Decompositions of amplituhedra

Karp, Steven N.; Williams, Lauren K.; Zhang, Yan X

doi:10.4171/aihpd/87

Geodesic

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Decompositions of amplituhedra

Karp, Steven N. ; Williams, Lauren K. ; Zhang, Yan X

Annales de l’Institut Henri Poincaré D, Tome 7 (2020) no. 3, pp. 303-363.

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Résumé

The (tree) amplituhedron $𝒜_{n, k, m}$ is the image in the Grassmannian ${Gr}_{k, k + m}$ of the totally nonnegative Grassmannian ${Gr}_{k, n}^{\geq 0}$ , under a (map induced by a) linear map which is totally positive. It was introduced by Arkani-Hamed and Trnka in 2013 in order to give a geometric basis for the computation of scattering amplitudes in planar $𝒩 = 4$ supersymmetric Yang–Mills theory. In the case relevant to physics ( $m = 4$ ), there is a collection of recursively-defined $4 k$ -dimensional BCFW cells in ${Gr}_{k, n}^{\geq 0}$ , whose images conjecturally "triangulate" the amplituhedron – that is, their images are disjoint and cover a dense subset of $𝒜_{n, k, 4}$ . In this paper, we approach this problem by first giving an explicit (as opposed to recursive) description of the BCFW cells. We then develop sign-variational tools which we use to prove that when $k = 2$ , the images of these cells are disjoint in $𝒜_{n, k, 4}$ . We also conjecture that for arbitrary even $m$ , there is a decomposition of the amplituhedron $𝒜_{n, k, m}$ involving precisely $M (k, n - k - m, \frac{m}{2})$ top-dimensional cells (of dimension $k m$ ), where $M (a, b, c)$ is the number of plane partitions contained in an $a \times b \times c$ box. This agrees with the fact that when $m = 4$ , the number of BCFW cells is the Narayana number $N_{n - 3, k + 1} = \frac{1}{n - 3} (\binom{n - 3}{k + 1}) (\binom{n - 3}{k})$ .

Accepté le : 2018-06-13
Publié le : 2020-08-22

MR Zbl

DOI : 10.4171/aihpd/87

Classification : 05-XX, 14-XX, 15-XX, 81-XX
Keywords: Amplituhedron, scattering amplitude, totally nonnegative Grassmannian, BCFW recursion, Narayana number, plane partition

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     author = {Karp, Steven N. and Williams, Lauren K. and Zhang, Yan X},
     title = {Decompositions of amplituhedra},
     journal = {Annales de l{\textquoteright}Institut Henri Poincar\'e D},
     pages = {303--363},
     volume = {7},
     number = {3},
     year = {2020},
     doi = {10.4171/aihpd/87},
     mrnumber = {4152617},
     zbl = {1470.81048},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.4171/aihpd/87/}
}

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Karp, Steven N.; Williams, Lauren K.; Zhang, Yan X. Decompositions of amplituhedra. Annales de l’Institut Henri Poincaré D, Tome 7 (2020) no. 3, pp. 303-363. doi : 10.4171/aihpd/87. http://geodesic.mathdoc.fr/articles/10.4171/aihpd/87/

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