论文标题
阳性的热带司法，高血压和M = 2 Amplituhedron
The positive tropical Grassmannian, the hypersimplex, and the m=2 amplituhedron
论文作者
论文摘要
从格拉斯曼尼亚人到高血压的矩图研究以及圆环轨道和矩阵多面体之间的关系可以追溯到Gelfand-Goresky-Goresky-Macpherson-Serganova的1987年基础工作。另一方面，Amplituhedron是一个非常新的对象，由Arkani-Hamed-Trnka定义，与$ \ Mathcal {n} = 4 $ Super Yang-Mills理论相关。在本文中，我们发现矩映射$μ：gr^{\ geq0} _ {k+1，n} \toδ_{k+1，n} $从正阳性的Grassmannian $ gr^{\ geq0} _ {\ geq0} _ {k+1，n} $ to hypersimplex，and amplitron， $ \ tilde {z}：gr^{\ geq0} _ {k，n} \ to \ mathcal {a} _ {n，k，2}（z）$来自$ gr^{\ geq0} _ {\ geq0} _ {\ geq0} _ {k，n} $ to $ m = 2 $ m = 2 $ m = 2 $ ampliturritron。我们认为这两个对象的阳性解剖是非正式的，它们是$δ_{k+1，n} $的细分（分别分别为$ \ nathcal {a} _ {n，k，2}（z）$），纳入阳性良好的正面良好细胞图像的不连接。 At first glance, $Δ_{k+1,n}$ 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 $Gr_{k,k+2}$.然而，我们猜想$δ_{k+1，n} $的positroid解剖与$ \ mathcal {a} _ {a} _ {n，k，2}（z）$的posiitroid dissections通过映射我们称为t-duality。我们证明了（无限）BCFW解剖类别的猜想，并提供其他实验证据。此外，我们证明了热带格拉曼尼亚阳性的次级风扇，是hypersimplex的常规阳性细分，并提出它还控制了Amplituhedron的T偶二氨基分区。在此过程中，我们证明，当且仅当所有二维面都是阳性多面体的情况下，曲霉的多层是阳性多层。为了将T二维概括为更高的$ m $，我们还为任何$ m $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $。
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 $μ:Gr^{\geq0}_{k+1,n}\toΔ_{k+1,n}$ 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 $Δ_{k+1,n}$ (respectively, $\mathcal{A}_{n,k,2}(Z)$) into a disjoint union of images of positroid cells of the positive Grassmannian. At first glance, $Δ_{k+1,n}$ 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 $Δ_{k+1,n}$ 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 $m$.
