126 problems

Let $\operatorname{Gr}_{k,n}^{\scriptscriptstyle\geq 0}$ be the positive Grassmannian, let its BCFW recurrence determine a collection of positroid cells, and let … be the amplituhe…

Let the tree-level amplitudes be decomposed into terms arising from BCFW recursions, and let $\operatorname{Gr}_{4,n}$ be the Grassmannian with its cluster algebra structure. A col…

Let $V(3,n)_{\geq 0}$ be the totally nonnegative part of the ABCT variety, and let the momentum amplituhedron be the positive geometry arising in the corresponding scattering-ampli…

Non-crossing chord diagram conjecture. The regions contributing to $A_n^{phi^4}$ are in bijection with the $C_{n/2-1}$ possible non-crossing chord diagrams. The region associated w…

Let $k\geq 1$ and $n\geq k+4$. Let $\mathcal{A}_{n,k,4}$ be the amplituhedron, and let $\mathcal{BCFW}_{n,k}$ denote the collection of $4k$-dimensional BCFW positroid cells in the…

Bern–Carrasco–Johansson colour–kinematics duality conjecture. There exists a choice of kinematic numerators for the trivalent diagrams entering $\scA_{n,L}$ such that whenever a tr…

The amplituhedron is a geometric region in momentum-twistor space whose canonical form is intended to produce scattering amplitudes at tree level and loop integrands at loop level.…

Let the on-mass-shell momentum-space scattering amplitudes of gravity and gluon scattering amplitudes in Yang–Mills theory be considered in perturbation theory. Gravity double-copy…

Pochhammer-like contour representation conjecture. The real integral form of $B_N$ can always be expressed alternatively by a Pochhammer-like contour integral in $M^c$.

Let $M_n^{(L)}$ denote the planar $n$-point scattering amplitude of maximally supersymmetric ${N}=4$ Yang–Mills theory at $L$ loops, normalized by the corresponding tree amplitude.…

Let a scattering problem in particle physics determine an appropriate moduli space and differential forms on it. A scattering amplitude, or correlation function, is expected to be…

Let $\Gamma_{L^\ast}$ be a graph in the class of loop BCFW graphs $\mathbf{\Gamma}^{\mathtt{BCFW}}_{k,n;L^\ast}$, and let … be the corresponding branch of the loop BCFW recursion.…

Let $\mathcal{M}_{k,n;L}$ and $\mathcal{A}_{k-2,n;L}$ denote respectively the $m=4$ momentum and momentum-twistor amplituhedra for $n$ particles, helicity $k$, and loop order…

The amplituhedron is the geometric space arising from the positive Grassmannian in the scattering-amplitude construction, and the BCFW recurrence relations are the Britto--Cachazo-…

Let $G$ be a finite graph, let $\boldsymbol{X}$ and $\boldsymbol{Y}$ denote the variables of its flat space wavefunction $\psi_G$, and let $\Omega_G$ be the canonical form of the c…

Let $n,k,m$ be positive integers with $k+m\leq n$, and let $Z$ be a generic $n\times(k+m)$ matrix all of whose minors are positive. Write $\mathcal{A}_{n,k,m}(Z)$ for the image of…

Reality conjecture. Under that positivity hypothesis, all leading singularities have real coordinates.

Amplituhedron canonical-form conjecture. When expressed in terms of lines in $\mathbb{P}^3$, the rational function is conjectured to be the canonical form of the positive geometry…

Let $A_{n,k,m}$ be the amplituhedron associated to a positive map $Z:{\mathbb{R}}^n\to{\mathbb{R}}^{k+m}$. A positive geometry is a pair $(X,X_{\geq 0})$ equipped with a canonical…

Parametric-region conjecture. For any $n$, the region ${\cal R}_{1}^{\alpha}(n)$ is parametrized by the following sets of inequalities:

Let $f(\mathbf{z})$ be a rational function defining a singularity hypersurface of a planar $\mathcal{N}=4$ super Yang–Mills scattering amplitude, and let…

Let $\operatorname{Gr}^{>0}_{4,n}$ be the totally positive Grassmannian, and let $f(\mathbf{z})=0$ describe a singularity hypersurface of a scattering amplitude. Scattering-amplitu…

Let $d538_{k,4,n}(Z)$ denote the tree amplituhedron associated with positive data $Z$. A positive geometry is a complex algebraic variety defined over $d53f$, together with a semia…

The symbol bootstrap studies scattering amplitudes through their singularities, collected into an alphabet, and a Grassmannian cluster algebra is the cluster algebra associated wit…

In Yang–Mills scattering amplitudes, consider the sign factors arising when amplitudes are Fourier transformed from momentum space to real twistor space for split-signature space-t…