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A Gluing Operator for the Ambitwistor String

Kai A. Roehrig, David Skinner

TL;DR

The paper introduces a non-local gluing operator $\Delta(z,w)$ within the genus-0 ambitwistor string and demonstrates how it sews together lower-point correlators to reconstruct tree-level CHY amplitudes and 1-loop loop integrands on a nodal sphere. By applying this operator to bi-adjoint $\phi^3$ theory, Yang-Mills, and gravity, the authors derive off-shell factorization structures and show that the resulting sphere correlators reproduce known CHY/Geyer results for loop integrands, with BRST invariance guiding the construction. The key technical advance is the explicit form of $\Delta$ and its coupling to both bosonic and fermionic gauge fields, which modifies the scattering equations to their off-shell/one-loop versions and yields a purely rational dependence on external data in the sphere framework. The work suggests a recursive, propagator-based worldsheet approach to higher-loop ambitwistor amplitudes, potentially providing a bridge to the standard string propagator and enabling systematic multi-loop constructions directly from sphere correlators. Overall, the gluing operator encapsulates the target-space propagator within a BRST-invariant worldsheet operator and offers a principled path to extend CHY and ambitwistor techniques to higher iterations of sewing and loop orders.

Abstract

We present a new operator in the ambitwistor string which glues together correlators with fewer points or of lower genus. It underpins the recursive construction of tree-level CHY scattering amplitudes by Dolan \& Goddard, as well as the computation of loop integrands on a Riemann sphere by Geyer et al. The gluing operator is a tractable object due to the finiteness of the spectrum. In particular, we demonstrate how it gives rise to the complete one-loop integrand in SYM and SUGRA. The operator is conjectured to be the path integral incarnation of the ambitwistor string propagator, and to coincide with the field theory limit of the standard string theory propagator.

A Gluing Operator for the Ambitwistor String

TL;DR

The paper introduces a non-local gluing operator within the genus-0 ambitwistor string and demonstrates how it sews together lower-point correlators to reconstruct tree-level CHY amplitudes and 1-loop loop integrands on a nodal sphere. By applying this operator to bi-adjoint theory, Yang-Mills, and gravity, the authors derive off-shell factorization structures and show that the resulting sphere correlators reproduce known CHY/Geyer results for loop integrands, with BRST invariance guiding the construction. The key technical advance is the explicit form of and its coupling to both bosonic and fermionic gauge fields, which modifies the scattering equations to their off-shell/one-loop versions and yields a purely rational dependence on external data in the sphere framework. The work suggests a recursive, propagator-based worldsheet approach to higher-loop ambitwistor amplitudes, potentially providing a bridge to the standard string propagator and enabling systematic multi-loop constructions directly from sphere correlators. Overall, the gluing operator encapsulates the target-space propagator within a BRST-invariant worldsheet operator and offers a principled path to extend CHY and ambitwistor techniques to higher iterations of sewing and loop orders.

Abstract

We present a new operator in the ambitwistor string which glues together correlators with fewer points or of lower genus. It underpins the recursive construction of tree-level CHY scattering amplitudes by Dolan \& Goddard, as well as the computation of loop integrands on a Riemann sphere by Geyer et al. The gluing operator is a tractable object due to the finiteness of the spectrum. In particular, we demonstrate how it gives rise to the complete one-loop integrand in SYM and SUGRA. The operator is conjectured to be the path integral incarnation of the ambitwistor string propagator, and to coincide with the field theory limit of the standard string theory propagator.

Paper Structure

This paper contains 13 sections, 121 equations, 2 figures.

Table of Contents

  1. Introduction
  2. A brief review of the ambitwistor string
  3. CHY formulae from the ambitwistor string
  4. The gluing operator for the bi-adjoint scalar
  5. Tree amplitudes in $\phi^3$ theory
  6. One loop amplitudes in $\phi^3$ theory
  7. The Yang-Mills gluing operator
  8. Neveu-Schwarz sector
  9. Ramond sector
  10. The gluing operator for gravity
  11. Conclusions
  12. Sphere Correlator with NS Gluing Operator Insertion
  13. Degeneration limit of the torus Szegó kernel

Figures (2)

  • Figure 1: Gluing two genus zero correlators, forming a new sphere.
  • Figure 2: The gluing operator identifies two points in a genus zero correlator, forming a nodal sphere.