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Title: The Positive Tropical Grassmannian, the Hypersimplex, and the m = 2 Amplituhedron
Abstract The positive Grassmannian $$Gr^{\geq 0}_{k,n}$$ is a cell complex consisting of all points in the real Grassmannian whose Plücker coordinates are non-negative. In this paper we consider the image of the positive Grassmannian and its positroid cells under two different maps: the moment map$$\mu $$ onto the hypersimplex [ 31] and the amplituhedron map$$\tilde{Z}$$ onto the amplituhedron [ 6]. For either map, we define a positroid dissection to be a collection of images of positroid cells that are disjoint and cover a dense subset of the image. Positroid dissections of the hypersimplex are of interest because they include many matroid subdivisions; meanwhile, positroid dissections of the amplituhedron can be used to calculate the amplituhedron’s ‘volume’, which in turn computes scattering amplitudes in $$\mathcal{N}=4$$ super Yang-Mills. We define a map we call T-duality from cells of $$Gr^{\geq 0}_{k+1,n}$$ to cells of $$Gr^{\geq 0}_{k,n}$$ and conjecture that it induces a bijection from positroid dissections of the hypersimplex $$\Delta _{k+1,n}$$ to positroid dissections of the amplituhedron $$\mathcal{A}_{n,k,2}$$; we prove this conjecture for the (infinite) class of BCFW dissections. We note that T-duality is particularly striking because the hypersimplex is an $(n-1)$-dimensional polytope while the amplituhedron $$\mathcal{A}_{n,k,2}$$ is a $2k$-dimensional non-polytopal subset of the Grassmannian $$Gr_{k,k+2}$$. Moreover, we prove that the positive tropical Grassmannian is the secondary fan for the regular positroid subdivisions of the hypersimplex, and prove that a matroid polytope is a positroid polytope if and only if all 2D faces are positroid polytopes. Finally, toward the goal of generalizing T-duality for higher $$m$$, we define the momentum amplituhedron for any even $$m$$.  more » « less
Award ID(s):
2152991
PAR ID:
10580274
Author(s) / Creator(s):
; ;
Publisher / Repository:
International Mathematics Research Notices
Date Published:
Journal Name:
International Mathematics Research Notices
Volume:
2023
Issue:
19
ISSN:
1073-7928
Page Range / eLocation ID:
16778 to 16836
Format(s):
Medium: X
Sponsoring Org:
National Science Foundation
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