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Triangulations and Canonical Forms of Amplituhedra: a fiber-based approach beyond polytopes | | Fatemeh Mohammadi
; Leonid Monin
; Matteo Parisi
; | | Date: |
14 Oct 2020 | | Abstract: | Any totally positive $(k+m) imes n$ matrix induces a map $pi_+$ from the
positive Grassmannian $Gr_+(k,n)$ to $Gr(k,k+m)$, whose image is the
amplituhedron $A_{n,k,m}$ and is endowed with a top-degree form called the
canonical form ${fOmega}(A_{n,k,m})$. This construction was introduced by
Arkani-Hamed and Trnka, where they showed that ${fOmega}(A_{n,k,4})$
encodes scattering amplitudes in $mathcal{N}=4$ super Yang-Mills theory.
Moreover, the computation of ${fOmega}(A_{n,k,m})$ is reduced to finding
the triangulations of $A_{n,k,m}$. However, while triangulations of polytopes
are fully captured by their secondary and fiber polytopes, the study of
triangulations of objects beyond polytopes is still underdeveloped.
We initiate the geometric study of subdivisions of $A_{n,k,m}$ in order to
establish the notion of secondary amplituhedron. For this purpose, we provide a
concrete birational parametrization of fibers of the projection $pi:
Gr(k,n)dasharrow mathcal{A}_{n,k,m}$. We then use this to explicitly
describe a rational top-degree form $omega_{n,k,m}$ (with simple poles) on the
fibers and compute ${fOmega}(A_{n,k,m})$ as a summation of certain residues
of $omega_{n,k,m}$. As main application of our approach, we develop a
well-structured notion of secondary amplituhedra for conjugate to polytopes,
i.e.~when $n-k-1=m$. We show that, in this case, each fiber of $pi$ is
parameterized by a projective space and its volume form $omega_{n,k,m}$ has
only poles on a hyperplane arrangement. Using such linear structures, for
amplituhedra which are cyclic polytopes or conjugate to polytopes, we show that
the Jeffrey-Kirwan residue computes ${fOmega}(A_{n,k,m})$ from the fiber
volume form $omega_{n,k,m}$. Finally, we propose a more general framework of
fiber positive geometries and analyze new families of examples such as fiber
and Grassmann polytopes. | | Source: | arXiv, 2010.07254 | | Services: | Forum | Review | PDF | Favorites | |
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