Combinatorial aspects of scattering amplitudes: amplituhedra, T-duality, and cluster algebras

Thesis

A paradigm shift has been occurring, according to which physical observables and their fundamental properties, symmetries and dualities emerge from underlying geometry and combinatorics. This has led to major new insights in both Physics and Mathematics. In the first part of this work, we discover a duality between two seemingly unrelated objects. On one side, the hypersimplex Delta_{k+1,n} – a polytope which has been studied in connection with the moment map, torus orbits in the Grassmannian, tropical geometry and cluster algebras. On the other side, the amplituhedron A_{n,k,2} – a subset of the Grassmannian (not a polytope!) which was introduced in the context of the physics of scattering amplitudes in planar N = 4 super Yang-Mills (SYM). We show these two objects are closely related by a combinatorial-geometric incarnation of T-duality from String Theory. Exploiting T-duality, we both draw striking connections between Delta_{k+1,n} and A_{n,k,2} and discover new properties of them. One of the main results is proving that positroid triangulations of the hypersimplex and of the amplituhedron are T-dual. Moreover, some of these triangulations can be obtained from BCFW recursions and the positive tropical Grassmannian Trop+Gr_{k+1,n} – both central in computations of scattering amplitudes. We then define the sought-after positive geometry of tree-level N = 4 SYM amplitudes in momentum space – the momentum amplituhedron. We conjecture that its positroid triangulations are T-dual to the ones of the m = 4 amplituhedron A_{n,k,4}, reflecting the physical Amplitudes/Wilson loops duality. In the last part, we pave the way for both connecting amplituhedra theory to cluster algebras and tropical geometry, and finding a geometric origin of cluster phenomena – increasingly emerging in scattering amplitudes. In particular, we show how cluster adjacency for Yangian invariants emerge purely from amplituhedra. We prove this for A_{n,k,2} in full generality and provide some results for the m = 4 amplituhedron. Finally, using the geometry of loop amplituhedra, we conjecture a new phenomenon for planar N = 4 SYM at loop level – LL-cluster adjacency – relating Leading Singularities and Landau Singularities, with checks and general proves at one loop.

Authors: Parisi, M

• DOI: 10.5287/ora-jnpegza9r
• Type of award: DPhil
• Level of award: Doctoral
• Awarding institution: University of Oxford
• Language: English
• Keywords: T-duality; scattering amplitudes; N=4 Super Yang-Mills Theory; cluster algebras; tropical geometry; positive geometries; hypersimplex; triangulations; tilings; positive grassmannian; amplituhedron; matroid and positroid theory
• Subjects: Scattering Amplitudes, N=4 Super Yang-Mills Theory, On-shell Methods, Grassmannians, Amplituhedron, Positive Geometries, Algebraic Combinatorics, Combinatorial Algebraic Geometry, Matroid and Positroid Theory, Tropical Geometry, and Cluster Algebras.