimensional (2D) (0,2) conformal field theories arising from twisted compactification of 4D N = 2 superconformal field theories on S-2.