Pure Spinor Partition Function and the Massive Superstring Spectrum

TL;DR

This paper computes the pure spinor partition function up to the fifth mass level in ten-dimensional superstring theory, revealing fermionic states and clarifying the Hilbert space structure.

Contribution

It provides an explicit calculation of the pure spinor partition function including spin dependence, and clarifies the nature of fermionic states in the bosonic pure spinor formalism.

Findings

Reproduces the massive superstring spectrum from the partition function.

Identifies fermionic states arising from non-globally defined functions in pure spinor space.

Clarifies the proper Hilbert space definition for pure spinor variables.

Abstract

We explicitly compute up to the fifth mass-level the partition function of ten-dimensional pure spinor worldsheet variables including the spin dependence. After adding the contribution from the (x^{\mu}, \theta^{\alpha}, p_{\alpha}) matter variables, we reproduce the massive superstring spectrum. Even though pure spinor variables are bosonic, the pure spinor partition function contains fermionic states which first appear at the second mass-level. These fermionic states come from functions which are not globally defined in pure spinor space, and are related to the b ghost in the pure spinor formalism. This result clarifies the proper definition of the Hilbert space for pure spinor variables.