How to Draw Tropical Planes

Herrmann, Sven; Jensen, Anders Nedergaard; Joswig, Michael; Sturmfels, Bernd

doi:10.37236/72

Cited by 76 publications

(134 citation statements)

References 20 publications

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“…However, before ending this section it is worth mentioning some of the results known in the mathematical literature which can help in the exploration of k = 3 Feynman diagrams and k = 3 amplitudes. In 2008, Herrmann et al [26] revisited the tropical Grassmannian G(3, 6) and carefully studied the tropical Grassmannian G(3, 7). In their work they computed all rays that define the corresponding spaces and the combinations that make up the facets.…”

Section: Tropical Grassmannians and Higher-k Feynman Diagramsmentioning

confidence: 99%

“…Recall that rays (or vertices when considering the intersection with a unit sphere) are in bijection with the kinematic invariants that make "propagators" while the facets are proposed to correspond to the new k = 3 Feynman diagrams. All the data collected in [26] is posted on the webpage: www.uni-math.gwdg.de/jensen/Research/G3_7/grassmann3_7.html Let us explain in more detail how to translate the tropical Grassmannian G(3, 7) data on the webpage to physics. The first important object to consider is the table of rays labeled R-vector.…”

Section: Tropical Grassmannians and Higher-k Feynman Diagramsmentioning

confidence: 99%

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Scattering equations: from projective spaces to tropical grassmannians

Cachazo

Early

Guevara

et al. 2019

J. High Energ. Phys.

261

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We introduce a natural generalization of the scattering equations, which connect the space of Mandelstam invariants to that of points on CP 1 , to higher-dimensional projective spaces CP k−1 . The standard, k = 2 Mandelstam invariants, s ab , are generalized to completely symmetric tensors s a 1 a 2 ...a k subject to a 'massless' condition s a 1 a 2 ···a k−2 b b = 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.

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Section: Tropical Grassmannians and Higher-k Feynman Diagramsmentioning

confidence: 99%

Section: Tropical Grassmannians and Higher-k Feynman Diagramsmentioning

confidence: 99%

Scattering equations: from projective spaces to tropical grassmannians

Cachazo

Early

Guevara

et al. 2019

J. High Energ. Phys.

261

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“…In fact there are several versions of it. Here we follow the original description given by Speyer and Sturmfels in their study of trop G(3, 6) [14] but adapted to the work of Herrmann, Jensen, Joswig, and Sturmfels [10] where the analysis of trop G(3, 7) is done.…”

Section: Biadjoint Amplitudes From the Tropical Grassmannian G(3 7)mentioning

confidence: 99%

“…In this work we extend the same analysis to n = 7, k = 3 biadjoint amplitudes and Trop G(3, 7). Luckily, the structure of Trop G(3, 7) has been carefully studied by Herrmann, Jensen, Joswig, and Sturmfels [10] and we make use of their results to carry out all our Feynman diagram computations which lead to the explicit form of all 360 × 360 biadjoint amplitudes m The strategy for determining the number of solutions to the X(3, 7) scattering equations is the following. We start with the dual version, i.e., the X(4, 7) scattering equations near a soft limit.…”

Section: Introductionmentioning

confidence: 99%

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

Cachazo

Rojas

2020

J. High Energ. Phys.

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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 CP 2 . 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. arXiv:1906.05979v1 [hep-th]

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“…This endows Dr(d, n) with a secondary fan structure as subfan of the secondary fan of ∆(d, n). In [11], the authors proved that for d = 3 the two fan structures coincide.…”

Section: Introductionmentioning

confidence: 99%

On local Dressians of matroids

Olarte

Panizzut

Schröter

2019

Algebraic and Geometric Combinatorics on Lattice Polytopes

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A b s t r ac t . We study the fan structure of Dressians Dr (d, n) and local Dressians Dr(M) for a given matroid M. In particular we show that the fan structure on Dr(M) given by the three term Plücker relations coincides with the structure as a subfan of the secondary fan of the matroid polytope P (M). As a corollary, we have that a matroid subdivision is determined by its 3-dimensional skeleton. We also prove that the Dressian of the sum of two matroids is isomorphic to the product of the Dressians of the matroids. Finally we focus on indecomposable matroids. We show that binary matroids are indecomposable, and we provide a non-binary indecomposable matroid as a counterexample for the converse. arXiv:1809.08965v1 [math.CO] 24 Sep 2018 Cambridge University Press, Cambridge, 1986.I n s t i t u t f ü r M at h e m at i k , F U B e r l i n , A r n i m a l l e e 2 , 1 4 1 9 5 B e r l i n , G e r m a n y , E -m a i l : o l a rt e @ z e dat . f u -b e r l i n . d e I n s t i t u t f ü r M at h e m at i k , T U B e r l i n , S t r . d e s 1 7 . J u n i 1 3 6 , 1 0 6 2 3 B e r l i n , G e r m a n y , E -m a i l : pa n i z z u t @ m at h . t u -b e r l i n . d e D e pa rt m e n t o f M at h e m at i c a l S c i e n c e s , B i n g h a m t o n U n i v e r s i t y , B i n g h a m t o n , N Y 1 3 9 0 2 , U S A , E -m a i l : s c h ro e t e r @ m at h . b i n g h a m t o n . e d u

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How to Draw Tropical Planes

Cited by 76 publications

References 20 publications

Scattering equations: from projective spaces to tropical grassmannians

Scattering equations: from projective spaces to tropical grassmannians

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

On local Dressians of matroids

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