Lie Polynomials and a Twistorial Correspondence for Amplitudes

Understand

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 Mobi\"us transformation.
• We introduce a twistorial correspondence between the cotangent bundle $T^*_D\mathcal{M}_{0,n}$, 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.