The Magic Number Conjecture for the $m=2$ amplituhedron and Parke-Taylor identities
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2024
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| author | Parisi, Matteo Sherman-Bennett, Melissa Tessler, Ran Williams, Lauren |
| author_facet | Parisi, Matteo Sherman-Bennett, Melissa Tessler, Ran Williams, Lauren |
| contents | The amplituhedron $A_{n,k,m}$ is a geometric object introduced in the context of scattering amplitudes in $N=4$ super Yang Mills. It generalizes the positive Grassmannian (when $n=k+m$), cyclic polytopes (when $k=1$), and the bounded complex of the cyclic hyperplane arrangement (when $m=1$). Of substantial interest are the tilings of the amplituhedron, which are analogous to triangulations of a polytope. Karp, Williams and Zhang (2020) observed that the known tilings of $A_{n,k,2}$ have cardinality ${n-2 \choose k}$ and the known tilings of $A_{n,k,4}$ have cardinality the Narayana number $\frac{1}{n-3}{n-3 \choose k+1}{n-3 \choose k}$; generalizing these observations, they conjectured that for even $m$ the tilings of $A_{n, k,m}$ have cardinality the MacMahon number, the number of plane partitions which fit inside a $k \times (n-k-m) \times \frac{m}{2}$ box. We refer to this prediction as the `Magic Number Conjecture'. In this paper we prove the Magic Number Conjecture for the $m=2$ amplituhedron: that is, we show that each tiling of $A_{n,k,2}$ has cardinality ${n-2 \choose k}$. We prove this by showing that all positroid tilings of the hypersimplex $Δ_{k+1,n}$ have cardinality ${n-2 \choose k}$, then applying T-duality. In addition, we give combinatorial necessary conditions for tiles to form a tiling of $A_{n,k,2}$; we give volume formulas for Parke-Taylor polytopes and certain positroid polytopes in terms of circular extensions of cyclic partial orders; and we prove new variants of the classical Parke-Taylor identities. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2404_03026 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The Magic Number Conjecture for the $m=2$ amplituhedron and Parke-Taylor identities Parisi, Matteo Sherman-Bennett, Melissa Tessler, Ran Williams, Lauren Combinatorics High Energy Physics - Theory Mathematical Physics Algebraic Geometry 05E14, 52B20, 52B40, 52C40, 05A99 The amplituhedron $A_{n,k,m}$ is a geometric object introduced in the context of scattering amplitudes in $N=4$ super Yang Mills. It generalizes the positive Grassmannian (when $n=k+m$), cyclic polytopes (when $k=1$), and the bounded complex of the cyclic hyperplane arrangement (when $m=1$). Of substantial interest are the tilings of the amplituhedron, which are analogous to triangulations of a polytope. Karp, Williams and Zhang (2020) observed that the known tilings of $A_{n,k,2}$ have cardinality ${n-2 \choose k}$ and the known tilings of $A_{n,k,4}$ have cardinality the Narayana number $\frac{1}{n-3}{n-3 \choose k+1}{n-3 \choose k}$; generalizing these observations, they conjectured that for even $m$ the tilings of $A_{n, k,m}$ have cardinality the MacMahon number, the number of plane partitions which fit inside a $k \times (n-k-m) \times \frac{m}{2}$ box. We refer to this prediction as the `Magic Number Conjecture'. In this paper we prove the Magic Number Conjecture for the $m=2$ amplituhedron: that is, we show that each tiling of $A_{n,k,2}$ has cardinality ${n-2 \choose k}$. We prove this by showing that all positroid tilings of the hypersimplex $Δ_{k+1,n}$ have cardinality ${n-2 \choose k}$, then applying T-duality. In addition, we give combinatorial necessary conditions for tiles to form a tiling of $A_{n,k,2}$; we give volume formulas for Parke-Taylor polytopes and certain positroid polytopes in terms of circular extensions of cyclic partial orders; and we prove new variants of the classical Parke-Taylor identities. |
| title | The Magic Number Conjecture for the $m=2$ amplituhedron and Parke-Taylor identities |
| topic | Combinatorics High Energy Physics - Theory Mathematical Physics Algebraic Geometry 05E14, 52B20, 52B40, 52C40, 05A99 |
| url | https://arxiv.org/abs/2404.03026 |