Some more amplituhedra are contractible

Blagojević, Pavle V. M.; Galashin, Pavel; Palić, Nevena; Ziegler, Günter M.

doi:10.1007/s00029-019-0462-2

Cited by 5 publications

(2 citation statements)

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“…We would like to remark that the fibers of the amplituhedron are expected to have nice topological properties. For example, in [BGPZ19] the authors showed that for the amplituhedra conjugate to polytopes, the fibers are homeomorphic to a ball.…”

Section: Amplituhedron Fibersmentioning

confidence: 99%

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Triangulations and Canonical Forms of Amplituhedra: A Fiber-Based Approach Beyond Polytopes

Mohammadi

Monin

Parisi

2021

Commun. Math. Phys.

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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 extend the projection π + to a rational map π : Gr(k, n)Gr(k, k +m) and provide a concrete birational parametrization of the fibers of π . We then use this to explicitly describe a rational top-degree form ω n,k,m (with simple poles) on the fibers and compute (A n,k,m ) as a summation of certain residues of ω n,k,m . As main application of our approach, we develop a well-structured notion of secondary amplituhedra for conjugate to polytopes, i.e. when n − k − 1 = m (even). We show that, in this case, each fiber of π is parametrized by a projective space and its volume form ω n,k,m has only poles on a hyperplane arrangement. Using such linear structures, for amplituhedra which are cyclic polytopes or conjugate to polytopes, we show that the Jeffrey-Kirwan residue computes (A n,k,m ) from the fiber volume form ω n,k,m . In particular, we give conceptual proofs of the statements of Ferro et al. (J Phys A Math Theor 52(4):045201, 2018). Finally, we propose a more general framework of fiber positive geometries and analyze new families of examples such as fiber polytopes and Grassmann polytopes.

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Section: Amplituhedron Fibersmentioning

confidence: 99%

“…Acknowledgements. We would like to thank Hugh Thomas for pointing out the reference [ADLRS00] and the anonymous referees for their insightful comments, especially for pointing out the reference [BGPZ19]. M.P.…”

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confidence: 99%