Multiloop Amplitudes and Vanishing Theorems using the Pure Spinor Formalism for the Superstring

TL;DR

This paper develops a super-Poincare covariant formalism for superstring amplitudes using pure spinor variables, enabling multiloop calculations and confirming key finiteness and duality conjectures.

Contribution

It introduces a covariant prescription for multiloop superstring amplitudes using pure spinor formalism, including the construction of a suitable b ghost and proof of amplitude vanishing and duality results.

Findings

Massless N-point multiloop amplitudes vanish for N<4.

Superstring theory is perturbatively finite.

Type IIB S-duality conjecture is confirmed.

Abstract

A ten-dimensional super-Poincare covariant formalism for the superstring was recently developed which involves a BRST operator constructed from superspace matter variables and a pure spinor ghost variable. A super-Poincare covariant prescription was defined for computing tree amplitudes and was shown to coincide with the standard RNS prescription. In this paper, picture-changing operators are used to define functional integration over the pure spinor ghosts and to construct a suitable $b$ ghost. A super-Poincare covariant prescription is then given for the computation of N-point multiloop amplitudes. One can easily prove that massless N-point multiloop amplitudes vanish for N<4, confirming the perturbative finiteness of superstring theory. One can also prove the Type IIB S-duality conjecture that $R^4$ terms in the effective action receive no perturbative contributions above one loop.