On world-sheet S-matrix of NSR string in static gauge

TL;DR

This paper derives the one-loop world-sheet S-matrix for the NSR string in static gauge, showing it matches the Green-Schwarz case, and discusses related models including heterotic strings and Tar T deformations.

Contribution

It provides a Nambu-like formulation of the NSR string with off-shell transverse fields after gauge fixing, extending previous work on superstring scattering amplitudes.

Findings

One-loop S-matrix for NSR string matches GS superstring results.

Elimination of auxiliary fields in static gauge is achieved.

Discusses extensions to heterotic string and Tar T deformations.

Abstract

As was shown in arXiv:1203.1054, expanding the Nambu action near the "long string" vacuum in the static gauge one finds that the one-loop 2 to 2 scattering amplitude of the $D-2=24$ transverse 2d fluctuations $X^i$ is given by a pure phase expression consistent with underlying integrability. Similar computation in the Green-Schwarz superstring was carried out in arXiv:2404.09658. Here we consider the case of the NSR string starting with its manifestly 2d covariant action given by $D$ scalar multiplets coupled to 2d supergravity. Eliminating the zweibein and gravitino fields is non-trivial, and a Nambu-like formulation of the spinning string involving only scalar coordinates and their 2d fermionic partners has not previously been available. We show how the auxiliary fields can be eliminated in static gauge after a specific choice of superconformal gauge for the fermions, while keeping the transverse fields $X^i$ and $\psi^i$ off-shell. The resulting one-loop S-matrix for $X^i$ is found, as expected, to be the same as in the GS superstring case. We also discuss similar actions for the heterotic string and a $T\bar T$ deformation of free 2d scalar multiplet.