Séminaire Lotharingien de Combinatoire, 91B.85 (2024), 12 pp.
Chaim Even-Zohar, Tsviqa Lakrec, Matteo Parisi, Melissa Sherman-Bennett, Ran Tessler and Lauren Williams
Cluster Algebras and Tilings for The m = 4 Amplituhedron
Abstract.
The amplituhedron AZn,k,m is the
image of the
positive Grassmannian Grk,n≥ 0 under the
map Z~: Grk,n≥
0 -> Grk,k+m
induced by a positive linear map
Z : Rn -> Rk+m.
It was originally introduced in physics in order to give a geometric
interpretation of scattering amplitudes. More specifically, one can
compute scattering amplitudes in N=4 SYM by
`tiling' the m=4
amplituhedron AZn,k,4
- that is, decomposing
AZn,k,4 into
`tiles' (closures of images of 4k-dimensional cells of
Grk,n≥ 0
on which Z~ is injective).
In this article we deepen both our understanding of tiles and tilings
of the m=4 amplituhedron and the connection with cluster algebras.
Firstly, we prove
the cluster adjacency conjecture for BCFW tiles of
AZn,k,4, which says that
facets of tiles are cut out by collections of compatible cluster variables
for Gr4,n. Secondly, we describe each BCFW tile as the
semialgebraic set in Grk,k+4 where certain cluster
variables have particular signs.
Finally, we prove
the BCFW tiling conjecture, which says that any way of
iterating the BCFW recurrence
gives rise to a tiling of the amplituhedron
AZn,k,4.
Along the way, we introduce a method to construct seeds for
Gr4,n comprised of
high-degree cluster variables, which may be of independent interest in the study of
cluster algebras.
Received: November 15, 2023.
Accepted: February 15, 2024.
Final version: April 1, 2024.
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