Abstract
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 mathcal {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^*_Dmathcal {M}_{0,n}, the bundle of forms with logarithmic singularities on the divisor D as the twistor space, and mathcal {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 mathcal {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 mathcal {K}_n introduced by ABHY.
Highlights
Colour-kinematics duality and the double copy [1,2] have had a powerful influence on recent developments in scattering amplitudes
We build on observations by Kapranov in an after dinner talk [5] concerning the relevance of Lie polynomials, both in the double copy and in the Parke–Taylor expressions that pervade the subject
We review the role played by Lie polynomials in the geometry of the moduli space M0,n of n points σi in CP1, both in describing the compact cycles in the homology Hn−3(M0,n − D), which is isomorphic to Lie(n−1), and dually the relative cocycles in H n−3(M0,n, D), represented by the top degree holomorphic Parke–Taylor forms
Summary
Colour-kinematics duality and the double copy [1,2] have had a powerful influence on recent developments in scattering amplitudes. We build on the recent work by Arkani-Hamed, Bai, He and Yan [6] that introduces differential forms in the space of kinematic invariants, Kn. We tie them together by means of a double fibration correspondence that leads to a Penrose-like transform for the formulae of Cachazo He and Yuan (CHY) arising from the scattering equations [7,8]. ABHY use the w as numerators so that given a set of conventional numerators N one can associate a scattering (n − 3)-form N These arise from our double fibration via a Penrose transform . We give an improved and extended definition of these and show how they tie into the geometry of the correspondence
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