Speaker
Description
This work is about a duality between two seemingly unrelated objects. The hypersimplex $\Delta_{k+1,n}$ -- a polytope of dimension $n-1$ in $\mathbb{R}^n$ -- has been the center of attention of both mathematicians and physicists, in connection with the moment map, torus orbits in the Grassmannian, tropical geometry and cluster algebras. Meanwhile, the amplituhedron $\mathcal{A}_{n,k,2}$ -- a full-dimensional subset of the Grassmannian $\mbox{Gr}_{k,k+2}$ (not a polytope!) -- was introduced by A.Hamed and Trnka in the context of the physics of scattering amplitudes in $\mathcal{N}=4$ super Yang-Mills theory (SYM). Surprisingly, as was first discovered in the work by Lukowski-Parisi-Williams [LPW], these two objects are closely related by a combinatorial-geometric incarnation of T-duality from String Theory, responsible for the Amplitudes/Wilson loops duality in $\mathcal{N}=4$ SYM.
In this work, exploiting T-duality, we both draw new striking connections between $\Delta_{k+1,n}$ and $\mathcal{A}_{n,k,2}$ and discover new properties of them. We show that inequalities cutting out positroid polytopes -- images of positroid cells in the hypersimplex -- translate into sign conditions characterizing the T-dual Grasstopes -- images of positroid cells in the amplituhedron. Moreover, we subdivide the amplituhedron into chambers, just as the hypersimplex can be subdivided into simplices - both enumerated by the Eulerian numbers, and related by T-duality. We use these properties to prove the main conjecture of [LPW]: a collection of positroid polytopes triangulates the hypersimplex if and only if the collection of T-dual Grasstopes triangulates the amplituhedron. As corollaries, some class of nice triangulations can be obtained from BCFW recursions and the positive tropical Grassmannian $\mbox{Trop}^+\mbox{Gr}_{k+1,n}$ -- both central in computations of scattering amplitudes.
Along the way, we also prove several more conjectures: Arkani-Hamed--Thomas--Trnka's conjecture that $\mathcal{A}_{n,k,2}$ can be characterized using sign-flips, Lukowski--Parisi--Spradlin--Volovich's conjectures about generalized triangles (Grasstopes in a triangulation of $\mathcal{A}_{n,k,2}$), and (a generalization of) $m=2$ cluster adjacency. Finally, we discover novel cluster-algebraic structures in the amplituhedron -- motivated by finding a geometric origin to cluster phenomena appearing in $\mathcal{N}=4$ SYM, and beyond.
This is based on joint work with M. Sherman-Bennett and L. K. Williams (Preprint, arXiv: 2104.08254).
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