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Triangulations and Canonical Forms of Amplituhedra: A Fiber-Based Approach Beyond Polytopes by Fatemeh Mohammadi; Leonid Monin; Matteo Parisi is a Mathematics article available to read on EtoBox.

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Any totally positive (k + m) × n matrix induces a map π + from the positive Grassmannian Gr + (k, n) to the Grassmannian Gr(k, k + m), whose image is the amplituhedron A n,k,m and is endowed with a top-degree form called the canonical form (A n,k,m ). This construction was introduced by Arkani-Hamed and Trnka (J High Energy Phys 2014(10):30, 2014), where they showed that (A n,k,4 ) encodes scattering amplitudes in N = 4 super Yang-Mills theory. One way to compute (A n,k,m ) is to subdivide A n,k,m into so-called generalized triangles and sum over their associated canonical forms. Hence, the physical computation of scattering amplitudes is reduced to finding the triangulations of A n,k,4 . However, while triangulations of polytopes are fully captured by their secondary and fiber polytopes (Billera and Sturmfels in Ann Math 527-549, 1992; Gelfand et al. in Discriminants, resultants, and multidimensional determinants, Birkahuser, Boston, 1994), the study of triangulations of objects beyond polytopes is still underdeveloped. In this work, we initiate the geometric study of subdivisions of A n,k,m in order to establish the notion of secondary amplituhedron. For this purpose, we first ex

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Author

Fatemeh Mohammadi; Leonid Monin; Matteo Parisi

Publisher

Springer Science and Business Media LLC

Published

2021

Language

EN

Field

Mathematics (Physical Sciences)