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Scattering equations, supergravity integrands, and pure spinors

Tim Adamo, Eduardo Casali

TL;DR

The paper develops a non-minimal pure spinor worldsheet framework to compute higher-genus amplitudes for type II supergravity, proposing that genus-g correlators yield the g-loop integrands localized on scattering equations. It extends the minimal formalism with additional pure spinor variables and a BRST charge, and constructs a covariant effective b-ghost to handle moduli integration. Through explicit genus-one and genus-two four-point checks, the authors demonstrate correct supersymmetric prefactors, factorization behavior, and IR consistency with field theory, supporting the conjecture that their worldsheet correlators reproduce gravity loop integrands. The work highlights deep connections to pure spinor string theory and worldline approaches, suggesting a path toward a covariant, loop-level description of supergravity amplitudes.

Abstract

The tree-level S-matrix of type II supergravity can be computed in scattering equation form by correlators in a worldsheet theory analogous to a chiral, infinite tension limit of the pure spinor formalism. By defining a non-minimal version of this theory, we give a prescription for computing correlators on higher genus worldsheets which manifest space-time supersymmetry. These correlators are conjectured to provide the loop integrands of supergravity scattering amplitudes, supported on the scattering equations. We give non-trivial evidence in support of this conjecture at genus one and two with four external states. Throughout, we find a close correspondence with the pure spinor formalism of superstring theory, particularly regarding regulators and zero-mode counting.

Scattering equations, supergravity integrands, and pure spinors

TL;DR

The paper develops a non-minimal pure spinor worldsheet framework to compute higher-genus amplitudes for type II supergravity, proposing that genus-g correlators yield the g-loop integrands localized on scattering equations. It extends the minimal formalism with additional pure spinor variables and a BRST charge, and constructs a covariant effective b-ghost to handle moduli integration. Through explicit genus-one and genus-two four-point checks, the authors demonstrate correct supersymmetric prefactors, factorization behavior, and IR consistency with field theory, supporting the conjecture that their worldsheet correlators reproduce gravity loop integrands. The work highlights deep connections to pure spinor string theory and worldline approaches, suggesting a path toward a covariant, loop-level description of supergravity amplitudes.

Abstract

The tree-level S-matrix of type II supergravity can be computed in scattering equation form by correlators in a worldsheet theory analogous to a chiral, infinite tension limit of the pure spinor formalism. By defining a non-minimal version of this theory, we give a prescription for computing correlators on higher genus worldsheets which manifest space-time supersymmetry. These correlators are conjectured to provide the loop integrands of supergravity scattering amplitudes, supported on the scattering equations. We give non-trivial evidence in support of this conjecture at genus one and two with four external states. Throughout, we find a close correspondence with the pure spinor formalism of superstring theory, particularly regarding regulators and zero-mode counting.

Paper Structure

This paper contains 13 sections, 93 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Non-minimal formalism
  3. Review of the minimal theory
  4. Non-minimal worldsheet action and BRST charge
  5. Effective $b$-ghost
  6. Zero modes, functional integrals, and regulators
  7. Amplitudes at Higher Genus
  8. Amplitude prescription
  9. Four-point function: Genus one
  10. Four-point function: Genus two
  11. Conclusions
  12. Worldsheet currents and OPEs
  13. Worldsheet action from a gauge-fixing procedure

Figures (2)

  • Figure 1: The genus two worldsheet at the boundary of the moduli space.
  • Figure 2: The degenerating worldsheet modeled on two tori $\Sigma_1, \Sigma_2$ connected by the annulus $A_t$. Solid dots denote scattering equations of the form $k\cdot P$, while crosses denote scattering equations of the form $P^2$.