Amplituhedron: Geometry of Scattering Amplitudes

• The amplituhedron is a geometric object that encodes scattering amplitudes in planar N=4 SYM theory using positive Grassmannian structures.
• Its construction employs canonical differential forms and pushforwards from cyclic polytopes, ensuring fundamental properties like locality and unitarity.
• Stratification and topology of the amplituhedron reveal intricate combinatorial insights, extending its applications to multi-loop and deformed calculations.

The amplituhedron is a geometric object that encodes scattering amplitudes in planar $\mathcal{N}=4$ supersymmetric Yang–Mills (SYM) theory as canonical differential forms on a positive region in a Grassmannian. The central principle is that geometric positivity and stratification in Grassmannian spaces subsume the analytic properties—locality, unitarity, and dual/conformal invariance—traditionally imposed in quantum field theory. Formally, the amplituhedron generalizes cyclic polytopes and the positive Grassmannian to a class of semialgebraic regions whose canonical volume forms reproduce, via explicit pushforward, the full integrand for (tree- and loop-level) scattering in planar $\mathcal{N}=4$ SYM (Arkani-Hamed et al., 2013). In what follows, the combinatorics, topology, stratification, convexity, and physical consequences of the amplituhedron are rigorously detailed.

1. Formal Definition and Construction

Let $G_+(k,n)$ denote the positive Grassmannian—the set of $k$-planes in $\mathbb{R}^n$ represented by $k\times n$ real matrices $C$ with all maximal $k\times k$ minors strictly positive. External data is given by an $(k+m)\times n$ matrix $Z$ whose maximal minors are strictly positive. The amplituhedron is the image

$\mathcal{N}=4$0

where $\mathcal{N}=4$1 is the Grassmannian of $\mathcal{N}=4$2-planes in $\mathcal{N}=4$3 (Arkani-Hamed et al., 2013). In the case $\mathcal{N}=4$4, this map is central to planar $\mathcal{N}=4$5 SYM.

For the extension to loop level with $\mathcal{N}=4$6 loops, one considers a family of $\mathcal{N}=4$7 (one for each loop), stacking them with $\mathcal{N}=4$8 into a $\mathcal{N}=4$9 matrix $G_+(k,n)$0, and forming $G_+(k,n)$1. Positivity must be imposed for all maximal minors of $G_+(k,n)$2, including mixed minors involving different $G_+(k,n)$3 and (optionally) $G_+(k,n)$4 rows (Franco et al., 2014).

In the $G_+(k,n)$5 case, the amplituhedron is equivalent to the complex of bounded faces for a cyclic hyperplane arrangement, and for $G_+(k,n)$6 it arises as a nonlinear analog of cyclic polytopes in $G_+(k,n)$7 (Karp et al., 2016, Koefler et al., 14 Jan 2025). For $G_+(k,n)$8, $G_+(k,n)$9 recovers the classical cyclic polytope (Arkani-Hamed et al., 2013).

2. Canonical Form and Physical Correspondence

For any positive geometry $k$0, the canonical form $k$1 is the unique (up to scale) top-degree rational differential form with logarithmic poles on all boundary divisors, such that the residue along any complete flag of boundaries yields $k$2 (Arkani-Hamed et al., 2013).

The integrand of planar $k$3 SYM is this canonical form on $k$4: $k$5 in local positive coordinates $k$6. The rational denominators $k$7 are affine-linear functions whose vanishing defines the boundary facets of the amplituhedron (Franco et al., 2014).

Alternatively, in bracket notation (e.g., for $k$8, $k$9, $\mathbb{R}^n$0),

$\mathbb{R}^n$1

with each bracket a $\mathbb{R}^n$2 or $\mathbb{R}^n$3 minor (Franco et al., 2014).

The poles of $\mathbb{R}^n$4 correspond exactly to co-dimension one boundaries of the amplituhedron, implemented by the vanishing of minors or bracket expressions, and with the numerator canceling would-be spurious poles incompatible with positivity (Franco et al., 2014, Arkani-Hamed et al., 2013).

3. Stratification and Boundary Structure

Every boundary of the amplituhedron is realized by sending a collection of Plücker minors to zero. The stratification is naturally two-staged:

• The initial decomposition ("$\mathbb{R}^n$5") is by positroid cells: determining which $\mathbb{R}^n$6 (or general $\mathbb{R}^n$7) minors vanish, as in the positroid stratification (Franco et al., 2014). For loops, this is the $\mathbb{R}^n$8-fold (product) structure, restricted by extended positivity.
• The secondary stratification ("$\mathbb{R}^n$9") is by vanishing non-minimal minors, subject to compatibility with Plücker relations and extended positivity.

Boundary labels record which minors are set to zero and determine the dimension. For example, in the four-point, $k\times n$0, two-loop case: $k\times n$1 with $k\times n$2 for $k\times n$3 down to $k\times n$4 respectively, totaling 1232 boundaries. The corresponding Euler characteristic is

$k\times n$5

(Franco et al., 2014).

Permutation labels provide an alternative combinatorial stratification corresponding to decorated permutations for positroid cells at tree level, and to more intricate combinatorial data for multi-loop amplituhedra.

4. Deformations, Topology, and Explicit Examples

A remarkable discovery is that, after formally relaxing all inter-minor constraints in stratification—specifically, allowing all non-minimal minors to be independently switched off—the "deformed amplituhedron" possesses a simplified topology: for $k\times n$6, $k\times n$7, the Euler characteristic for the deformed object is always 2 for $k\times n$8 loops. This suggests a conjectured uniform topological structure for the deformed positive Grassmannian in these cases, though the geometric reason remains elusive (Franco et al., 2014).

Explicit tabulations for two- and three-loop cases at low $k\times n$9 are given, with the Euler characteristic switching from $C$0 (undeformed) to $C$1 (deformed) at three loops, emphasizing the impact of stratification combinatorics on topology.

5. Amplituhedron–Integrand Correspondence

The structure of the canonical form is tightly controlled by the amplituhedron stratification. The poles of the integrand match the boundary strata ("labels") of the amplituhedron: each denominator corresponds to a minor vanishing and, through extended positivity and Plücker relations, allows systematic enumeration of all codimension strata both from the geometric and analytic sides (Franco et al., 2014).

This correspondence provides nontrivial tests of the amplituhedron/scattering amplitude duality, as the full list of boundaries and their multiplicities extracted from the integrand coincides with that derived geometrically via minors.

6. Alternative Descriptions and Theoretical Generalizations

The combinatorial and topological structure of the amplituhedron admits alternative characterizations:

• Sign variation and cyclic permutations: In the $C$2 amplituhedron, sign variation characterizes the cell decomposition and the topology (PL homeomorph to a closed ball), realized as the complex