Lie polynomials and a twistorial correspondence for amplitudes

• Published in: Letters in mathematical physics Vol. 111; no. 6; p. 147
• Main Authors: Frost, Hadleigh, Mason, Lionel
• Format: Journal Article
• Language: English
• Published: Dordrecht Springer Netherlands 01.12.2021; Springer Nature B.V
• Subjects: Amplitudes; Complex Systems; Geometry; Group Theory and Generalizations; Homology; Invariants; Mathematical and Computational Physics; Momentum; Physics; Physics and Astronomy; Polynomials; Riemann manifold; Scattering; Theoretical; Twistor theory; 81T13; Lie polynomials; Scattering amplitudes
• ISSN: 0377-9017; 1573-0530; 1573-0530
• DOI: 10.1007/s11005-021-01483-1

We review Lie polynomials as a mathematical framework that underpins the structure of the so-called double copy relationship between gauge and gravity theories (and a network of other theories besides). We explain how Lie polynomials naturally arise in the geometry and cohomology of M 0 , n , the moduli space of n points on the Riemann sphere up to Mobiüs transformation. We introduce a twistorial correspondence between the cotangent bundle T D ∗ M 0 , n , the bundle of forms with logarithmic singularities on the divisor D as the twistor space, and K n the space of momentum invariants of n massless particles subject to momentum conservation as the analogue of space–time. This gives a natural framework for Cachazo He and Yuan (CHY) and ambitwistor-string formulae for scattering amplitudes of gauge and gravity theories as being the corresponding Penrose transform. In particular, we show that it gives a natural correspondence between CHY half-integrands and scattering forms, certain n - 3 -forms on K n , introduced by Arkani-Hamed, Bai, He and Yan (ABHY). We also give a generalization and more invariant description of the associahedral n - 3 -planes in K n introduced by ABHY.