Tropical Grassmannians Research Articles

Planar matrices and arrays of Feynman diagrams: poles for higher k

Planar arrays of tree diagrams were introduced as a generalization of Feynman diagrams that enable the computation of biadjoint amplitudes mn(k) for k > 2. In this follow-up work, we investigate the poles of mn(k) from the perspective of such arrays. For general k, we characterize the underlying polytope as a Flag Complex and propose a computation of the amplitude-based solely on the knowledge of the poles, whose number is drastically less than the number of the full arrays. As an example, we first provide all the poles for the cases (k, n) = (3, 7), (3, 8), (3, 9), (3, 10), (4, 8) and (4, 9) in terms of their planar arrays of degenerate Feynman diagrams. We then implement simple compatibility criteria together with an addition operation between arrays and recover the full collections/arrays for such cases. Along the way, we implement hard and soft kinematical limits, which provide a map between the poles in kinematic space and their combinatoric arrays. We use the operation to give a proof of a previously conjectured combinatorial duality for arrays in (k, n) and (n − k, n). We also outline the relation to boundary maps of the hypersimplex Δ k,n and rays in the tropical Grassmannian Tr(k,n) .

Massively parallel computation of tropical varieties, their positive part, and tropical Grassmannians

We present a massively parallel framework for computing tropicalizations of algebraic varieties which can make use of symmetries using the workflow management system GPI-Space and the computer algebra system Singular. We determine the tropical Grassmannian TGr0(3,8). Our implementation works efficiently on up to 840 cores, computing the 14763 orbits of maximal cones under the canonical S8-action in about 20 minutes.Relying on our result, we show that the Gröbner structure of TGr0(3,8) refines the 16-dimensional skeleton of the coarsest fan structure of the Dressian Dr(3,8), except for 23 orbits of special cones, for which we construct explicit obstructions to the realizability of their tropical linear spaces. Moreover, we propose algorithms for identifying maximal-dimensional cones which belong to positive tropicalizations of algebraic varieties. We compute the positive Grassmannian TGr+(3,8) and compare it to the cluster complex of the classical Grassmannian Gr(3,8).

The Positive Tropical Grassmannian, the Hypersimplex, and them= 2 Amplituhedron

AbstractThe positive Grassmannian $Gr^{\geq 0}_{k,n}$ is a cell complex consisting of all points in the real Grassmannian whose Plücker coordinates are non-negative. In this paper we consider the image of the positive Grassmannian and its positroid cells under two different maps: the moment map$\mu $ onto the hypersimplex [ 31] and the amplituhedron map$\tilde{Z}$ onto the amplituhedron [ 6]. For either map, we define a positroid dissection to be a collection of images of positroid cells that are disjoint and cover a dense subset of the image. Positroid dissections of the hypersimplex are of interest because they include many matroid subdivisions; meanwhile, positroid dissections of the amplituhedron can be used to calculate the amplituhedron’s ‘volume’, which in turn computes scattering amplitudes in $\mathcal{N}=4$ super Yang-Mills. We define a map we call T-duality from cells of $Gr^{\geq 0}_{k+1,n}$ to cells of $Gr^{\geq 0}_{k,n}$ and conjecture that it induces a bijection from positroid dissections of the hypersimplex $\Delta _{k+1,n}$ to positroid dissections of the amplituhedron $\mathcal{A}_{n,k,2}$; we prove this conjecture for the (infinite) class of BCFW dissections. We note that T-duality is particularly striking because the hypersimplex is an $(n-1)$-dimensional polytope while the amplituhedron $\mathcal{A}_{n,k,2}$ is a $2k$-dimensional non-polytopal subset of the Grassmannian $Gr_{k,k+2}$. Moreover, we prove that the positive tropical Grassmannian is the secondary fan for the regular positroid subdivisions of the hypersimplex, and prove that a matroid polytope is a positroid polytope if and only if all 2D faces are positroid polytopes. Finally, toward the goal of generalizing T-duality for higher $m$, we define the momentum amplituhedron for any even $m$.

Smoothly splitting amplitudes and semi-locality

In this paper, we study a novel behavior developed by certain tree-level scalar scattering amplitudes, including the biadjoint, NLSM, and special Galileon, when a subset of kinematic invariants vanishes without producing a singularity. This behavior exhibits properties which we call smooth splitting and semi-locality. The former means that an amplitude becomes the product of exactly three amputated Berends-Giele currents, while the latter means that any two currents share one external particle. We call these smooth splittings 3-splits. In fact, there are exactly left(begin{array}{c}n {}3end{array}right)-n such 3-splits of an n-particle amplitude, one for each tripod in a polygon; as they cannot be obtained from standard factorization, they are a new phenomenon in Quantum Field Theory. In fact, the resulting splitting is analogous to the one first seen in Cachazo-Early-Guevara-Mizera (CEGM) amplitudes which generalize standard cubic scalar amplitudes from their Tr G(2, n) formulation to Tr G(k, n), where Tr G(k, n) is the tropical Grassmannian. Along the way, we show how smooth splittings naturally lead to the discovery of mixed amplitudes in the NLSM and special Galileon theories and to novel BCFW-like recursion relations for NLSM amplitudes.

Singularities of eight- and nine-particle amplitudes from cluster algebras and tropical geometry

We further exploit the relation between tropical Grassmannians and Gr(4, n) cluster algebras in order to make and refine predictions for the singularities of scattering amplitudes in planar mathcal{N} = 4 super Yang-Mills theory at higher multiplicity n ≥ 8. As a mathematical foundation that provides access to square-root symbol letters in principle for any n, we analyse infinite mutation sequences in cluster algebras with general coefficients. First specialising our analysis to the eight-particle amplitude, and comparing it with a recent, closely related approach based on scattering diagrams, we find that the only additional letters the latter provides are the two square roots associated to the four-mass box. In combination with a tropical rule for selecting a finite subset of variables of the infinite Gr(4, 9) cluster algebra, we then apply our results to obtain a collection of 3, 078 rational and 2, 349 square-root letters expected to appear in the nine-particle amplitude. In particular these contain the alphabet found in an explicit 2-loop NMHV symbol calculation at this multiplicity.

Minimal Kinematics: An All k and n Peek into Trop<sup>+</sup>G(k,n)

In this note we present a formula for the Cachazo-Early-Guevara-Mizera (CEGM) generalized biadjoint amplitudes for all $k$ and $n$ on what we call the minimal kinematics. We prove that on the minimal kinematics, the scattering equations on the configuration space of $n$ points on $\mathbb{CP}^{k-1}$ has a unique solution, and that this solution is in the image of a Veronese embedding. The minimal kinematics is an all $k$ generalization of the one recently introduced by Early for $k=2$ and uses a choice of cyclic ordering. We conjecture an explicit formula for $m_n^{(k)}(\mathbb{I},\mathbb{I})$ which we have checked analytically through $n=10$ for all $k$. The answer is a simple rational function which has only simple poles; the poles have the combinatorial structure of the circulant graph ${\rm C}_n^{(1,2,\dots, k-2)}$. Generalized biadjoint amplitudes can also be evaluated using the positive tropical Grassmannian ${\rm Tr}^+{\rm G}(k,n)$ in terms of generalized planar Feynman diagrams. We find perfect agreement between both definitions for all cases where the latter is known in the literature. In particular, this gives the first strong consistency check on the $90\,608$ planar arrays for ${\rm Tr}^+{\rm G}(4,8)$ recently computed by Cachazo, Guevara, Umbert and Zhang. We also introduce another class of special kinematics called planar-basis kinematics which generalizes the one introduced by Cachazo, He and Yuan for $k=2$ and uses the planar basis recently introduced by Early for all $k$. Based on numerical computations through $n=8$ for all $k$, we conjecture that on the planar-basis kinematics $m_n^{(k)}(\mathbb{I},\mathbb{I})$ evaluates to the multidimensional Catalan numbers, suggesting the possibility of novel combinatorial interpretations. For $k=2$ these are the standard Catalan numbers.

Birational sequences and the tropical Grassmannian

We introduce iterated sequences for Grassmannians, a new class of Fang–Fourier–Littelmann's birational sequences and explain how they give rise to points in the tropical Grassmannian. For Gr(2,Cn) we show that the associated valuations induce toric degenerations. We describe recursively the vertices of the corresponding Newton–Okounkov polytopes, which are particular vertices of a hypercube and hence integral. We show further that every toric degenerations of Gr(2,Cn) constructed using the tropical Grassmannian due to Speyer and Sturmfels can be recovered by iterated sequences.

Positive Configuration Space

We define and study the totally nonnegative part of the Chow quotient of the Grassmannian, or more simply the nonnegative configuration space. This space has a natural stratification by positive Chow cells, and we show that nonnegative configuration space is homeomorphic to a polytope as a stratified space. We establish bijections between positive Chow cells and the following sets: (a) regular subdivisions of the hypersimplex into positroid polytopes, (b) the set of cones in the positive tropical Grassmannian, and (c) the set of cones in the positive Dressian. Our work is motivated by connections to super Yang–Mills scattering amplitudes, which will be discussed in a sequel.

The positive Dressian equals the positive tropical Grassmannian

The Dressian and the tropical Grassmannian parameterize abstract and realizable tropical linear spaces; but in general, the Dressian is much larger than the tropical Grassmannian. There are natural positive notions of both of these spaces – the positive Dressian, and the positive tropical Grassmannian (which we introduced roughly fifteen years ago in [J. Algebraic Combin. 22 (2005), pp. 189–210]) – so it is natural to ask how these two positive spaces compare. In this paper we show that the positive Dressian equals the positive tropical Grassmannian. Using the connection between the positive Dressian and regular positroidal subdivisions of the hypersimplex, we use our result to give a new “tropical” proof of da Silva’s 1987 conjecture (first proved in 2017 by Ardila-Rincón-Williams) that all positively oriented matroids are realizable. We also show that the finest regular positroidal subdivisions of the hypersimplex consist of series-parallel matroid polytopes, and achieve equality in Speyer’s f f -vector theorem. Finally we give an example of a positroidal subdivision of the hypersimplex which is not regular, and make a connection to the theory of tropical hyperplane arrangements.

Wall-Crossing for Newton–Okounkov Bodies and the Tropical Grassmannian

Abstract Tropical geometry and the theory of Newton–Okounkov bodies are two methods that produce toric degenerations of an irreducible complex projective variety. Kaveh and Manon showed that the two are related. We give geometric maps between the Newton–Okounkov bodies corresponding to two adjacent maximal-dimensional prime cones in the tropicalization of $X$. Under a technical condition, we produce a natural “algebraic wall-crossing” map on the underlying value semigroups (of the corresponding valuations). In the case of the tropical Grassmannian $Gr(2,m)$, we prove that the algebraic wall-crossing map is the restriction of a geometric map. In an appendix by Nathan Ilten, he explains how the geometric wall-crossing phenomenon can also be derived from the perspective of complexity-one $T$-varieties; Ilten also explains the connection to the “combinatorial mutations” studied by Akhtar–Coates–Galkin–Kasprzyk.

Tropical Grassmannians, cluster algebras and scattering amplitudes

We provide a cluster-algebraic approach to the computation of the recently introduced generalised biadjoint scalar amplitudes related to Grassmannians Gr(k, n). A finite cluster algebra provides a natural triangulation for the tropical Grassmannian whose volume computes the scattering amplitudes. Using this method one can construct the entire colour-ordered amplitude via mutations starting from a single term.

Notes on biadjoint amplitudes, Trop G(3, 7) and X(3, 7) scattering equations

In these notes we use the recently found relation between facets of tropical Grassmannians and generalizations of Feynman diagrams to compute all “biadjoint amplitudes” for n = 7 and k = 3. We also study scattering equations on X (3, 7), the configuration space of seven points on CP2. We prove that the number of solutions is 1272 in a two-step process. In the first step we obtain 1162 explicit solutions to high precision using near-soft kinematics. In the second step we compute the matrix of 360 ×360 biadjoint amplitudes obtained by using the facets of Trop G(3, 7), subtract the result from using the 1162 solutions and compute the rank of the resulting matrix. The rank turns out to be 110, which proves that the number of solutions in addition to the 1162 explicit ones is exactly 110.

Initial ideals of Pfaffian ideals

We explore the relationship between secant ideals and initial ideals of I 2 , n , the ideal of the Grassmannian, Gr ( 2 , ℂ n ) . The ( r − 1 ) -secant of I 2 , n is the ideal generated by the 2 r × 2 r subpfaffians of a generic n × n skew-symmetric matrix. It has been conjectured that for a weight vector ω in the tropical Grassmannian, the secant of the initial ideal of I 2 , n with respect to ω is equal to the initial ideal of the secant. We show that this conjecture is not true in general. Using the correspondence between weight vectors in the tropical Grassmannian and binary leaf-labeled trees, we also give necessary and sufficient conditions for the conjecture to hold in terms of the topology of the tree associated to ω . In the course of proving this result, we show that the ideal in ω ( I 2 , n { r } ) is always prime, thus giving a new class of prime initial ideals of the Pfaffian ideals.

Tropicalization of positive Grassmannians

We introduce combinatorial objects which are parameterized by the positive part of the tropical Grassmannian $$\mathrm{Gr}(k,n)$$. Our method is to relate the Grassmannian to configuration spaces of flags. By work of the first author, and of Goncharov and Shen, configuration spaces of flags naturally tropicalize to give configurations of points in the affine building, which we call higher laminations. We use higher laminations to give two dual objects that are parameterized by the positive tropicalization of $$\mathrm{Gr}(k,n)$$: equivalence classes of higher laminations; or certain restricted subset of higher laminations. This extends results of Speyer and Sturmfels on the tropicalization of $$\mathrm{Gr}(2,n)$$, and of Speyer and Williams on the tropicalization of $$\mathrm{Gr}(3,6)$$ and $$\mathrm{Gr}(3,7)$$. We also analyze the $${\mathcal {X}}$$-variety associated to the Grassmannian, and give an interpretation of its positive tropicalization.

Scattering equations: from projective spaces to tropical grassmannians

We introduce a natural generalization of the scattering equations, which connect the space of Mandelstam invariants to that of points on ℂℙ1, to higher-dimensional projective spaces ℂℙk − 1. The standard, k = 2 Mandelstam invariants, sab, are generalized to completely symmetric tensors {mathrm{s}}_{a_1{a}_2dots {a}_k} subject to a ‘massless’ condition {mathrm{s}}_{a_1{a}_2dots {a}_{k-2}bb}=0 and to ‘momentum conservation’. The scattering equations are obtained by constructing a potential function and computing its critical points. We mainly concentrate on the k = 3 case: study solutions and define the generalization of biadjoint scalar amplitudes. We compute all ‘biadjoint amplitudes’ for (k, n) = (3, 6) and find a direct connection to the tropical Grassmannian. This leads to the notion of k = 3 Feynman diagrams. We also find a concrete realization of the new kinematic spaces, which coincides with the spinor-helicity formalism for k = 2, and provides analytic solutions analogous to the MHV ones.

Computing Tropical Points and Tropical Links

We present an algorithm for computing zero-dimensional tropical varieties based on triangular decomposition and Newton polygon methods. From it, we derive algorithms for computing points on and links of higher-dimensional tropical varieties, using intersections with affine hyperplanes to reduce the dimension to zero. We use the algorithms to show that the tropical Grassmannians {mathcal {G}}_{3,8} and {mathcal {G}}_{4,8} are not simplicial.

Toric Degenerations of Gr(2, n) and Gr(3, 6) via Plabic Graphs

We establish an explicit bijection between the toric degenerations of the Grassmannian Gr(2, n) arising from maximal cones in tropical Grassmannians and the ones coming from plabic graphs corresponding to Gr(2, n). We show that a similar statement does not hold for Gr(3, 6).

Matroids from hypersimplex splits

A class of matroids is introduced which is very large as it strictly contains all paving matroids as special cases. As their key feature these split matroids can be studied via techniques from polyhedral geometry. It turns out that the structural properties of the split matroids can be exploited to obtain new results in tropical geometry, especially on the rays of the tropical Grassmannians.

Cluster algebras of type $$D_4$$ D 4 , tropical planes, and the positive tropical Grassmannian

We show that the number of combinatorial types of clusters of type \(D_4\) modulo reflection-rotation is exactly equal to the number of combinatorial types of generic tropical planes in \(\mathbb {TP}^5\). This follows from a result of Sturmfels and Speyer which classifies these generic tropical planes into seven combinatorial classes using a detailed study of the tropical Grassmannian \({{\mathrm{Gr}}}(3,6)\). Speyer and Williams show that the positive part \({{\mathrm{Gr}}}^+(3,6)\) of this tropical Grassmannian is combinatorially equivalent to a small coarsening of the cluster fan of type \(D_4\). We provide a structural bijection between the rays of \({{\mathrm{Gr}}}^+(3,6)\) and the almost positive roots of type \(D_4\) which makes this connection more precise. This bijection allows us to use the pseudotriangulations model of the cluster algebra of type \(D_4\) to describe the equivalence of “positive” generic tropical planes in \(\mathbb {TP}^5\), giving a combinatorial model which characterizes the combinatorial types of generic tropical planes using automorphisms of pseudotriangulations of the octogon.

Dressians, tropical Grassmannians, and their rays

Abstract The Dressian Dr(k,n) parametrizes all tropical linear spaces, and it carries a natural fan structure as a subfan of the secondary fan of the hypersimplex Δ(k,n). We explore the combinatorics of the rays of Dr(k,n), that is, the most degenerate tropical planes, for arbitrary k and n. This is related to a new rigidity concept for configurations of n - k points in the tropical k - 1-torus. Additional conditions are given for k = 3. On the way, we compute the entire fan Dr(3,8).