Past Twistor Workshop

The duality between color and kinematics present in scattering amplitudes of Yang-Mills theory strongly suggest the existence of a hidden kinematic Lie algebra that controls the gauge theory. While associated BCJ numerators are known on closed forms to any multiplicity at tree level, the kinematic algebra has only been partially explored for the simplest of four-dimensional amplitudes: up to the MHV sector. In this paper we introduce a framework that allows us to characterize the algebra beyond the MHV sector. This allows us to both constrain some of the ambiguities of the kinematic algebra, and better control the generalized gauge freedom that is associated with the BCJ numerators. Specifically, in this paper, we work in dimension-agnostic notation and determine the kinematic algebra valid up to certain O((εi⋅εj)2) terms that in four dimensions compute the next-to-MHV sector involving two scalars. The kinematic algebra in this sector is simple, given that we introduce tensor currents that generalize standard Yang-Mills vector currents. These tensor currents controls the generalized gauge freedom, allowing us to generate multiple different versions of BCJ numerators from the same kinematic algebra. The framework should generalize to other sectors in Yang-Mills theory.

In this paper we define a new object, the momentum amplituhedron, which is the long sought-after positive geometry for tree-level scattering amplitudes in N=4 super Yang-Mills theory in spinor helicity space. Inspired by the construction of the ordinary amplituhedron, we introduce bosonized spinor helicity variables to represent our external kinematical data, and restrict them to a particular positive region. The momentum amplituhedron Mn,k is then the image of the positive Grassmannian via a map determined by such kinematics. The scattering amplitudes are extracted from the canonical form with logarithmic singularities on the boundaries of this geometry.

 Massless scattering amplitudes in four-dimensional Minkowski spacetime can be Mellin transformed to correlation functions on the celestial sphere at null infinity called celestial amplitudes. We study various properties of massless four-point scalar and gluon celestial amplitudes such as conformal partial wave decomposition, crossing relations and optical theorem. As a byproduct, we derive the analog of the single and double soft limits for all gluon celestial amplitudes.

Cluster Adjacency is a geometric principle which defines a subclass of multiple polylogarithms with analytic properties compatible with that of scattering amplitudes and Feynman loop integrals. We use this principle to a priori remove the redundances in the perturbative bootstrap approach and efficiently compute the four-loop NMHV heptagon. Moreover, cluster adjacency is naturally applied to the space of $A_n$ polylogarithms and generates numerous structures therein to be explored further.

Following recent papers by Nima Arkani-Hamed.

Using non-minimal pure spinor superspace, Cederwall has constructed BRST-invariant actions for D=10 super-Born-Infeld and D=11 supergravity which are quartic in the superfields. But since the superfields have explicit dependence on the non-minimal pure spinor variables, it is non-trivial to show these actions correctly describe super-Born-Infeld and supergravity. In this talk, I will expand solutions to the equations of motion from the pure spinor action for D=10 abelian super Born-Infeld to leading order around the linearized solutions and show that they correctly describe the interactions expected. If I have time, I will explain how to generalize these ideas to D=11 supergravity.

In this informal talk, I describe the kinds of functions relevant to scattering amplitudes in perturbative, four-dimensional quantum field theories. In particular, I will argue that generic amplitudes are non-polylogarithmic (beyond one loop), but that there is an upper bound to their geometric complexity. Moreover, I show a veritable `bestiary' of examples which saturate this bound in complexity---including three, all-loop families of integrals defined in massless $\phi^4$ theory which can, at best, be represented as dilogarithms integrated over (2L-2)-dimensional Calabi-Yau manifolds. 

This will be an informal discussion developing the details of the Amplituhedron for the loop integrand.

We discuss kinematic algebras associated to the scattering equations that arise in the description of the scattering of massless particles.  We describe their role in the BCJ duality between colour and kinematics in gauge theory, and its relation to gravity. We find that the scattering equations are a consistency condition for a self-dual-type vertex and identify an extension of the anti-self-dual vertex, such that the two vertices are not conjugate in general. Both vertices correspond to the structure constants of Lie algebras. We give a prescription for the use of the generators of these Lie algebras in trivalent graphs that leads to a natural set of BCJ numerators. In particular, we write BCJ numerators for each contribution to the amplitude associated to a solution of the scattering equations. This leads to a decomposition of the determinant of a certain kinematic matrix, which appears naturally in the amplitudes, in terms of trivalent graphs. We also present the kinematic analogues of colour traces, according to these algebras, and the associated decomposition of that determinant.

We discuss kinematic algebras associated to the scattering equations that arise in the description of the scattering of massless particles.  We describe their role in the BCJ duality between colour and kinematics in gauge theory, and its relation to gravity. We find that the scattering equations are a consistency condition for a self-dual-type vertex and identify an extension of the anti-self-dual vertex, such that the two vertices are not conjugate in general. Both vertices correspond to the structure constants of Lie algebras. We give a prescription for the use of the generators of these Lie algebras in trivalent graphs that leads to a natural set of BCJ numerators. In particular, we write BCJ numerators for each contribution to the amplitude associated to a solution of the scattering equations. This leads to a decomposition of the determinant of a certain kinematic matrix, which appears naturally in the amplitudes, in terms of trivalent graphs. We also present the kinematic analogues of colour traces, according to these algebras, and the associated decomposition of that determinant.

This talk will give an introduction to the recent paper by Arkani Hamed et. al. arxiv:1212:5605.