The Covariant Superstring on K3

TL;DR

This paper extends the pure spinor formalism to compactification on a K3 surface, simplifying the cohomology calculation and connecting with the hybrid formalism through harmonic superspace techniques.

Contribution

It introduces a novel approach to compactify the pure spinor formalism on K3 using harmonic superspace and homological algebra, providing new tools for analyzing physical states.

Findings

Reduction of cohomology calculation to a small Hilbert space

Facilitation of physical state conditions via harmonic superspace

Establishment of contact with the hybrid formalism

Abstract

We compactify the pure spinor formalism on a K3 surface. The pure spinor splits into a six-dimensional pure spinor, a projective superspace harmonic, and 6 non-covariant variables. A homological algebra argument reduces the calculation of the cohomology of the Berkovits differential to a "small" Hilbert space which is the string-theoretic analogue of projective superspace. The description of the physical state conditions is facilitated by lifting to the full harmonic superspace, which is accomplished by the introduction of the missing harmonics as non-minimal variables. Finally, contact with the hybrid formalism is made by returning to the small Hilbert space and fermionizing the projective parameter.