Mini-workshop: Scattering amplitudes, cluster algebras, and positive geometries. Abstracts from the mini-workshop held December 5–11, 2021 (hybrid meeting). (English) Zbl 1506.00037
Summary: Cluster algebras were developed by Fomin and Zelevinsky about twenty years ago. While the initial motivation came from within algebra (total positivity, canonical bases), it quickly became clear that cluster algebras possess deep links to a host of other subjects in mathematics and physics. In a separate vein, starting about ten years ago, Arkani-Hamed and his collaborators began a program of reformulating the bases of quantum field theory, motivated by a desire to discover the basic rules of quantum mechanics and spacetime as arising from deeper mathematical principles. Their approach to the fundamental problem of particle scattering amplitudes entails encoding the solution in geometrical objects, “positive geometries” and “amplituhedra”. Surprisingly, cluster algebras have been found to be tightly woven into the mathematics needed to describe these geometries. The purpose of this workshop is to explore the various connections between cluster algebras, scattering amplitudes, and positive geometries.
MSC:
| 00B05 | Collections of abstracts of lectures |
| 00B25 | Proceedings of conferences of miscellaneous specific interest |
| 81-06 | Proceedings, conferences, collections, etc. pertaining to quantum theory |
| 13-06 | Proceedings, conferences, collections, etc. pertaining to commutative algebra |
| 81U10 | \(n\)-body potential quantum scattering theory |
| 13F60 | Cluster algebras |
| 14M15 | Grassmannians, Schubert varieties, flag manifolds |
| 51M20 | Polyhedra and polytopes; regular figures, division of spaces |
Software:
HomotopyContinuationReferences:
| [1] | S. He, Z. Li, Q. Yang, Kinematics, cluster algebras and Feynman integrals, arXiv:2112.11842 [hep-th]. |
| [2] | N. Arkani-Hamed, Y. Bai, and T. Lam, Positive geometries and canonical forms, JHEP November 2017, Article 39. · Zbl 1383.81273 |
| [3] | N. Arkani-Hamed, S. He, and T. Lam, Stringy canonical forms, JHEP February 2021, Article 69. · Zbl 1460.83083 |
| [4] | Nima Arkani-Hamed and Jaroslav Trnka. The amplituhedron. Journal of High Energy Physics, 2014(10):30, Oct 2014. · Zbl 1468.81075 |
| [5] | Ruth Britto, Freddy Cachazo, Bo Feng, and Edward Witten. Direct proof of the tree-level scattering amplitude recursion relation in Yang-Mills theory. Phys. Rev. Lett., 94(18):181602, 4, 2005. |
| [6] | Michael P. Carr and Satyan L. Devadoss. Coxeter complexes and graph-associahedra. Topol-ogy Appl., 153(12):2155-2168, 2006. · Zbl 1099.52001 |
| [7] | Pavel Galashin. Critical varieties in the Grassmannian. arXiv:2102.13339, 2021. |
| [8] | Pavel Galashin. Poset associahedra. arXiv:2110.07257, 2021. |
| [9] | Pavel Galashin. Totally nonnegative critical varieties. arXiv:2110.08548, 2021. |
| [10] | Pavel Galashin and Pavlo Pylyavskyy. Ising model and the positive orthogonal Grassman-nian. Duke Math. J., 169(10):1877-1942, 2020. · Zbl 1446.82008 |
| [11] | R. Kenyon. The Laplacian and Dirac operators on critical planar graphs. Invent. Math., 150(2):409-439, 2002. · Zbl 1038.58037 |
| [12] | Thomas Lam. Electroid varieties and a compactification of the space of electrical networks. Adv. Math., 338:549-600, 2018. · Zbl 1396.05105 |
| [13] | Suho Oh, Alexander Postnikov, and David E. Speyer. Weak separation and plabic graphs. Proc. Lond. Math. Soc. (3), 110(3):721-754, 2015. · Zbl 1309.05182 |
| [14] | Matteo Parisi, Melissa Sherman-Bennett, and Lauren Williams. The m = 2 amplituhedron and the hypersimplex: signs, clusters, triangulations, Eulerian numbers. arXiv:2104.08254, 2021. |
| [15] | Alexander Postnikov. Total positivity, Grassmannians, and networks. Preprint, http:// math.mit.edu/ apost/papers/tpgrass.pdf, 2006. |
| [16] | D. Agostini, T. Brysiewicz, C. Fevola, L. Kühne, B. Sturmfels, and S. Telen: Likelihood Degenerations. arXiv preprint arXiv:2107.10518 (2021). |
| [17] | P. Breiding and S. Timme: HomotopyContinuation.jl: A Package for Homotopy Contin-uation in Julia, Math. Software -ICMS 2018, 458-465, Springer International Publishing (2018). · Zbl 1396.14003 |
| [18] | F. Cachazo, N. Early, A. Guevara and S. Mizera: Scattering equations: from projective spaces to tropical Grassmannians, J. High Energy Phys. 39 (2019). · Zbl 1445.81064 |
| [19] | F. Cachazo, S. He and E. Y. Yuan: Scattering equations and Kawai-Lewellen-Tye orthog-onality, Physical Review D 90 (2014) 065001. |
| [20] | F. Cachazo, B. Umbert and Y. Zhang: Singular solutions in soft limits, J. High Energy Phys. 148 (2020). |
| [21] | F. Catanese, S. Hoşten, A. Khetan and B. Sturmfels: The maximum likelihood degree, American Journal of Mathematics 128 (2006) 671-697. · Zbl 1123.13019 |
| [22] | E. Gross and J. Rodriguez: Maximum likelihood geometry in the presence of data zeros, ISSAC 2014 -39th Internat. Symposium on Symbolic and Algebraic Computation, 232-239, ACM, New York (2014). · Zbl 1325.68285 |
| [23] | J. Huh: The maximum likelihood degree of a very affine variety, Compos. Math. 149 (2013) 1245-1266. · Zbl 1282.14007 |
| [24] | J. Huh and B. Sturmfels: Likelihood geometry, Combinatorial Algebraic Geometry, Lecture Notes in Mathematics 2108, Springer Verlag, 63-117 (2014). · Zbl 1328.14004 |
| [25] | D. Maclagan and B. Sturmfels: Introduction to Tropical Geometry, Graduate Studies in Mathematics, Vol. 161, American Mathematical Society (2015). · Zbl 1321.14048 |
| [26] | B. Sturmfels and S. Telen: Likelihood equations and scattering amplitudes, Algebraic Sta-tistics, to appear, arXiv:2012.05041. |
| [27] | N. Arkani-Hamed, S. He, T. Lam, and H. Thomas. “Binary geometries, generalized particles and strings, and cluster algebras.” arXiv preprint arXiv:1912.11764 (2019). |
| [28] | N. Arkani-Hamed, T. Lam, and M. Spradlin. “Positive configuration space.” Communica-tions in Mathematical Physics (2021): 1-46. |
| [29] | F. Cachazo and N. Early. Minimal Kinematics: An All k and n Peek into T rop + G(k, n). |
| [30] | Symmetry, Integrability and Geometry: Methods and Applications 17, no. 0 (2021): 78-22. |
| [31] | F. Cachazo and N. Early. Planar Kinematics: Cyclic Fixed Points, Mirror Superpotential, k-Dimensional Catalan Numbers, and Root Polytopes. arXiv:2010.09708 (2020). |
| [32] | N. Early. Planarity in Generalized Scattering Amplitudes: PK Polytope, Generalized Root Systems and Worldsheet Associahedra. arXiv:2106.07142 (2021). |
| [33] | I. M. Gelfand, M. Graev, and A. Postnikov. Combinatorics of hypergeometric functions associated with positive roots. In The Arnold-Gelfand mathematical seminars, pp. 205-221. Birkhauser Boston, 1997. · Zbl 0876.33011 |
| [34] | S. Karp. “Moment curves and cyclic symmetry for positive Grassmannians.” Bulletin of the London Mathematical Society 51, no. 5 (2019): 900-916. · Zbl 1443.14048 |
| [35] | R. Marsh and K. Rietsch. The B-model connection and mirror symmetry for Grassmanni-ans. Advances in Mathematics 366 (2020): 107027. · Zbl 1453.14104 |
| [36] | K. Rietsch. A mirror construction for the totally nonnegative part of the Peterson variety. Nagoya Mathematical Journal 183 (2006): 105-142. · Zbl 1111.14048 |
| [37] | K. Rietsch and L. Williams. Newton-Okounkov bodies, cluster duality, and mirror symmetry for Grassmannians. Duke Mathematical Journal 168, no. 18 (2019): 3437-3527. · Zbl 1439.14142 |
| [38] | N. Arkani-Hamed and J. Trnka, The Amplituhedron, JHEP 10 (2014), 030. · Zbl 1468.81075 |
| [39] | N. Arkani-Hamed, Y. Bai, S. He and G. Yan, Scattering Forms and the Positive Geometry of Kinematics, Color and the Worldsheet, JHEP 05 (2018), 096. · Zbl 1391.81200 |
| [40] | G. t’Hooft, A planar diagram theory for strong interactions, Nuclear Physics B 72 (3): 461, (1974). |
| [41] | V.V. Fock and A.B. Goncharov, Dual Teichüller and lamination spaces. Handbook of Te-ichmüller theory, 1(11), 647-684, (2007). · Zbl 1162.32009 |
| [42] | R. Penner, Decorated Teichmüller theory (Vol. 1), European Mathematical Society, (2012). · Zbl 1243.30003 |
| [43] | S. Fomin, M. Shapiro and D. Thurston, Cluster algebras and triangulated surfaces. Part I: Cluster complexes, Acta Mathematica, 201(1), 83-146, (2008). · Zbl 1263.13023 |
| [44] | Arkani-Hamed, N., et al. Binary Geometries, Generalized Particles and Strings, and Cluster Algebras. ArXiv:1912.11764 [Hep-Th], 2020. |
| [45] | Brown, Francis C. S. Multiple Zeta Values and Periods of Moduli Spaces M 0,N . Annales Scientifiques de l’École Normale Supérieure, vol. 42, no. 3, 2009, pp. 371-489. · Zbl 1216.11079 |
| [46] | Nima Arkani-Hamed, Song He, Thomas Lam and Hugh Thomas, Binary geometries, gen-eralized particles and strings, and cluster algebras, arXiv:1912.11764. |
| [47] | Ibrahim Assem, Thomas Brüstle, Gabrielle Charbonneau-Jodoin and Pierre-Guy Plamon-don, Gentle algebras arising from surface triangulations Algebra Number Theory 4 (2010), no. 2, 201-229. · Zbl 1242.16011 |
| [48] | Philippe Caldero and Frédéric Chapoton, Cluster algebras as Hall algebras of quiver repre-sentations, Comment. Math. Helv. 81 (2006), no. 3, 595-616. · Zbl 1119.16013 |
| [49] | Salomón Dominguez and Christof Geiss, A Caldero-Chapoton formula for generalized clus-ter categories, J. Algebra 399 (2014), 887-893. · Zbl 1316.13033 |
| [50] | Zirô Koba and Holger Bech Nielsen, Reaction amplitude for n-mesons: A generalization of the Veneziano-Bardakçi-Ruegg-Virasoro model, Nuclear Physics B10 (1969) 633-665. |
| [51] | Markus Reineke, Every projective variety is a quiver Grassmannian, Algebr. Represent. Theory 16 (2013), no. 5, 1313-1314. · Zbl 1284.16015 |
| [52] | Nima Arkani-Hamed and Jaroslav Trnka. The amplituhedron. J. High Energy Phys., (10):33, 2014. · Zbl 1468.81075 |
| [53] | A. M. Bloch, H. Flaschka, and T. Ratiu. A convexity theorem for isospectral manifolds of Jacobi matrices in a compact Lie algebra. Duke Math. J., 61(1):41-65, 1990. · Zbl 0715.58004 |
| [54] | Arkady Berenstein, Sergey Fomin, and Andrei Zelevinsky. Parametrizations of canonical bases and totally positive matrices. Adv. Math., 122(1):49-149, 1996. · Zbl 0966.17011 |
| [55] | Yuntao Bai, Song He, and Thomas Lam. The amplituhedron and the one-loop Grassmannian measure. J. High Energy Phys., (1):112, front matter+41, 2016. · Zbl 1388.81763 |
| [56] | Anthony M. Bloch and Steven N. Karp. Gradient flows, adjoint orbits, and the topology of totally nonnegative flag varieties. arXiv:2109.04558. |
| [57] | Anthony M. Bloch and Steven N. Karp. On two notions of total positivity for partial flag varieties. In preparation. · Zbl 1516.14089 |
| [58] | Sergey Fomin and Andrei Zelevinsky. Double Bruhat cells and total positivity. J. Amer. Math. Soc., 12(2):335-380, 1999. · Zbl 0913.22011 |
| [59] | Pavel Galashin, Steven N. Karp, and Thomas Lam. The totally nonnegative Grassmannian is a ball. Adv. Math. (to appear), arXiv:1707.02010. |
| [60] | Pavel Galashin, Steven N. Karp, and Thomas Lam. The totally nonnegative part of G/P is a ball. Adv. Math., 351:614-620, 2019. · Zbl 1440.14224 |
| [61] | G. Lusztig. Total positivity in reductive groups. In Lie theory and geometry, volume 123 of Progr. Math., pages 531-568. Birkhäuser Boston, Boston, MA, 1994. · Zbl 0845.20034 |
| [62] | G. Lusztig. Total positivity in partial flag manifolds. Represent. Theory, 2:70-78, 1998. Towards cluster super-algebras Marcus Spradlin (joint work with S. James Gates, Jr., S.-N. Hazel Mak, Anastasia Volovich) · Zbl 0895.14014 |
| [63] | This talk is based on the preprint [1]. The workshop focuses on connections be-tween scattering amplitudes and cluster algebras. There have recently been several apparently distinct approaches to the construction of cluster superalgebras (see for example [2, 3, 4, 5, 6, 7]), and it is natural to wonder whether these make any appearance in physics. In the talk I review the definition proposed by Ovsienko References |
| [64] | S. J. Gates, S. N. Hazel Mak, M. Spradlin and A. Volovich, Cluster Superalgebras and Stringy Integrals, arXiv:2111.08186 (2021). |
| [65] | V. Ovsienko, Cluster superalgebras, arXiv:1503.01894 (2015). |
| [66] | L. Li, J. Mixco, B. Ransingh and A. K. Srivastava, An Introduction to Supersymmetric Cluster Algebras, arXiv:1708.03851 (2017). |
| [67] | E. Shemyakova and T. Voronov, On super Plücker embedding and cluster algebras, arXiv:1906.12011 (2019). |
| [68] | V. Ovsienko and M. Shapiro, Cluster algebras with Grassmann variables, Electronic Re-search Announcements 26 (2019), 1. · Zbl 1442.13081 |
| [69] | G. Musiker, N. Ovenhouse and S. W. Zhang, An Expansion Formula for Decorated Super-Teichmüller Spaces, arXiv:2102.09143 (2021). |
| [70] | G. Musiker, N. Ovenhouse and S. W. Zhang, Double Dimer Covers on Snake Graphs from Super Cluster Expansions, arXiv:2110.06497 (2021). |
| [71] | D. Damgaard, L. Ferro, T. Lukowski, M. Parisi, The momentum amplituhedron, J. High Energy Phys. 8 (2019). · Zbl 1421.81148 |
| [72] | L. Ferro, T. Lukowski, R. Moerman, From momentum amplituhedron boundaries to ampli-tude singularities and back, J. High Energy Phys. 7 (2020). · Zbl 1451.81358 |
| [73] | R. Moerman, L. Williams, Grass trees and forests: Enumeration of Grassmannian trees and forests, with applications to the momentum amplituhedron, arXiv:2112.02061. References |
| [74] | N. Arkani-Hamed, Y. Bai and T. Lam, JHEP 11, 039 (2017) doi:10.1007/JHEP11(2017)039 [arXiv:1703.04541 [hep-th]]. · Zbl 1383.81273 · doi:10.1007/JHEP11(2017)039 |
| [75] | N. Arkani-Hamed and J. Trnka, JHEP 10, 030 (2014) doi:10.1007/JHEP10(2014)030 [arXiv:1312.2007 [hep-th]]. · Zbl 1468.81075 · doi:10.1007/JHEP10(2014)030 |
| [76] | D. Damgaard, L. Ferro, T. Lukowski and M. Parisi, JHEP 08, 042 (2019) doi:10.1007/JHEP08(2019)042 [arXiv:1905.04216 [hep-th]]. · Zbl 1421.81148 · doi:10.1007/JHEP08(2019)042 |
| [77] | Y. t. Huang, R. Kojima, C. Wen and S. Q. Zhang, [arXiv:2111.03037 [hep-th]]. |
| [78] | S. He, C. K. Kuo and Y. Q. Zhang, [arXiv:2111.02576 [hep-th]]. |
| [79] | N. Arkani-Hamed, Y. Bai, S. He and G. Yan, JHEP 05, 096 (2018) doi:10.1007/JHEP05(2018)096 [arXiv:1711.09102 [hep-th]]. · Zbl 1391.81200 · doi:10.1007/JHEP05(2018)096 |
| [80] | T. Lukowski, [arXiv:1908.00386 [hep-th]]. |
| [81] | T. Lukowski and R. Moerman, Comput. Phys. Commun. 259, 107653 (2021) doi:10.1016/j.cpc.2020.107653 [arXiv:2002.07146 [hep-th]]. · Zbl 1530.81132 · doi:10.1016/j.cpc.2020.107653 |
| [82] | R. Moerman and L. K. Williams, [arXiv:2112.02061 [math.CO]]. |
| [83] | A. Postnikov, [arXiv:1806.05307 [math.CO]]. |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
