Decompositions of amplituhedra
Autor: | Karp, Steven N., Williams, Lauren K., Zhang, Yan X |
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Rok vydání: | 2017 |
Předmět: | |
Zdroj: | Ann. Inst. Henri Poincar\'e D 7 (2020), no. 3, 303-363 |
Druh dokumentu: | Working Paper |
DOI: | 10.4171/AIHPD/87 |
Popis: | The (tree) amplituhedron A(n,k,m) is the image in the Grassmannian Gr(k,k+m) of the totally nonnegative part of Gr(k,n), 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 N=4 supersymmetric Yang-Mills theory. In the case relevant to physics (m=4), there is a collection of recursively-defined 4k-dimensional BCFW cells in the totally nonnegative part of Gr(k,n), 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, m/2) top-dimensional cells (of dimension km), where M(a,b,c) is the number of plane partitions contained in an a x b x 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). Comment: 46 pages; appendix written with Hugh Thomas |
Databáze: | arXiv |
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