The Amplituhedron: Structure, Combinatorics, and Positive Geometry

The amplituhedron is a remarkably elegant geometric object that was discovered in 2013 in the context of studying scattering amplitudes in theoretical particle physics (specifically, in planar N = 4 super Yang-Mills theory). It presents a radically different way to calculate fundamental quantities in quantum field theory, in which long-standing principles such as unitarity and locality, which have been postulates of QFT for nearly a century, emerge from the more fundamental notion of positivity. In addition, it gives a simplified way of expressing scattering amplitudes, without any need for Feynman diagrams or virtual particles.

Surprisingly, the best tools to study the amplituhedron turned out to come from combinatorics. Specifically, the amplituhedron An,k is the projection of the non-negative Grassmannian Gr≥0 k,n ⊂ Grk,n to Grk,k+4 by a linear map Rn → Rk+4 that satisfies certain positivity constraints. As the non-negative Grassmannian is a well-studied object in algebraic combinatorics, this has given birth to a growing interface between these fields. The conference we suggest will foster collaboration and knowledge exchange among experts in these areas.

Following the introduction of the amplituhedron, analogous objects have been discovered in various physical contexts, which are collectively called positive geometries. For N = 4 SYM theory, these include the loop amplituhedron, the momentum amplituhedron and the correlahedron. Amplituhedra also exist for scattering amplitudes of other theories, such as ABJM theory and ϕ3 theory. Furthermore, applications to cosmology and the conformal bootstrap program have also been suggested. Some positive geometries motivated by pure mathematics have also been proposed. The proposed conference will explore these ideas.