Lie polynomials and a twistorial correspondence for amplitudes

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 $$\mathcal {M}_{0,n}$$ 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 $$T^*_D\mathcal {M}_{0,n}$$ T D ∗ M 0 , n , the bundle of forms with logarithmic singularities on the divisor D as the twistor space, and $$\mathcal {K}_n$$ 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 $$n-3$$ n - 3 -forms on $$\mathcal {K}_n$$ K n , introduced by Arkani-Hamed, Bai, He and Yan (ABHY). We also give a generalization and more invariant description of the associahedral $$n-3$$ n - 3 -planes in $$\mathcal {K}_n$$ K n introduced by ABHY.

Twistor constructions for higher-spin extensions of (self-dual) Yang-Mills

We present the inverse Penrose transform (the map from spacetime to twistor space) for self-dual Yang-Mills (SDYM) and its higher-spin extensions on a flat background. The twistor action for the higher-spin extension of SDYM (HS-SDYM) is of $$ \mathcal{BF} $$ BF -type. By considering a deformation away from the self-dual sector of HS-SDYM, we discover a new action that describes a higher-spin extension of Yang-Mills theory (HS-YM). The twistor action for HS-YM is a straightforward generalization of the Yang-Mills one.

On the supercritical defocusing NLW outside a ball

We study a defocusing semilinear wave equation, with a power nonlinearity $$|u|^{p-1}u$$ | u | p - 1 u , defined outside the unit ball of $$\mathbb {R}^{n}$$ R n , $$n\ge 3$$ n ≥ 3 , with Dirichlet boundary conditions. We prove that if $$p>n+3$$ p > n + 3 and the initial data are nonradial perturbations of large radial data, there exists a global smooth solution. The solution is unique among energy class solutions satisfying an energy inequality. The main tools used are the Penrose transform and a Strichartz estimate for the exterior linear wave equation perturbed with a large, time dependent potential.

Bulk interactions and boundary dual of higher-spin-charged particles

We consider higher-spin gravity in (Euclidean) AdS4, dual to a free vector model on the 3d boundary. In the bulk theory, we study the linearized version of the Didenko-Vasiliev black hole solution: a particle that couples to the gauge fields of all spins through a BPS-like pattern of charges. We study the interaction between two such particles at leading order. The sum over spins cancels the UV divergences that occur when the two particles are brought close together, for (almost) any value of the relative velocity. This is a higher-spin enhancement of supergravity’s famous feature, the cancellation of the electric and gravitational forces between two BPS particles at rest. In the holographic context, we point out that these “Didenko-Vasiliev particles” are just the bulk duals of bilocal operators in the boundary theory. For this identification, we use the Penrose transform between bulk fields and twistor functions, together with its holographic dual that relates twistor functions to boundary sources. In the resulting picture, the interaction between two Didenko-Vasiliev particles is just a geodesic Witten diagram that calculates the correlator of two boundary bilocals. We speculate on implications for a possible reformulation of the bulk theory, and for its non-locality issues.