ABHY Associahedra and Newton polytopes of $F$-polynomials for finite type cluster algebras

Bazier-Matte, Véronique; Chapelier-Laget, Nathan; Douville, Guillaume; Mousavand, Kaveh; Thomas, Hugh; Yıldırım, Emine

doi:10.48550/arxiv.1808.09986

Cited by 26 publications

(78 citation statements)

References 20 publications

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“…For this case Λ = (1, 1, 2). We choose subsets using the three permutations of Λ as 4,6] may be obtained from the first one by replacing [1,5] by [4,6]. We thus have the grading…”

Section: General Potentialsmentioning

confidence: 99%

“…This furnishes two further sets, namely, [1,3], [4,6] and [1,4], [4,6] of projectives and hence two more terms to the scattering amplitude. The repeated set [1,3], [1,4] is not to be counted again. As a third example, let us consider the thirteenth diagram in Fig.…”

Section: General Potentialsmentioning

confidence: 99%

“…• is obtained again from the first one, by replacing [1,4] by [3,5]. The choice of projectives for the pseudo-periodic quiver is…”

Section: General Potentialsmentioning

confidence: 99%

“…The ABHY programme provided a geometric way of writing scattering amplitudes of a biadjoint cubic scalar field theory in four dimensions as the volume form of the dual to the associahedron, a polyhedron associated to the combinatorics of grouping the scattering particles according to the vertices of Feynman diagrams they give rise to. This relationship between the scattering amplitudes and the associahedron led to the connection of the former with the algebraic geometry of Grassmannians, tropical geometry [2], cluster algebras [3,4] and representation theory of quivers and related subjects [5]. The ABHY scheme has been generalized in various directions.…”

Section: Introductionmentioning

confidence: 99%

“…have three sets of projectives {[1,3],[1,4]}, {[1,3],[1,5]} and {[1,4],[1,5]} arising from this. They contribute in Fig.12.…”

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confidence: 99%

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Towards a categorification of scattering amplitudes

Barmeier¹,

Oak²,

Pal³

et al. 2021

Preprint

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Categorification of scattering amplitudes for planar Feynman diagrams in scalar field theories with a polynomial potential is reported. Amplitudes for cubic theories are directly written down in terms of projectives of hearts of intermediate t-structures restricted to the cluster category of quiver representations, without recourse to geometry. It is shown that for theories with φ m+2 potentials those corresponding to m-cluster categories are to be used. The case of generic polynomial potentials is treated and our results suggest the existence of a generalization of higher cluster categories which we call pseudo-periodic categories. An algorithm to obtain the projectives of hearts of intermediate t-structures for these types is presented.

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“…For this case Λ = (1, 1, 2). We choose subsets using the three permutations of Λ as 4,6] may be obtained from the first one by replacing [1,5] by [4,6]. We thus have the grading…”

Section: General Potentialsmentioning

confidence: 99%

Section: General Potentialsmentioning

confidence: 99%

“…• is obtained again from the first one, by replacing [1,4] by [3,5]. The choice of projectives for the pseudo-periodic quiver is…”

Section: General Potentialsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…have three sets of projectives {[1,3],[1,4]}, {[1,3],[1,5]} and {[1,4],[1,5]} arising from this. They contribute in Fig.12.…”

mentioning

confidence: 99%

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Towards a categorification of scattering amplitudes

Barmeier¹,

Oak²,

Pal³

et al. 2021

Preprint

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On positive geometries of quartic interactions: Stokes polytopes, lower forms on associahedra and world-sheet forms

Aneesh

Banerjee

Jagadale

et al. 2020

J. High Energ. Phys.

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In [1], two of the present authors along with P. Raman attempted to extend the Amplituhedron program for scalar field theories [2] to quartic scalar interactions. In this paper we develop various aspects of this proposal. Using recent seminal results in Representation theory [3,4], we show that projectivity of scattering forms and existence of kinematic space associahedron completely capture planar amplitudes of quartic interaction. We generalise the results of [1] and show that for any n-particle amplitude, the positive geometry associated to the projective scattering form is a convex realisation of Stokes polytope which can be naturally embedded inside one of the ABHY associahedra defined in [2,5]. For a special class of Stokes polytopes with hyper-cubic topology, we show that they have a canonical convex realisation in kinematic space as boundaries of kinematic space associahedra.We then use these kinematic space geometric constructions to write world-sheet forms for φ 4 theory which are forms of lower rank on the CHY moduli space. We argue that just as in the case of bi-adjoint φ 3 scalar amplitudes, scattering equations can be used as diffeomorphisms between certain n−4

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Moduli space of paired punctures, cyclohedra and particle pairs on a circle

Zhang

2019

J. High Energ. Phys.

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In this paper, we study a new moduli space M c n+1 , which is obtained from M 0,2n+2 by identifying pairs of punctures. We find that this space is tiled by 2 n−1 n! cyclohedra, and construct the canonical form for each chamber. We also find the corresponding Koba-Nielsen factor can be viewed as the potential of the system of n+1 pairs of particles on a circle, which is similar to the original case of M 0,n where the system is n−3 particles on a line. We investigate the intersection numbers of chambers equipped with Koba-Nielsen factors. Then we construct cyclohedra in kinematic space and show that the scattering equations serve as a map between the interior of worldsheet cyclohedron and kinematic cyclohedron. Finally, we briefly discuss string-like integrals over such moduli space.

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ABHY Associahedra and Newton polytopes of $F$-polynomials for finite type cluster algebras

Cited by 26 publications

References 20 publications

Towards a categorification of scattering amplitudes

Towards a categorification of scattering amplitudes

On positive geometries of quartic interactions: Stokes polytopes, lower forms on associahedra and world-sheet forms

Moduli space of paired punctures, cyclohedra and particle pairs on a circle

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