The Mostly BRST Exact Operator in Superstrings

TL;DR

This paper investigates the use of the mostly BRST (mBRST) exact operator for fixing conformal Killing groups in superstring theory, demonstrating its effectiveness in calculating tree-level amplitudes and deriving it via Faddeev-Popov gauge fixing.

Contribution

It extends the application of the mBRST exact operator to general tree-level superstring amplitudes and provides a Faddeev-Popov derivation independent of external string number.

Findings

Successfully re-calculated the three-gluon amplitude using the operator.

Demonstrated the operator's emergence through Faddeev-Popov gauge fixing.

Confirmed the operator's utility in fixing conformal Killing vectors in superstring amplitudes.

Abstract

A careful gauge fixing of the conformal killing group (CKG) on genus zero surfaces in bosonic string theory gives non-vanishing two point amplitudes that match the corresponding field theory expressions arXiv:1906.06051, arXiv:1909.03672 . An important ingredient for gauge fixing two point amplitudes in arXiv:1909.03672 was the mostly BRST (mBRST) exact operator. The utility of this operator in gauge fixing CKG perhaps is not just limited to two point amplitudes - we can insert a mBRST exact operator to fix a conformal killing vector (CKV) instead of fixing the position of a vertex operator in a general tree level bosonic string amplitude arXiv:2108.05628, arXiv:2109.08433. Using the mBRST exact operator, written in the pure spinor variables, we get the expected two point superstring amplitudes arXiv:2012.03802. In this work we explore if it is possible to use this operator for fixing CKG for general tree level amplitudes in superstrings. We find it to be the case by explicitly re-calculating the three gluon amplitude in open strings by making use of this operator. Finally, we show that this operator directly emerges by following a Faddeev-Popov gauge fixing method in bosonic string theory. This derivation is independent of the number of external strings, thus proving the above claim.