Two-loop superstring five-point amplitudes I: Construction via chiral splitting and pure spinors

Abstract

The full two-loop amplitudes for five massless states in Type II and Heterotic superstrings are constructed in terms of convergent integrals over the genus-two moduli space of compact Riemann surfaces and integrals of Green functions and Abelian differentials on the surface. The construction combines elements from the BRST cohomology of the pure spinor formulation and from chiral splitting with the help of loop momenta and homology invariance. The α ′ → 0 limit of the resulting superstring amplitude is shown to be in perfect agreement with the previously known amplitude computed in Type II supergravity. Investigations of the α ′ expansion of the Type II amplitude and comparisons with predictions from S-duality are relegated to a first companion paper. A construction from first principles in the RNS formulation of the genus-two amplitude with five external NS states is relegated to a second companion paper.

Two-loop superstring five-point amplitudes. Part I. Construction via chiral splitting and pure spinors

“…The contribution from even spin structures produces the even parity part of the chiral amplitude with external NS states. This even parity part will be shown to agree with the bosonic supermultiplet components of the chiral amplitude in pure-spinor superspace derived in [29]. The chiral genus-two five-point amplitude additionally has an odd parity part,…”

“…In both cases Fbos stands for the chiral amplitude of the (compactified) bosonic string. Its construction for the five-point function was outlined in [29], and the corresponding result may be carried over the present work without modification.…”

“…In recent work, the genus-two amplitude for five external massless string states was obtained for both the Type II and Heterotic strings in [29]. The construction produces a consistent expression for the chiral amplitude by an amalgam of results from chiral splitting in terms of loop momenta and homology shift invariance of the RNS formulation [3,30] (see also [31]) and results from zero-mode counting and the BRST cohomology in the pure-spinor formalism [8].…”

“…The physical string amplitudes are given in terms of convergent integrals over the genus-two moduli space of compact Riemann surfaces. The moduli-space integrands in turn are themselves integrals over the vertex points of combinations of Green functions and Abelian differentials [29]. The overall normalization of the Type II amplitude was fixed using methods analogous to those used in [25,32] for the four-point amplitude.…”

10D Super-Yang-Mills Scattering Amplitudes From Its Pure Spinor Action

“…This should not be a surprise at all. In fact, the pure spinor formalism for superstrings [9] is nowadays arguably the most powerful and efficient framework for computing string scattering amplitudes as compared to the traditional Ramond-Neveu-Schwarz and Green-Schwarz formalisms [10,11,12,13,14].…”