Abstract
Plabic graphs are intimately connected to the positroid stratification of the positive Grassmannian. The duals to these graphs are quivers, and it is possible to associate to them cluster algebras. For the top-cell graph of Gr +(k, n), this cluster algebra is the homogeneous coordinate ring of the corresponding positroid variety. We prove that the same statement holds for plabic graphs describing lower dimensional cells. In this way we obtain a map from the positroid strata onto cluster subalgebras of Gr +(k, n). We explore some of the consequences of this map for tree-level scattering amplitudes in \( \mathcal{N} \) =4 super Yang-Mills theory.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
N. Arkani-Hamed et al., Scattering Amplitudes and the Positive Grassmannian, arXiv:1212.5605 [INSPIRE].
N. Arkani-Hamed, F. Cachazo, C. Cheung and J. Kaplan, A Duality For The S Matrix, JHEP 03 (2010) 020 [arXiv:0907.5418] [INSPIRE].
A. Postnikov, Total positivity, Grassmannians and networks, math/0609764 [INSPIRE].
N. Beisert, J. Broedel and M. Rosso, On Yangian-invariant regularization of deformed on-shell diagrams in \( \mathcal{N} \) =4 super-Yang-Mills theory, J. Phys. A 47 (2014) 365402 [arXiv:1401.7274] [INSPIRE].
J. Broedel, M. de Leeuw and M. Rosso, A dictionary between R-operators, on-shell graphs and Yangian algebras, JHEP 06 (2014) 170 [arXiv:1403.3670] [INSPIRE].
J. Broedel, M. de Leeuw and M. Rosso, Deformed one-loop amplitudes in N =4 super-Yang-Mills theory, arXiv:1406.4024 [INSPIRE].
S. Franco, D. Galloni and A. Mariotti, Bipartite Field Theories, Cluster Algebras and the Grassmannian, arXiv:1404.3752 [INSPIRE].
S. Franco, Cluster Transformations from Bipartite Field Theories, Phys. Rev. D 88 (2013) 105010 [arXiv:1301.0316] [INSPIRE].
Y.-T. Huang and C. Wen, ABJM amplitudes and the positive orthogonal grassmannian, JHEP 02 (2014) 104 [arXiv:1309.3252] [INSPIRE].
N. Kanning, T. Lukowski and M. Staudacher, A shortcut to general tree-level scattering amplitudes in \( \mathcal{N} \) =4 SYM via integrability, Fortsch. Phys. 62 (2014) 556 [arXiv:1403.3382] [INSPIRE].
A. Amariti and D. Forcella, Scattering Amplitudes and Toric Geometry, JHEP 09 (2013) 133 [arXiv:1305.5252] [INSPIRE].
B. Leclerc, Cluster structures on strata of flag varieties, arXiv:1402.4435.
G. Muller and D.E. Speyer, Cluster Algebras of Grassmannians are Locally Acyclic, arXiv:1401.5137.
J.S. Scott, Grassmannians and cluster algebras, Proc. London Math. Soc. 92 (2003) 345 math/0311148.
J. Golden, A.B. Goncharov, M. Spradlin, C. Vergu and A. Volovich, Motivic Amplitudes and Cluster Coordinates, JHEP 01 (2014) 091 [arXiv:1305.1617] [INSPIRE].
J. Golden, M.F. Paulos, M. Spradlin and A. Volovich, Cluster Polylogarithms for Scattering Amplitudes, arXiv:1401.6446 [INSPIRE].
L.K. Williams, Shelling totally nonnegative flag varieties, J. Reine Angew. Math. 609 (2007) 1.
S. Fomin and A. Zelevinsky, Cluster algebras. I. Foundations, J. Amer. Math. Soc. 15 (2002) 497.
S. Fomin and A. Zelevinsky, Cluster algebras. II. Finite type classification, Invent. Math. 154 (2003) 63.
A. Berenstein, S. Fomin and A. Zelevinsky, Cluster algebras. III. Upper bounds and double Bruhat cells, Duke Math. J. 126 (2005) 1.
S. Fomin and A. Zelevinsky, Cluster algebras. IV. Coefficients, Compos. Math. 143 (2007) 112.
L.K. Williams, Cluster algebras: an introduction, Bull. Amer. Math. Soc. (N.S.) 51 (2014) 1.
S. Fomin, Cluster algebras portal, http://www.math.lsa.umich.edu/~fomin/cluster.html.
Open Access
This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.
Author information
Authors and Affiliations
Corresponding author
Additional information
ArXiv ePrint: 1406.7273
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0), which permits use, duplication, adaptation, distribution, and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Paulos, M.F., Schwab, B.U.W. Cluster algebras and the positive Grassmannian. J. High Energ. Phys. 2014, 31 (2014). https://doi.org/10.1007/JHEP10(2014)031
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP10(2014)031