arXiv subjects

Nick Early

Publications and source records attributed to Nick Early.

At least 19 recordsLinked to original sources

arXiv2026

Noncrossing Duality and the Geometry of Positive Tropical Linear Spaces

While the positive Grassmannian is deeply understood through the rich combinatorics of plabic graphs and positroid cells, its tropical counterpart, the positive tropical Grassmannian Trop$_{>0}G(k,n)$, has lacked a comparable structural framework for general $k$. Both the global face structure of Trop$_{>0}G(k,n)$ and the internal metric geometry of the tropical linear spaces it parametrizes have remained largely uncharted. This paper develops a systematic algebraic and polyhedral foundation that resolves this gap. The engine of our framework is a fundamental tropical duality, analogous to the duality between cluster variables (or more precisely, their $u$-coordinates) and $\mathbf{g}$-vectors, pairing two families of objects introduced by the first author: the planar basis of tropical Plücker vectors and the planar cross-ratios on the positive configuration space. We prove that this duality links the fan structure of the positive tropical Grassmannian to the noncrossing fan of Santos, Stump, and Welker, yielding a global bijection between integer points of $Trop_{>0}G(k,n)$ and noncrossing tableaux. We then study how this discrete combinatorial data controls the continuous metric geometry of positive tropical linear spaces. We realize the bounded complex of an integer positive tropical linear space as the subdifferential of a central roof function on the hypersimplex, and use this realization to embed it into a dilate of the fundamental alcoved simplex. The dilation factor, and hence the geometric diameter of the complex, is governed by a single invariant, the planar kinematics ($K) weight, which we show equals the number of columns in the associated noncrossing tableau. The results of this work are applied in our parallel work on scaffolds for higher tropical Grassmannians.

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arXiv2026

Scaffolds for Higher Tropical Grassmannians: Foundations

Scaffolds are the one-dimensional skeleta of high-dimensional flag simplicial complexes of nonpositive curvature. They generalize the phylogenetic trees of Trop G(2,n) to arbitrary $k$, drawing together SL(k)-web bases, affine buildings, the combinatorics of the positive tropical Grassmannian and low-dimensional topology. We prove that scaffolds model points in all tropical Grassmannians via a $k$-point distance function. In this paper, we study in detail CAT(0) planar graphs, which are positive scaffolds for the tropical Grassmannian of three-planes. CAT(0) planar graphs are directed versions of the diskoids of Fontaine-Kamnitzer-Kuperberg, planar dual to SL(3)-webs. Our main result is the construction of a unique representation of any given integer positive tropical Plucker vector by a normal CAT(0) planar graph. We show that any normal CAT(0) planar graph embeds into the tropical linear space as a Lam-Postnikov membrane, and embeds into the Keel-Tevelev membrane within the affine building. We show that Early's planar basis expansion can be computed directly from the strand combinatorics of the dual web, and connect this expansion to Petersen-Pylyavskyy-Speyer's noncrossing tableaux, explored further in our companion paper.

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arXiv2026

Positive del Pezzo Geometry

Real, complex, and tropical algebraic geometry join forces in a new branch of mathematical physics called positive geometry. We develop the positive geometry of del Pezzo surfaces and their moduli spaces, viewed as very affine varieties. Their connected components are derived from polyhedral spaces with Weyl group symmetries. We study their canonical forms and scattering amplitudes, and we solve the likelihood equations.

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arXiv2026

Classification of prime modules of quantum affine algebras corresponding to 2-column tableaux

Finite dimensional simple modules of quantum affine algebras of type A correspond to semistandard Young tableaux of rectangular shapes. In this paper, we classify all prime modules corresponding to 2-column semistandard Young tableaux, up to a conjectural property. Moreover, we give a conjectural sufficient condition for a module corresponding to a tableau with more than two columns to be prime.

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arXiv2025

When alcoved polytopes add

Alcoved polytopes are characterized by the property that all facet normal directions are parallel to the roots $e_i-e_j$. Unlike other prominent families of polytopes, like generalized permutahedra, alcoved polytopes are not closed under Minkowski sums. We nonetheless show that the Minkowski sum of a collection of alcoved polytopes is alcoved if and only if each pairwise sum is alcoved. This implies that the type fan of alcoved polytopes is determined by its two-dimensional cones. Moreover, we provide a complete characterization of when the Minkowski sum of alcoved simplices is again alcoved via a graphical criterion on pairs of ordered set partitions. Our characterization reduces to checking conditions on restricted partitions of length at most six. In particular, we show how the Minkowski sum decompositions of the two most well-known families of alcoved polytopes, the associahedron and the cyclohedron, fit in our framework. Additionally, inspired by the physical construction of one-loop scattering amplitudes, we present a new infinite family of alcoved polytopes, called $\widehat{D}_n$ polytopes. We conclude by drawing a connection to matroidal blade arrangements and the Dressian.

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arXiv2025

The CEGM NLSM

Studying quantum field theories through geometric principles has revealed deep connections between physics and mathematics, including the discovery by Cachazo, Early, Guevara and Mizera (CEGM) of a generalization of biadjoint scalar amplitudes. However, extending this to generalizations of other quantum field theories remains a central challenge. Recently it has been discovered that the nonlinear sigma model (NLSM) emerges after a certain zero-preserving deformation from $\text{tr}(ϕ^3)$. In this work, we find a much richer story of zero-preserving deformations in the CEGM context, yielding generalized NLSM amplitudes. We prove an explicit formula for the residual embedding of an $n$-point NLSM amplitude in a mixed $n+2$ point generalized NLSM amplitude, which provides a strong consistency check on our generalization. We show that the dimension of the space of pure kinematic deformations is $\gcd(k,n)-1$, we introduce a deformation-compatible modification of the Global Schwinger Parameterization, and we include a new proof, using methods from matroidal blade arrangements, of the linear independence for the set of planar kinematic invariants for CEGM amplitudes. Our framework is compatible with string theory through recent generalizations of the Koba-Nielsen string integral to any positive configuration space $X^+(k,n)$, where the usual Koba-Nielsen string integral corresponds to $X(2,n) = \mathcal{M}_{0,n}$.

hep-th↗

arXiv2025

Honeycomb Tessellations and Graded Permutohedral Blades

This paper investigates enumerative aspects of permutohedral blades, which provide a generalization of the notion of the tropical hyperplane arrangement. Blade provide the combinatorial underpinning of generalized biadjoint scalar scattering amplitudes in work of Cachazo, Early, Guevara and Mizera (CEGM). We construct a graded basis for a vector space of indicator functions of blades, with the grading determined by the dimension of the support of the function. We prove a Minkowski sum decomposition law into lines and tripods and we explore connections to the cohomology ring of certain moduli spaces and a non-planar analog of the square move for plabic graphs. We give a closed formula for the graded dimension of the basis. It is shown in an Appendix by Donghyun Kim that the coefficients appearing in the numerator of the generating function for the graded dimension are symmetric, and that they sum to $\frac{(2j)!}{j!}$. Unimodality is still open.

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arXiv2024

The Chirotropical Grassmannian

Recent developments in particle physics have revealed deep connections between scattering amplitudes and tropical geometry. From the heart of this relationship emerged the chirotropical Grassmannian $\text{Trop}^χ\text{G}(k,n)$ and the chirotropical Dressian $\text{Dr}^χ(k,n)$, polyhedral fans built from uniform realizable chirotopes that encode the combinatorial structure of Generalized Feynman Diagrams. We prove that $\text{Trop}^χ\text{G}(3,n) = \text{Dr}^χ(3,n)$ for $n = 6,7,8$, and develop algorithms to compute these objects from their rays modulo lineality. Using these algorithms, we compute all chirotropical Grassmannians $\text{Trop}^χ\text{G}(3,n)$ for $n = 6,7,8$ across all isomorphism classes of chirotopes. We prove that each chirotopal configuration space $X^χ(3,6)$ is diffeomorphic to a polytope and propose an associated canonical logarithmic differential form. Finally, we show that the equality between chirotropical Grassmannian and Dressian fails for $(k,n) = (4,8)$.

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arXiv2024

Generalized permutohedra in the kinematic space

In this note, we study the permutohedral geometry of the poles of a certain differential form introduced in recent work of Arkani-Hamed, Bai, He and Yan. There it was observed that the poles of the form determine a family of polyhedra which have the same face lattice as that of the permutohedron. We realize that family explicitly, proving that it in fact fills out the configuration space of a particularly well-behaved family of generalized permutohedra, the zonotopal generalized permutohedra, that are obtained as the Minkowski sums of line segments parallel to the root directions $e_i-e_j$. Finally we interpret Mizera's formula for the biadjoint scalar amplitude $m(\mathbb{I}_n,\mathbb{I}_n)$, restricted to a certain dimension $n-2$ subspace of the kinematic space, as a sum over the boundary components of the standard root cone, which is the conical hull of the roots $e_1-e_2,\ldots, e_{n-2}-e_{n-1}$.

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arXiv2024

Tropical Geometry, Quantum Affine Algebras, and Scattering Amplitudes

The goal of this paper is to make a connection between tropical geometry, representations of quantum affine algebras, and scattering amplitudes in physics. The connection allows us to study important and difficult questions in these areas: (1) We give a systematic construction of prime modules (including prime non-real modules) of quantum affine algebras using tropical geometry. We also introduce new objects which generalize positive tropical Grassmannians. (2) We propose a generalization of Grassmannian string integrals in physics, in which the integrand is no longer a finite, but rather an infinite product indexed by prime modules of a quantum affine algebra. We give a general formula of $u$-variables using prime tableaux (corresponding to prime modules of quantum affine algebras of type $A$) and Auslander-Reiten quivers of Grassmannian cluster categories. (3) We study limit $g$-vectors of cluster algebras. This is another way to obtain prime non-real modules of quantum affine algebras systematically. Using limit $g$-vectors, we construct new examples of non-real modules of quantum affine algebras.

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arXiv2024

Color-Dressed Generalized Biadjoint Scalar Amplitudes: Local Planarity

The biadjoint scalar theory has cubic interactions and fields transforming in the biadjoint representation of ${\rm SU}(N)\times {\rm SU}\big({\tilde N}\big)$. Amplitudes are "color" decomposed in terms of partial amplitudes computed using Feynman diagrams which are simultaneously planar with respect to two orderings. In 2019, a generalization of biadjoint scalar amplitudes based on generalized Feynman diagrams (GFDs) was introduced. GFDs are collections of Feynman diagrams derived by incorporating an additional constraint of "local planarity" into the construction of the arrangements of metric trees in combinatorics. In this work, we propose a natural generalization of color orderings which leads to color-dressed amplitudes. A generalized color ordering (GCO) is defined as a collection of standard color orderings that is induced, in a precise sense, from an arrangement of projective lines on $\mathbb{RP}^2$. We present results for $n\leq 9$ generalized color orderings and GFDs, uncovering new phenomena in each case. We discover generalized decoupling identities and propose a definition of the "colorless" generalized scalar amplitude. We also propose a notion of GCOs for arbitrary $\mathbb{RP}^{k-1}$, discuss some of their properties, and comment on their GFDs. In a companion paper, we explore the definition of partial amplitudes using CEGM integral formulas.

hep-th↗

arXiv2024

Minimal Kinematics on $\mathcal{M}_{0,n}$

Minimal kinematics identifies likelihood degenerations where the critical points are given by rational formulas. These rest on the Horn uniformization of Kapranov-Huh. We characterize all choices of minimal kinematics on the moduli space $\mathcal{M}_{0,n}$. These choices are motivated by the CHY model in physics and they are represented combinatorially by 2-trees. We compute 2-tree amplitudes, and we explore extensions to non-planar on-shell diagrams, here identified with the hypertrees of Castravet-Tevelev.

math.AG↗

arXiv2023

Moduli Space Tilings and Lie-Theoretic Color Factors

A detailed understanding of the moduli spaces $X(k,n)$ of $n$ points in projective $k-1$ space is essential to the investigation of generalized biadjoint scalar amplitudes, as discovered by Cachazo, Early, Guevara and Mizera (CEGM) in 2019. But in math, conventional wisdom says that it is completely hopeless due to the arbitrarily high complexity of realization spaces of oriented matroids. In this paper, we nonetheless find a path forward. We present a Lie-theoretic realization of color factors for color-dressed generalized biadjoint scalar amplitudes, formulated in terms of certain tilings of the real moduli space $X(k,n)$ and collections of logarithmic differential forms, resolving an important open question from recent work by Cachazo, Early and Zhang. The main idea is to replace the realization space decomposition of $X(k,n)$ with a large class of overlapping tilings whose topologies are individually relatively simple. So we obtain a collection of color-dressed amplitudes, each of which satisfies $U(1)$ decoupling separately. The essential complexity appears when they are all superposed.

hep-th↗

arXiv2023

Generalized Color Orderings: CEGM Integrands and Decoupling Identities

In a recent paper, we defined generalized color orderings (GCO) and Feynman diagrams (GFD) to compute color-dressed generalized biadjoint amplitudes. In this work, we study the Cachazo-Early-Guevara-Mizera (CEGM) representation of generalized partial amplitudes and ``decoupling" identities. This representation is a generalization of the Cachazo-He-Yuan (CHY) formulation as an integral over the configuration space $X(k,n)$ of $n$ points on $\mathbb{CP}^{k-1}$ in generic position. Unlike the $k=2$ case, Parke-Taylor-like integrands are not enough to compute all partial amplitudes for $k>2$. Here we give a set of constraints that integrands associated with GCOs must satisfy and use them to construct all $(3,n<9)$ integrands, all $(3,9)$ integrands up to four undetermined constants, and $95 \%$ of $(4,8)$ integrands up to 24 undetermined constants. $k=2$ partial amplitudes are known to satisfy identities. Among them, the so-called $U(1)$ decoupling identities are the simplest ones. These are characterized by a label $i$ and a color ordering in $X(2,|[n]\setminus \{i\}|)$. Here we introduce decoupling identities for $k>2$ determined combinatorially using GCOs. Moreover, we identify the natural analog of $U(1)$ identities as those characterized by a pair of labels $i\neq j$, and a pair of GCOs, one in $X(k,|[n]\setminus \{i\}|)$ and the other in $X(k-1,|[n]\setminus \{j\}|)$. We call them {\it double extension} identities. We also provide explicit connections among different ways of representing GCOs, such as configurations of lines, configurations of points, and reorientation classes of uniform oriented matroids (chirotopes).

hep-th↗

arXiv2023

Biadjoint Scalars and Associahedra from Residues of Generalized Amplitudes

In the Grassmannian formulation of the S-matrix for planar $\mathcal{N}=4$ Super Yang-Mills, $N^{k-2}MHV$ scattering amplitudes for $k$ negative and $n-k$ positive helicity gluons can be expressed, by an application of the global residue theorem, as a signed sum over a collection of $(k-2)(n-k-2)$-dimensional residues. These residues are supported on certain positroid subvarieties of the Grassmannian $G(k,n)$. In this paper, we replace the Grassmannian $G(3,n)$ with its torus quotient, the moduli space of $n$ points in the projective plane in general position, and planar $\mathcal{N}=4$ SYM with generalized biadjoint scalar amplitudes $m^{(3)}_n$ as introduced by Cachazo-Early-Guevara-Mizera (CEGM). Whereas in the Grassmannian formulation residues of the Parke-Taylor form correspond to individual BCFW, or on-shell diagrams, we show that each such $(n-5)$-dimensional residue of $m^{(3)}_n$ is an entire biadjoint scalar partial amplitude $m^{(2)}_n$, that is a sum over all tree-level Feynman diagrams for a fixed planar order. We propose a generalization which would give rise to identifications of $m^{(2)}_n$ inside $m^{(k)}_n$ for $k\ge 4$, via $(k-2)(n-k-2)$-dimensional residues. Our proof for $k=3$ uses the CEGM formula for $m^{(3)}_n$; it predicts a new Minkowski sum realization of the associahedron in terms of certain positroid polytopes in the second hypersimplex $Δ_{2,n}$.

hep-th↗

arXiv2023

Factorization for Generalized Biadjoint Scalar Amplitudes via Matroid Subdivisions

We study the problem of factorization for residues of generalized biadjoint scalar scattering amplitudes $m^{(k)}_n$, introduced by Cachazo, Early, Guevara and Mizera (CEGM), involving multi-dimensional residues which factorize generically into $k$-ary products of lower-point generalized biadjoint amplitudes of the same type $m^{(k)}_{n_1}\cdots m^{(k)}_{n_k}$, where $n_1+\cdots +n_k = n+k(k-1)$, noting that smaller numbers of factors arise as special cases. Such behavior is governed geometrically by regular matroid subdivisions of hypersimplices and cones in the positive tropical Grassmannian, and combinatorially by collections of compatible decorated ordered set partitions, considered modulo cyclic rotation. We make a proposal for conditions under which this happens and we develop $k=3,4$ in detail. We conclude briefly to propose a novel formula to construct coarsest regular matroid subdivisions of all hypersimplices $Δ_{k,n}$ and rays of the positive tropical Grassmannian, which should be of independent interest.

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arXiv2022

Weighted blade arrangements and the positive tropical Grassmannian

In this paper, we continue our study of blade arrangements and the positroidal subdivisions which are induced by them on $Δ_{k,n}$. A blade is a tropical hypersurface which is generated by a system of $n$ affine simple roots of type $SL_n$ that enjoys a cyclic symmetry. When placed at the center of a simplex, a blade induces a decomposition into $n$ maximal cells which are known as Pitman-Stanley polytopes. We introduce a complex $(B_{k,n},\partial)$ of weighted blade arrangements and we prove that the positive tropical Grassmannian surjects onto the top component of the complex, such that the induced weights on blades in the faces $Δ_{2,n-(k-2)}$ of $Δ_{k,n}$ are (1) nonnegative and (2) their support is weakly separated. We finally introduce a hierarchy of elementary weighted blade arrangements for all hypersimplices which is minimally closed under the boundary maps $\partial$, and apply our result to classify up to isomorphism type all rays of the positive tropical Grassmannian $\text{Trop}_+ G(3,n)$ for $n\le 9$.

math.CO↗

arXiv2022

Planarity in Generalized Scattering Amplitudes: PK Polytope, Generalized Root Systems and Worldsheet Associahedra

In this paper we study the role of planarity in generalized scattering amplitudes, through several closely interacting structures in combinatorics, algebraic and tropical geometry. The generalized biadjoint scalar amplitude, introduced recently by Cachazo-Early-Guevara-Mizera (CEGM), is a rational function of homogeneous degree $-(k-1)(n-k-1)$ in $\binom{n}{k}-n$ independent variables; its poles can be constructed directly from the rays of the positive tropical Grassmannian. We introduce for each pair of integers $(k,n)$ with $2\le k\le n-2$ a system of generalized positive roots which arises as a specialization of the planar basis of kinematic invariants. We prove that the higher root polytope $\mathcal{R}^{(k)}_{n-k}$ has volume the k-dimensional Catalan number $C^{(k)}_{n-k}$, via a flag unimodular triangulation into simplices, in bijection with noncrossing collections of $k$-element subsets. We also give a bijection between certain positroidal subdivisions, called tripods, of the hypersimplex $Δ_{3,n}$ and noncrossing pairs of 3-element subsets that are not weakly separated. We show that the facets of the Planar Kinematics (PK) polytope, introduced recently by Cachazo and the author, are exactly the $\binom{n}{k}-n$ generalized positive roots. We show that the PK specialization of the generalized biadjoint amplitude evaluates to $C^{(k)}_{n-k}$. Looking forward, we give defining equations and conjecture explicit solutions using $(\mathbb{CP}^{n-k-1})^{\times (k-1)}$ via a notion of compatibility degree for noncrossing collections, for a two parameter family of generalized worldsheet associahedra $\mathcal{W}^+_{k,n}$. These specialize when $k=2$ to a certain dihedrally invariant partial compactification of the configuration space $M_{0,n}$ of $n$ distinct points in $\mathbb{CP}^{1}$. Many detailed examples are given throughout to motivate future work.

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