One-loop superstring six-point amplitudes and anomalies in pure spinor superspace

We present the massless six-point one-loop amplitudes in the open and closed superstring using brst cohomology arguments from the pure spinor formalism. the hexagon gauge anomaly is traced back to a class of kinematic factors in pure spinor superspace which were recently introduced as brst pseudo-invariants. this complements previous work where brst invariance arguments were used to derive the non-anomalous part of the amplitude. the associated worldsheet functions are non-singular and demonstrated to yield total derivatives on moduli space upon gauge variation. these cohomology considerations yield an efficient organizing principle for closed-string amplitudes that match expectations from s-duality in the low-energy limit.