Related papers: Amplituhedra and origami

We study the planar orthogonal drawing style within the framework of partial representation extension. Let $(G,H,{\Gamma}_H )$ be a partial orthogonal drawing, i.e., G is a graph, $H\subseteq G$ is a subgraph and ${\Gamma}_H$ is a planar…

This paper concerns random bipartite planar maps which are defined by assigning weights to their faces. The paper presents a threefold contribution to the theory. Firstly, we prove the existence of the local limit for all choices of weights…

The all-loop integrand for scattering amplitudes in planar N = 4 SYM is determined by an "amplitude form" with logarithmic singularities on the boundary of the amplituhedron. In this note we provide strong evidence for a new striking…

The Barrett-Crane models of Lorentzian quantum gravity are a family of spin foam models based on the Lorentz group. We show that for various choices of edge and face amplitudes, including the Perez-Rovelli normalization, the amplitude for…

We initiate the systematic study of \emph{local positive spaces} which arise in the context of the Amplituhedron construction for scattering amplitudes in planar maximally supersymmetric Yang-Mills theory. We show that all local positive…

A drawing in the plane ($\mathbb{R}^2$) of a graph $G=(V,E)$ equipped with a function $\gamma: V \rightarrow \mathbb{N}$ is \emph{$x$-bounded} if (i) $x(u) <x(v)$ whenever $\gamma(u)<\gamma(v)$ and (ii) $\gamma(u)\leq\gamma(w)\leq…

The amplituhedron $A_{n,k,m}(Z)$ is the image of the positive Grassmannian $Gr_{k,n}^{\geq 0}$ under the map ${Z}: Gr_{k,n}^{\geq 0} \to Gr_{k,k+m}$ induced by a positive linear map $Z:\mathbb{R}^n \to \mathbb{R}^{k+m}$. Motivated by a…

The Gram spectrahedron $\text{Gram}(f)$ of a form $f$ with real coefficients parametrizes the sum of squares decompositions of $f$, modulo orthogonal equivalence. For $f$ a sufficiently general positive binary form of arbitrary degree, we…

For weighted $L^1$ space on the unit sphere of $\RR^{d+1}$, in which the weight functions are invariant under finite reflection groups, a maximal function is introduced and used to prove the almost everywhere convergence of orthogonal…

Rigidly and flat-foldable quadrilateral mesh origami is the class of quadrilateral mesh crease patterns with one fundamental property: the patterns can be folded from flat to fully-folded flat by a continuous one-parameter family of…

The geometric structure of S-matrix encapsulated by the "Amplituhedron program" has begun to reveal itself even in non-supersymmetric quantum field theories. Starting with the seminal work of Arkani-Hamed, Bai, He and Yan it is now…

In this note we show that $\beta'_*$, the $\beta$-function slopes in the electric and magnetic theories are equal at the corresponding infrared fixed points. This follows from the scaling of the correlators of the trace of the energy…

In this paper we study the orthogonal momentum amplituhedron $\mathcal{O}_k$, a recently introduced positive geometry that encodes the tree-level scattering amplitudes in ABJM theory. We generate the full boundary stratification of…

Consider an oriented curve $\Gamma$ in a domain $D$ in the plane $\boldsymbol R^2$. Thinking of $D$ as a piece of paper, one can make a curved folding in the Euclidean space $\boldsymbol R^3$. This can be expressed as the image of an…

We consider the conjecture by Aichholzer, Aurenhammer, Hurtado, and Krasser that any two points sets with the same cardinality and the same size convex hull can be triangulated in the "same" way, more precisely via \emph{compatible…

In this paper, we will show methods to interpret some rigid origami with higher degree vertices as the limit case of structures with degree-4 supplementary angle vertices. The interpretation is based on separating each crease into two…

We build upon the prior works of [1-3] to study tree-level planar amplitudes for a massless scalar field theory with polynomial interactions. Focusing on a specific example, where the interaction is given by $\lambda_3\phi^{3}\ +\lambda_4…

Let $F$ be a totally real field and $K$ a finite abelian CM extension of $F$. Using class field theory, we show that our previous result giving a strong form of the Brumer-Stark conjecture implies the minus part of the equivariant Tamagawa…

In the Horndeski's most general scalar-tensor theories, we derive the three-point correlation function of scalar non-Gaussianities generated during single-field inflation in the presence of slow-variation corrections to the leading-order…

We extend the argument presented by Benincasa, Boucher-Veronneau, and Cachazo to show that graviton tree amplitudes are well behaved under large complex deformations of the momenta of a pair of like-helicity gravitons. This shows that BCFW…