Anatomy of the Amplituhedron

TL;DR

This work advances the geometry of the amplituhedron by developing a detailed stratification framework for loop-based, four-point amplitudes in planar $\mathcal{N}=4$ SYM, introducing mini and full stratifications, and a combinatorial implementation via graphs and hyper perfect matchings that elegantly encode extended positivity. It demonstrates a precise correspondence between geometric boundaries and integrand singularities, including a complete two-loop analysis and initial three-loop explorations, providing strong support for the amplituhedron conjecture. A key novelty is the deformed amplituhedron, in which relaxed relations among non-minimal minors yield surprisingly simple topologies across loop orders, suggesting new ways to triangulate and understand amplitudes. The results deepen the link between positivity, permutations, and combinatorics, and point toward broad future directions, such as extensions to higher $k$ and $n$, connections to cluster algebras, and potential applications to ABJM-like theories.

Abstract

We initiate a comprehensive investigation of the geometry of the amplituhedron, a recently found geometric object whose volume calculates the integrand of scattering amplitudes in planar N=4 SYM theory. We do so by introducing and studying its stratification, focusing on four-point amplitudes. The new stratification exhibits interesting combinatorial properties and positivity is neatly captured by permutations. As explicit examples, we find all boundaries for the two and three loop amplitudes and related geometries. We recover the stratifications of some of these geometries from the singularities of the corresponding integrands, providing a non-trivial test of the amplituhedron/scattering amplitude correspondence. We finally introduce a deformation of the stratification with remarkably simple topological properties.

Figures (15)

• Figure 1: A natural decomposition of the poset associated to the stratification. $\Gamma_0$ corresponds to $2\times 2$ minors and $\Gamma_1$ corresponds to non-minimal ones.
• Figure 2: Boundaries of $G_{+}(2,4)$. The parentheses indicate which Plücker coordinates are turned on. The top level has all 6 coordinates turned on and has dimension 4, the bottom level has only one coordinate turned on and has dimension 0.
• Figure 3: Boundary structure of $G_{+}(2,4)$ and the graphs associated to each boundary. For each graph we indicate the set of non-vanishing Plücker coordinates.
• Figure 4: The seven perfect matchings for the bipartite graph associated to the top-dimensional cell of $G_+(2,4)$. Edges in the perfect matchings are shown in red. The graph is embedded into a disk, whose boundary is shown in gray.
• Figure 5: Perfect orientations corresponding to the perfect matchings shown in Figure \ref{['fig:sqbpms']}. The edges of the perfect matchings are shown in red, the source set is labeled underneath each graph in green and the Plücker coordinate associated to each perfect flow is in blue. The last two perfect orientations have the same sources and hence contribute to the same Plücker coordinate.