Amplitudes at Strong Coupling as Hyper-Kähler Scalars.

Alday and Maldacena conjectured an equivalence between string amplitudes in AdS_{5}×S^{5} and null polygonal Wilson loops in planar N=4 super-Yang-Mills (SYM) theory. At strong coupling this identifies SYM amplitudes with areas of minimal surfaces in anti-de Sitter space. For minimal surfaces in AdS_{3}, we find that the nontrivial part of these amplitudes, the remainder function, satisfies an integrable system of nonlinear differential equations, and we give its Lax form. The result follows from a new perspective on "Y systems," which defines a new psuedo-hyper-Kähler structure directly on the space of kinematic data, via a natural twistor space defined by the Y-system equations. The remainder function is the (pseudo-)Kähler scalar for this geometry. This connection to pseudo-hyper-Kähler geometry and its twistor theory provides a new ingredient for extending recent conjectures for nonperturbative amplitudes using structures arising at strong coupling.

A Lie Bracket for the Momentum Kernel.

We prove results for the study of the double copy and tree-level colour/kinematics duality for tree-level scattering amplitudes using the properties of Lie polynomials. We show that the 'S-map' that was defined to simplify super-Yang-Mills multiparticle superfields is in fact a Lie bracket. A generalized KLT map from Lie polynomials to their dual is obtained by studying our new Lie bracket; the matrix elements of this map yield a recently proposed 'generalized KLT matrix', and this reduces to the usual KLT matrix when its entries are restricted to a basis. Using this, we give an algebraic proof for the cancellation of double poles in the KLT formula for gravity amplitudes. We further study Berends-Giele recursion for biadjoint scalar tree amplitudes that take values in Lie polynomials. Field theory amplitudes are obtained from these 'Lie polynomial amplitudes' using numerators characterized as homomorphisms from the free Lie algebra to kinematic data. Examples are presented for the biadjoint scalar, Yang-Mills theory and the nonlinear sigma model. That these theories satisfy the Bern-Carrasco-Johansson amplitude relations follows from the structural properties of Lie polynomial amplitudes that we prove.

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 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.

Maximal-Helicity-Violating Scattering of Gluons and Gravitons in Chiral Strong Fields.

We present all-multiplicity formulas for the tree-level scattering of gluons and gravitons in the maximal helicity violating (MHV) helicity configuration, calculated in certain chiral strong fields. The strong backgrounds we consider are self-dual plane waves in gauge theory and general relativity, which are treated exactly and admit a well-defined S matrix. The gauge theory background-coupled MHV amplitude is simply a dressed analog of the familiar Parke-Taylor formula, but the gravitational version has nontrivial new structures due to graviton tails. Both formulas have just one residual integral rather than the n-2 expected at n points from space-time perturbation theory; this simplification arises from the integrability of self-dual backgrounds and their corresponding twistor description. The resulting formulas pass several consistency checks and limit to the well-known expressions for MHV scattering of gluons and gravitons when the background becomes trivial.

Polarized Scattering Equations for 6D Superamplitudes.

We introduce a spinorial version of the scattering equations, the polarized scattering equations, that incorporates spinor polarization data. They underpin new formulas for tree-level scattering amplitudes in six dimensions that directly extend to maximal supersymmetry. We find new ingredients for integrands for super Yang-Mills theory, gravity, M5 and D5 branes. We explain how the polarized scattering equations and supersymmetry representations arise from an ambitwistor string with target given by a supertwistor description of 6D superambitwistor space. On reduction to four dimensions, the polarized scattering equations give rise to massive analogues of the 4D refined scattering equations for amplitudes on the Coulomb branch. They give a quite distinct framework from that of Cachazo et al.; in particular, the formulas do not change character from even to odd numbers of particles.

Loop Integrands for Scattering Amplitudes from the Riemann Sphere.

The scattering equations on the Riemann sphere give rise to remarkable formulas for tree-level gauge theory and gravity amplitudes. Adamo, Casali, and Skinner conjectured a one-loop formula for supergravity amplitudes based on scattering equations on a torus. We use a residue theorem to transform this into a formula on the Riemann sphere. What emerges is a framework for loop integrands on the Riemann sphere that promises to have a wide application, based on off-shell scattering equations that depend on the loop momentum. We present new formulas, checked explicitly at low points, for supergravity and super-Yang-Mills amplitudes and for n-gon integrands at one loop. Finally, we show that the off-shell scattering equations naturally extend to arbitrary loop order, and we give a proposal for the all-loop integrands for supergravity and planar super-Yang-Mills theory.

Ambitwistor strings in four dimensions.

We develop ambitwistor string theories for four dimensions to obtain new formulas for tree-level gauge and gravity amplitudes with arbitrary amounts of supersymmetry. Ambitwistor space is the space of complex null geodesics in complexified Minkowski space, and in contrast to earlier ambitwistor strings, we use twistors rather than vectors to represent this space. Although superficially similar to the original twistor string theories of Witten, Berkovits, and Skinner, these theories differ in the assignment of world sheet spins of the fields, rely on both twistor and dual twistor representatives for the vertex operators, and use the ambitwistor procedure for calculating correlation functions. Our models are much more flexible, no longer requiring maximal supersymmetry, and the resulting formulas for amplitudes are simpler, having substantially reduced moduli. These are supported on the solutions to the scattering equations refined according to helicity and can be checked by comparison with corresponding formulas of Witten and of Cachazo and Skinner.

Twistor theory at fifty: from contour integrals to twistor strings.

We review aspects of twistor theory, its aims and achievements spanning the last five decades. In the twistor approach, space-time is secondary with events being derived objects that correspond to compact holomorphic curves in a complex threefold-the twistor space. After giving an elementary construction of this space, we demonstrate how solutions to linear and nonlinear equations of mathematical physics-anti-self-duality equations on Yang-Mills or conformal curvature-can be encoded into twistor cohomology. These twistor correspondences yield explicit examples of Yang-Mills and gravitational instantons, which we review. They also underlie the twistor approach to integrability: the solitonic systems arise as symmetry reductions of anti-self-dual (ASD) Yang-Mills equations, and Einstein-Weyl dispersionless systems are reductions of ASD conformal equations. We then review the holomorphic string theories in twistor and ambitwistor spaces, and explain how these theories give rise to remarkable new formulae for the computation of quantum scattering amplitudes. Finally, we discuss the Newtonian limit of twistor theory and its possible role in Penrose's proposal for a role of gravity in quantum collapse of a wave function.