The geometry of pure spinor space

TL;DR

This paper explores the complex geometric structure of ten-dimensional pure spinor space, deriving its Kähler metric and Laplacian, and discusses implications for Calabi-Yau properties and future research directions.

Contribution

It provides an explicit covariant metric and Laplacian for pure spinor space, clarifying its geometric structure beyond simple cone models.

Findings

Derived the Kähler structure and metric of pure spinor space.

Provided a covariant expression for the Laplacian.

Discussed potential extensions to eleven-dimensional theories.

Abstract

We investigate the complex geometry of D=10 pure spinor space. The K\"ahler structure and the corresponding metric giving rise to the desired Calabi-Yau property are determined, and an explicit covariant expression for the Laplacian is given. The metric is not that of a cone obtained by embedding pure spinor space in a flat space of unconstrained spinors. Some directions for future studies, concerning regularisation and generalisation to eleven dimensions, are briefly discussed.