Cohomology foundations of one-loop amplitudes in pure spinor superspace

TL;DR

This paper develops a cohomology-based framework using pure spinor BRST techniques to efficiently represent and analyze one-loop amplitudes in ten-dimensional superstring and super-Yang--Mills theories, revealing anomalies and pseudo-invariants.

Contribution

It introduces the concept of pseudo-cohomology and pseudo-invariant superfields to systematically handle anomalies in pure spinor superspace representations of one-loop amplitudes.

Findings

Identifies the hexagon gauge anomaly in the framework.

Defines pseudo-cohomology to analyze BRST anomalies.

Establishes a network of pseudo-invariant superfields.

Abstract

We describe a pure spinor BRST cohomology framework to compactly represent ten-dimensional one-loop amplitudes involving any number of massless open- and closed-string states. The method of previous work to construct scalar and vectorial BRST invariants in pure spinor superspace signals the appearance of the hexagon gauge anomaly when applied to tensors. We study the systematics of the underlying BRST anomaly by defining the notion of pseudo-cohomology. This leads to a rich network of pseudo-invariant superfields of arbitrary tensor rank whose behavior under traces and contractions with external momenta is determined from cohomology manipulations. Separate papers will illustrate the virtue of the superfields in this work to represent one-loop amplitudes of the superstring and of ten-dimensional super-Yang--Mills theory.