Two point amplitude for closed superstrings

TL;DR

This paper develops a consistent method for computing tree-level two-point amplitudes in closed superstrings within the pure spinor formalism, overcoming previous challenges by using nonstandard ghost-number vertex operators.

Contribution

It introduces a novel prescription for closed superstring two-point amplitudes that replaces open-string BRST charges with closed-string ones and employs nonstandard ghost-number vertex operators.

Findings

Successfully computes closed-string two-point amplitudes at tree level.

Shows the prescription is essentially unique due to BRST cohomology constraints.

Extends the pure spinor formalism to closed strings with a new vertex operator approach.

Abstract

We present a prescription for computing the tree-level two-point amplitude of closed strings in the pure spinor superstring formalism, thereby completing the analysis of such superstring amplitudes. The construction relies on fixing the residual conformal Killing group using a mostly BRST-exact operator that has been successfully applied in the open-string case. Earlier attempts at a straightforward extension to closed strings--treating them na\"ively as products of open strings--fail. Nevertheless, we show that a consistent prescription can be obtained by replacing the open-string BRST charge with the closed-string BRST charge. The key idea is to employ closed-string vertex operators with nonstandard ghost-number assignments, rather than the conventional ghost-number (1,1) vertices. Furthermore, since the pure spinor BRST cohomology for closed strings vanishes at total (left plus right) ghost number four or higher, we find that the resulting prescription is essentially unique.