Tropical Fans Research Articles

Adjacency for scattering amplitudes from the Gröbner fan

Scattering amplitudes in planar N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 4 super Yang-Mills theory exhibit singularities which reflect various aspects of the cluster algebras associated to the Grassmannians Gr(4, n) and their tropical counterparts. Here we investigate the potential origins of such structures and examine the extent to which they can be recovered from the Gröbner structure of the underlying Plücker ideals, focussing on the Grassmannians corresponding to finite cluster algebras.Starting from the Plücker ideal, we describe how the polynomial cluster variables are encoded in non-prime initial ideals associated to certain maximal cones of the positive tropical fan. Following [1] we show that extending the Plücker ideal by such variables leads to a Gröbner fan with a single maximal Gröbner cone spanned by the positive tropical rays. The associated initial ideal encodes the compatibility relations among the full set of cluster variables. Thus we find that the Gröbner structure naturally encodes both the symbol alphabet and the cluster adjacency relations exhibited by scattering amplitudes without invoking the cluster algebra at all.As a potential application of these ideas we then examine the kinematic ideal associated to non-dual conformal massless scattering written in terms of spinor helicity variables. For five-particle scattering we find that the ideal can be identified with the Plücker ideal for Gr(3, 6) and the corresponding tropical fan contains a number of non-prime ideals which encode all additional letters of the two-loop pentagon function alphabet present in various calculations of massless five-point finite remainders.

Lorentzian Fans

Abstract We introduce the notion of Lorentzian fans, which form a special class of tropical fans that are particularly well-suited for proving Alexandrov–Fenchel-type inequalities. To demonstrate the utility of Lorentzian fans, we prove a practical characterization of them in terms of their two-dimensional star fans. We also show that Lorentzian fans are closed under many common tropical fan operations, and we discuss how the Lorentzian property descends to the underlying tropical variety, allowing us to deduce Alexandrov–Fenchel-type inequalities in the general setting of tropical intersection theory on tropical fan varieties.

Tropical fans and normal complexes: Putting the “volume” back in “volume polynomials”

Associated to any divisor in the Chow ring of a simplicial tropical fan, we construct a family of polytopal complexes, called normal complexes, which we propose as an analogue of the well-studied notion of normal polytopes from the setting of complete fans. We describe certain closed convex polyhedral cones of divisors for which the “volume” of each divisor in the cone—that is, the degree of its top power—is equal to the volume of the associated normal complexes. For the Bergman fan of any matroid with building set, we prove that there exists an open family of such cones of divisors with nonempty interiors. We view the theory of normal complexes developed in this paper as a polytopal model underlying the combinatorial Hodge theory pioneered by Adiprasito, Huh, and Katz.

Tropical fans, scattering equations and amplitudes

We describe a family of tropical fans related to Grassmannian cluster algebras. These fans are related to the kinematic space of massless scattering processes in a number of ways. For each fan associated to the Grassmannian Gr(k, n) there is a notion of a generalised ϕ3 amplitude and an associated set of scattering equations which further generalise the Gr(k, n) scattering equations that have been recently introduced. Here we focus mostly on the cases related to finite Grassmannian cluster algebras and we explain how face variables for the cluster polytopes are simply related to the scattering equations. For the Grassmannians Gr(4, n) the tropical fans we describe are related to the singularities (or symbol letters) of loop amplitudes in planar mathcal{N} = 4 super Yang-Mills theory. We show how each choice of tropical fan leads to a natural class of polylogarithms, generalising the notion of cluster adjacency and we describe how the currently known loop data fit into this classification.

Algebraic singularities of scattering amplitudes from tropical geometry

We address the appearance of algebraic singularities in the symbol alphabet of scattering amplitudes in the context of planar mathcal{N} = 4 super Yang-Mills theory. We argue that connections between cluster algebras and tropical geometry provide a natural language for postulating a finite alphabet for scattering amplitudes beyond six and seven points where the corresponding Grassmannian cluster algebras are finite. As well as generating natural finite sets of letters, the tropical fans we discuss provide letters containing square roots. Remarkably, the minimal fan we consider provides all the square root letters recently discovered in an explicit two-loop eight-point NMHV calculation.

Characteristic classes of affine varieties and Plücker formulas for affine morphisms

An enumerative problem on a variety V is usually solved by reduction to intersection theory in the cohomology of a compactification of V . However, if the problem is invariant under a “nice” group action on V (so that V is spherical), then many authors suggested a better home for intersection theory: the direct limit of the cohomology rings of all equivariant compactifications of V . We call this limit the affine cohomology of V and construct affine characteristic classes of subvarieties of a complex torus, taking values in the affine cohomology of the torus. This allows us to make the first steps in computing affine Thom polynomials . Classical Thom polynomials count how many fibers of a generic proper map of a smooth variety have a prescribed collection of singularities and our affine version addresses the same question for generic polynomial maps of affine algebraic varieites. This notion is also motivated by developing an intersection-theoretic approach to tropical correspondence theorems: they can be reduced to the computation of affine Thom polynomials, because the fundamental class of a variety in the affine cohomology is encoded by the tropical fan of this variety. The first concrete answer that we obtain is the affine version of what were, historically speaking, the first three Thom polynomials – the Plücker formulas for the degree and the number of cusps and nodes of a projectively dual curve. This, in particular, characterizes toric varieties whose projective dual is a hypersurface, computes the tropical fan of the variety of double tangent hyperplanes to a toric variety, and describes the Newton polytope of the hypersurface of non-Morse polynomials of a given degree. We also make a conjecture on the general form of affine Thom polynomials; a key ingredient is the n -ary fan, generalizing the secondary polytope.

Tropical intersection products on smooth varieties

In analogy to [AR07,Chapter9], ee define an intersection product of tropical cycles on tropical linear spaces L^n_k , i.e. on tropical fans of the type \mathrm{max}\{0,x_1,...,x_n\}^{n-k} \cdot ℝ^n . Afterwards we use this result to obtain an intersection product of cycles on every smooth tropical variety, i.e. on every tropical variety that arises from gluing such tropical linear spaces. In contrast to classical algebraic geometry these products always yield well-defined cycles, not cycle classes only. Using these intersection products we are able to define the pull-back of a tropical cycle along a morphism between smooth tropical varieties. In the present article we stick to the definitions, notions and concepts introduced in [AR07].

Tropical fans and the moduli spaces of tropical curves

AbstractWe give a rigorous definition of tropical fans (the ‘local building blocks for tropical varieties’) and their morphisms. For a morphism of tropical fans of the same dimension we show that the number of inverse images (counted with suitable tropical multiplicities) of a point in the target does not depend on the chosen point; a statement that can be viewed as one of the important first steps of tropical intersection theory. As an application we consider the moduli spaces of rational tropical curves (both abstract and in some ℝr) together with the evaluation and forgetful morphisms. Using our results this gives new, easy and unified proofs of various tropical independence statements, e.g. of the fact that the numbers of rational tropical curves (in any ℝr) through given points are independent of the points.

Anomalous heavy mineral size relationships in a humid alluvial fan, Colombia; implications for placer formation

In humid tropical fan deposits of the late Pleistocene age in southwestern Colombia, the mean gold grain size is anomalously coarser than the associated mean magnetite grain size. This anomaly is even greater when equivalent diameter mean sizes are compared, which indicates that the gold is deposited under different flow conditions than the magnetite. Entrainment sorting concentrated the gold in the deeper parts of the channel during prolonged intervals when the flow velocity was above the entrainment velocity of magnetite and near that of gold (30-60 cm/sec). Magnetite was preferentially concentrated in sand bars in more shallow reaches of the channel where flow velocities were slightly lower (15-35 cm/sec). After the gold was concentrated, most of the magnetite was added to the deposit of cobbles when magnetite-rich sand bars migrated over the cobble deposits. In most cases, later flood events stripped away these sand bars and deposited more cobbles, resulting in a thick sequence of cobble gravel with gold and magnetite concentrated in the cobble matrix at certain intervals. The frequent high flow events in humid tropical fans are conducive to this type of placer development.