Compactification of the Heterotic Pure Spinor Superstring II

TL;DR

This paper investigates heterotic pure spinor superstring compactifications on Calabi-Yau orbifolds, identifying the necessary pure spinor components and the structure of the Hilbert space to correctly reproduce the spectrum in lower dimensions.

Contribution

It introduces a 'small' Hilbert space of untwisted variables and relates the cohomology of a reduced differential to the spectrum in compactified dimensions.

Findings

Correct spectrum is obtained with specific untwisted pure spinor components.

The small Hilbert space approach simplifies the description of untwisted sector states.

The mismatch in pure spinor components relates to the projective measure in harmonic superspace.

Abstract

We study compactifications of the heterotic pure spinor superstring to six and four dimensions focusing on two simple Calabi-Yau orbifolds. We show that the correct spectrum can be reproduced only if, in the twisted sector, there remain exactly 5 and 2 pure spinor components untwisted, respectively. This naturally defines a "small" Hilbert space of untwisted variables. We point out that the cohomology of the reduced differential on this small Hilbert space can be used to describe the states in the untwisted sector, provided certain auxiliary constraints are defined. In dimension six, the mismatch between the number of pure spinor components in the small Hilbert space and the number of components of a six-dimensional pure spinor is interpreted as providing the projective measure on the analytic subspace (in the projective description) of harmonic superspace.