Positive Configuration Space

Arkani-Hamed, Nima; Lam, Thomas; Spradlin, Marcus

doi:10.1007/s00220-021-04041-x

Positive Configuration Space

Journal Article · 11 April 2021 · Communications in Mathematical Physics

DOI:https://doi.org/10.1007/s00220-021-04041-x· OSTI ID:1851255

Arkani-Hamed, Nima ^[1]; ; Spradlin, Marcus

OSTI

Abstract

We define and study the totally nonnegative part of the Chow quotient of the Grassmannian, or more simply thenonnegative configuration space. This space has a natural stratification bypositive 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.

Research Organization:: Institute for Advanced Study, Princeton, NJ (United States); Brown Univ., Providence, RI (United States)

Sponsoring Organization:: USDOE Office of Science (SC)

DOE Contract Number:: SC0009988; SC0010010;

OSTI ID:: 1851255

Journal Information:: Communications in Mathematical Physics, Journal Name: Communications in Mathematical Physics Journal Issue: 2 Vol. 384; ISSN 0010-3616

Publisher:: Springer

Country of Publication:: United States

Language:: English

References (31)

An Algebraic Cell Decomposition of the Nonnegative Part of a Flag Variety Rietsch, Konstanze Journal of Algebra, Vol. 213, Issue 1 https://doi.org/10.1006/jabr.1998.7665	journal	March 1999
Total Positivity in Reductive Groups Lusztig, G. Lie Theory and Geometry https://doi.org/10.1007/978-1-4612-0261-5_20	book	January 1994
Motivic amplitudes and cluster coordinates Golden, J. K.; Goncharov, A. B.; Spradlin, M. Journal of High Energy Physics, Vol. 2014, Issue 1 https://doi.org/10.1007/JHEP01(2014)091	journal	January 2014
Tropical Grassmannians, cluster algebras and scattering amplitudes Drummond, James; Foster, Jack; Gürdoğan, Ömer Journal of High Energy Physics, Vol. 2020, Issue 4 https://doi.org/10.1007/JHEP04(2020)146	journal	April 2020
Notes on biadjoint amplitudes, Trop G(3, 7) and X(3, 7) scattering equations Cachazo, Freddy; Rojas, Jairo M. Journal of High Energy Physics, Vol. 2020, Issue 4 https://doi.org/10.1007/JHEP04(2020)176	journal	April 2020
Scattering equations: from projective spaces to tropical grassmannians Cachazo, Freddy; Early, Nick; Guevara, Alfredo Journal of High Energy Physics, Vol. 2019, Issue 6 https://doi.org/10.1007/JHEP06(2019)039	journal	June 2019
How tropical are seven- and eight-particle amplitudes? Henke, Niklas; Papathanasiou, Georgios Journal of High Energy Physics, Vol. 2020, Issue 8 https://doi.org/10.1007/JHEP08(2020)005	journal	August 2020
The Amplituhedron Arkani-Hamed, Nima; Trnka, Jaroslav Journal of High Energy Physics, Vol. 2014, Issue 10 https://doi.org/10.1007/JHEP10(2014)030	journal	October 2014
Positive geometries and canonical forms Arkani-Hamed, Nima; Bai, Yuntao; Lam, Thomas Journal of High Energy Physics, Vol. 2017, Issue 11 https://doi.org/10.1007/JHEP11(2017)039	journal	November 2017
Generalized planar Feynman diagrams: collections Borges, Francisco; Cachazo, Freddy Journal of High Energy Physics, Vol. 2020, Issue 11 https://doi.org/10.1007/JHEP11(2020)164	journal	November 2020
Alcoved Polytopes, I Lam, Thomas; Postnikov, Alexander Discrete & Computational Geometry, Vol. 38, Issue 3 https://doi.org/10.1007/s00454-006-1294-3	journal	October 2007
The Tropical Totally Positive Grassmannian Speyer, David; Williams, Lauren Journal of Algebraic Combinatorics, Vol. 22, Issue 2 https://doi.org/10.1007/s10801-005-2513-3	journal	September 2005
Cluster algebras of type $$D_4$$ D 4 , tropical planes, and the positive tropical Grassmannian Brodsky, Sarah B.; Ceballos, Cesar; Labbé, Jean-Philippe Beiträge zur Algebra und Geometrie / Contributions to Algebra and Geometry, Vol. 58, Issue 1 https://doi.org/10.1007/s13366-016-0316-4	journal	October 2016
Combinatorial geometries, convex polyhedra, and schubert cells Gelfand, I. M.; Goresky, R. M.; MacPherson, R. D. Advances in Mathematics, Vol. 63, Issue 3 https://doi.org/10.1016/0001-8708(87)90059-4	journal	March 1987
Positroids and Schubert matroids Oh, Suho Journal of Combinatorial Theory, Series A, Vol. 118, Issue 8 https://doi.org/10.1016/j.jcta.2011.06.006	journal	November 2011
Bridge graphs and Deodhar parametrizations for positroid varieties Karpman, Rachel Journal of Combinatorial Theory, Series A, Vol. 142 https://doi.org/10.1016/j.jcta.2016.03.005	journal	August 2016
Grassmannian Geometry of Scattering Amplitudes Arkani-Hamed, Nima; Bourjaily, Jacob; Cachazo, Freddy https://doi.org/10.1017/CBO9781316091548	book	May 2016
Positroids and non-crossing partitions Ardila, Federico; Rincón, Felipe; Williams, Lauren Transactions of the American Mathematical Society, Vol. 368, Issue 1 https://doi.org/10.1090/tran/6331	journal	May 2015
Positroid varieties: juggling and geometry Knutson, Allen; Lam, Thomas; Speyer, David E. Compositio Mathematica, Vol. 149, Issue 10 https://doi.org/10.1112/S0010437X13007240	journal	August 2013
The twist for positroid varieties: THE TWIST FOR POSITROID VARIETIES Muller, Greg; Speyer, David E. Proceedings of the London Mathematical Society, Vol. 115, Issue 5 https://doi.org/10.1112/plms.12056	journal	July 2017
Weak separation and plabic graphs Oh, Suho; Postnikov, Alexander; Speyer, David E. Proceedings of the London Mathematical Society, Vol. 110, Issue 3 https://doi.org/10.1112/plms/pdu052	journal	February 2015
Tropical Linear Spaces Speyer, David E. SIAM Journal on Discrete Mathematics, Vol. 22, Issue 4 https://doi.org/10.1137/080716219	journal	January 2008
On local Dressians of matroids Olarte, Jorge Alberto; Panizzut, Marta; Schröter, Benjamin Summer Workshop on Lattice Polytopes, Algebraic and Geometric Combinatorics on Lattice Polytopes https://doi.org/10.1142/9789811200489_0020	conference	June 2019
Geometry of Chow quotients of Grassmannians Keel, Sean; Tevelev, Jenia Duke Mathematical Journal, Vol. 134, Issue 2 https://doi.org/10.1215/S0012-7094-06-13422-1	journal	August 2006
Introduction to Toric Varieties. (AM-131) Fulton, William Annals of Mathematics Studies series, Vol. 131 https://doi.org/10.1515/9781400882526	book	December 1993
The tropical Grassmannian Speyer, David; Sturmfels, Bernd Advances in Geometry, Vol. 4, Issue 3 https://doi.org/10.1515/advg.2004.023	journal	January 2004
Dressians, tropical Grassmannians, and their rays Herrmann, Sven; Joswig, Michael; Speyer, David E. Forum Mathematicum, Vol. 26, Issue 6 https://doi.org/10.1515/forum-2012-0030	journal	January 2014
Multiple zeta values and periods of moduli spaces $\overline{\mathfrak{M}}_{0,n}$ Brown, Francis C. S. Annales scientifiques de l'École normale supérieure, Vol. 42, Issue 3 https://doi.org/10.24033/asens.2099	journal	January 2009
How to Draw Tropical Planes Herrmann, Sven; Jensen, Anders; Joswig, Michael The Electronic Journal of Combinatorics, Vol. 16, Issue 2 https://doi.org/10.37236/72	journal	February 2009
Positivity for cluster algebras Lee, Kyungyong; Schiffler, Ralf Annals of Mathematics https://doi.org/10.4007/annals.2015.182.1.2	journal	July 2015
Totally nonnegative Grassmannian and Grassmann polytopes Lam, Thomas Current Developments in Mathematics, Vol. 2014, Issue 1 https://doi.org/10.4310/CDM.2014.v2014.n1.a2	journal	January 2014

Similar Records

Positive Configuration Space

Journal Article · 2021 · Communications in Mathematical Physics · OSTI ID:1776190

Positive geometries and canonical forms

Journal Article · 2017 · Journal of High Energy Physics (Online) · OSTI ID:1502489

Related Subjects

Physics

Positive Configuration Space

Citation Formats

References (31)

Similar Records

Related Subjects