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 $S^2$.
Highlights
The pure spinor formalism [1] is a reformulation of superstring theory which has the virtue that it can be quantized while preserving manifest covariance with respect to ten-dimensional super-Poincaresymmetry
We will find that the partition function can be expressed as a linear combination of two gk characters. These results suggest that the chiral algebra of ð0;2Þ g receives two types of contributions: one from states arising from the reduction of the 4D N 1⁄4 2 chiral algebra and one capturing contributions from a surface defect of the 4D SCFT T g that is wrapped along S2
We have found that the states in the βγ system with target Xg organize into a direct sum of irreducible modules of an affine gsymmetry algebra
Summary
The pure spinor formalism [1] is a reformulation of superstring theory which has the virtue that it can be quantized while preserving manifest covariance with respect to ten-dimensional super-Poincaresymmetry. We will argue that the pure spinor ghost sector possesses a hidden affine e6 symmetry at level −3, albeit with a choice of stress tensor different from the Sugawara one. The existence of an e6 finite-dimensional Lie algebra action on the zero modes of the pure spinor ghost sector was first observed in [6,7]. We will find it enlightening to consider a more general family of βγ systems whose target spaces, Xg, have enlarged symmetry g 1⁄4 d4ð1⁄4 soð8ÞÞ; e6, and e7. An amusing consequence is that the fields, ghosts, antifields, and antighosts of ten-dimensional supersymmetric Yang-Mills theory organize into the 27 and 27 of e6
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