TY - JOUR

T1 - The positive tropical Grassmannian, the hypersimplex, and the m=2 amplituhedron

AU - Lukowski, Tomasz

AU - Parisi, Matteo

AU - Williams, Lauren K.

N1 - © The Author(s) 2023. Published by Oxford University Press. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted reuse, distribution, and reproduction in any medium, provided the original work is properly cited.

PY - 2023/3/7

Y1 - 2023/3/7

N2 - The study of the moment map from the Grassmannian to the hypersimplex, and the relation between torus orbits and matroid polytopes, dates back to the foundational 1987 work of Gelfand-Goresky-MacPherson-Serganova. On the other hand, the amplituhedron is a very new object, defined by Arkani-Hamed-Trnka in connection with scattering amplitudes in $\mathcal{N}=4$ super Yang-Mills theory. In this paper we discover a striking duality between the moment map $\mu:Gr^{\geq0}_{k+1,n}\to\Delta_{k+1,n}$ from the positive Grassmannian $Gr^{\geq0}_{k+1,n}$ to the hypersimplex, and the amplituhedron map $\tilde{Z}:Gr^{\geq0}_{k,n}\to\mathcal{A}_{n,k,2}(Z)$ from $Gr^{\geq0}_{k,n}$ to the $m=2$ amplituhedron. We consider the positroid dissections of both objects, which informally, are subdivisions of $\Delta_{k+1,n}$ (respectively, $\mathcal{A}_{n,k,2}(Z)$) into a disjoint union of images of positroid cells of the positive Grassmannian. At first glance, $\Delta_{k+1,n}$ and $\mathcal{A}_{n,k,2}(Z)$ seem very different - the former is an $(n-1)$-dimensional polytope, while the latter is a $2k$-dimensional non-polytopal subset of $Gr_{k,k+2}$. Nevertheless, we conjecture that positroid dissections of $\Delta_{k+1,n}$ are in bijection with positroid dissections of $\mathcal{A}_{n,k,2}(Z)$ via a map we call T-duality. We prove this conjecture for the (infinite) class of BCFW dissections and give additional experimental evidence. Moreover, we prove that the positive tropical Grassmannian is the secondary fan for the regular positroid subdivisions of the hypersimplex, and propose that it also controls the T-dual positroid subdivisions of the amplituhedron. Along the way, we prove that a matroid polytope is a positroid polytope if and only if all two-dimensional faces are positroid polytopes. Towards the goal of generalizing T-duality for higher $m$, we also define the momentum amplituhedron for any even $m$.

AB - The study of the moment map from the Grassmannian to the hypersimplex, and the relation between torus orbits and matroid polytopes, dates back to the foundational 1987 work of Gelfand-Goresky-MacPherson-Serganova. On the other hand, the amplituhedron is a very new object, defined by Arkani-Hamed-Trnka in connection with scattering amplitudes in $\mathcal{N}=4$ super Yang-Mills theory. In this paper we discover a striking duality between the moment map $\mu:Gr^{\geq0}_{k+1,n}\to\Delta_{k+1,n}$ from the positive Grassmannian $Gr^{\geq0}_{k+1,n}$ to the hypersimplex, and the amplituhedron map $\tilde{Z}:Gr^{\geq0}_{k,n}\to\mathcal{A}_{n,k,2}(Z)$ from $Gr^{\geq0}_{k,n}$ to the $m=2$ amplituhedron. We consider the positroid dissections of both objects, which informally, are subdivisions of $\Delta_{k+1,n}$ (respectively, $\mathcal{A}_{n,k,2}(Z)$) into a disjoint union of images of positroid cells of the positive Grassmannian. At first glance, $\Delta_{k+1,n}$ and $\mathcal{A}_{n,k,2}(Z)$ seem very different - the former is an $(n-1)$-dimensional polytope, while the latter is a $2k$-dimensional non-polytopal subset of $Gr_{k,k+2}$. Nevertheless, we conjecture that positroid dissections of $\Delta_{k+1,n}$ are in bijection with positroid dissections of $\mathcal{A}_{n,k,2}(Z)$ via a map we call T-duality. We prove this conjecture for the (infinite) class of BCFW dissections and give additional experimental evidence. Moreover, we prove that the positive tropical Grassmannian is the secondary fan for the regular positroid subdivisions of the hypersimplex, and propose that it also controls the T-dual positroid subdivisions of the amplituhedron. Along the way, we prove that a matroid polytope is a positroid polytope if and only if all two-dimensional faces are positroid polytopes. Towards the goal of generalizing T-duality for higher $m$, we also define the momentum amplituhedron for any even $m$.

U2 - 10.1093/imrn/rnad010

DO - 10.1093/imrn/rnad010

M3 - Article

SN - 1073-7928

JO - International Mathematical Research Notices

JF - International Mathematical Research Notices

M1 - rnad010

ER -