Geodesic
Notes on Worldsheet-Like Variables for Cluster Configuration Spaces
Symmetry, integrability and geometry: methods and applications, Tome 19 (2023) Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

We continue the exploration of various appearances of cluster algebras in scattering amplitudes and related topics in physics. The cluster configuration spaces generalize the familiar moduli space ${\mathcal M}_{0,n}$ to finite-type cluster algebras. We study worldsheet-like variables, which for classical types have also appeared in the study of the symbol alphabet of Feynman integrals. We provide a systematic derivation of these variables from $Y$-systems, which allows us to express the dihedral coordinates in terms of them and to write the corresponding cluster string integrals in compact forms. We mainly focus on the $D_n$ type and show how to reach the boundaries of the configuration space, and write the saddle-point equations in terms of these variables. Moreover, these variables make it easier to study various topological properties of the space using a finite-field method. We propose conjectures about quasi-polynomial point count, dimensions of cohomology, and the number of saddle points for the $D_n$ space up to $n=10$, which greatly extend earlier results.
Keywords: cluster algebras, generalized associahedra, $Y$-systems, string amplitudes. 
@article{SIGMA_2023_19_a44,
     author = {Song He and Yihong Wang and Yong Zhang and Peng Zhao},
     title = {Notes on {Worldsheet-Like} {Variables} for {Cluster} {Configuration} {Spaces}},
     journal = {Symmetry, integrability and geometry: methods and applications},
     year = {2023},
     volume = {19},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SIGMA_2023_19_a44/}
}
TY  - JOUR
AU  - Song He
AU  - Yihong Wang
AU  - Yong Zhang
AU  - Peng Zhao
TI  - Notes on Worldsheet-Like Variables for Cluster Configuration Spaces
JO  - Symmetry, integrability and geometry: methods and applications
PY  - 2023
VL  - 19
UR  - http://geodesic.mathdoc.fr/item/SIGMA_2023_19_a44/
LA  - en
ID  - SIGMA_2023_19_a44
ER  - 
%0 Journal Article
%A Song He
%A Yihong Wang
%A Yong Zhang
%A Peng Zhao
%T Notes on Worldsheet-Like Variables for Cluster Configuration Spaces
%J Symmetry, integrability and geometry: methods and applications
%D 2023
%V 19
%U http://geodesic.mathdoc.fr/item/SIGMA_2023_19_a44/
%G en
%F SIGMA_2023_19_a44
Song He; Yihong Wang; Yong Zhang; Peng Zhao. Notes on Worldsheet-Like Variables for Cluster Configuration Spaces. Symmetry, integrability and geometry: methods and applications, Tome 19 (2023). http://geodesic.mathdoc.fr/item/SIGMA_2023_19_a44/

[1] Abhishek M., Hegde S., Jatkar D.P., Saha A.P., “Double soft theorem for generalised biadjoint scalar amplitudes”, SciPost Phys., 10 (2021), 036, 39 pp., arXiv: 2008.07271 | DOI | MR

[2] Agostini D., Brysiewicz T., Fevola C., Kühne L., Sturmfels B., Telen S., “Likelihood degenerations (with an appendix by Thomas Lam)”, Adv. Math., 414 (2023), 108863, 39 pp., arXiv: 2107.10518 | DOI | MR

[3] Arkani-Hamed N., Bai Y., He S., Yan G., “Scattering forms and the positive geometry of kinematics, color and the worldsheet”, J. High Energy Phys., 2018:5 (2018), 096, 76 pp., arXiv: 1711.09102 | DOI | MR

[4] Arkani-Hamed N., Bai Y., Lam T., “Positive geometries and canonical forms”, J. High Energy Phys., 2017:11 (2017), 039, 122 pp., arXiv: 1703.04541 | DOI | MR

[5] Arkani-Hamed N., He S., Lam T., “Cluster configuration spaces of finite type”, SIGMA, 17 (2021), 092, 41 pp., arXiv: 2005.11419 | DOI | MR | Zbl

[6] Arkani-Hamed N., He S., Lam T., “Stringy canonical forms”, J. High Energy Phys., 2021:2 (2021), 069, 59 pp., arXiv: 1912.08707 | DOI | MR

[7] Arkani-Hamed N., He S., Lam T., Thomas H., “Binary geometries, generalized particles and strings, and cluster algebras”, Phys. Rev. D, 107 (2023), 066015, 8 pp., arXiv: 1912.11764 | DOI | MR

[8] Arkani-Hamed N., He S., Salvatori G., Thomas H., “Causal diamonds, cluster polytopes and scattering amplitudes”, J. High Energy Phys., 2022:11 (2022), 049, 21 pp., arXiv: 1912.12948 | DOI | MR

[9] Arkani-Hamed N., Lam T., Spradlin M., “Non-perturbative geometries for planar $\mathcal{N} = 4$ SYM amplitudes”, J. High Energy Phys., 2021:3 (2021), 065, 14 pp., arXiv: 1912.08222 | DOI | MR

[10] Borges F., Cachazo F., “Generalized planar Feynman diagrams: collections”, J. High Energy Phys., 2020:11 (2020), 164, 27 pp., arXiv: 1910.10674 | DOI | MR | Zbl

[11] Breiding P., Timme S., “HomotopyContinuation.jl: a package for homotopy continuation in Julia”, Mathematical Software, Lecture Notes in Comput. Sci, 10931, Springer, Cham, 2018, 458–465, arXiv: 1711.10911 | DOI | Zbl

[12] Brown F.C.S., “Multiple zeta values and periods of moduli spaces $\overline{\mathfrak{M}}_{0,n}$”, Ann. Sci. Éc. Norm. Supér., 42 (2009), 371–489, arXiv: math.AG/0606419 | DOI | MR | Zbl

[13] Cachazo F., Early N., “Minimal kinematics: an all $k$ and $n$ peek into ${\rm Trop}^+{\rm G}(k,n)$”, SIGMA, 17 (2021), 078, 22 pp., arXiv: 2003.07958 | DOI | MR | Zbl

[14] Cachazo F., Early N., Planar kinematics: cyclic fixed points, mirror superpotential, $k$-dimensional Catalan numbers, and root polytopes, arXiv: 2010.09708

[15] Cachazo F., Early N., Guevara A., Mizera S., “Scattering equations: from projective spaces to tropical Grassmannians”, J. High Energy Phys., 2019:6 (2019), 039, 32 pp., arXiv: 1903.08904 | DOI | MR

[16] Cachazo F., Guevara A., Umbert B., Zhang Y., Planar matrices and arrays of Feynman diagrams, arXiv: 1912.09422

[17] Cachazo F., He S., Yuan E.Y., “Scattering of massless particles in arbitrary dimensions”, Phys. Rev. Lett., 113 (2014), 171601, 4 pp., arXiv: 1307.2199 | DOI

[18] Cachazo F., Mizera S., Zhang G., “Scattering equations: real solutions and particles on a line”, J. High Energy Phys., 2017:3 (2017), 151, 22 pp., arXiv: 1609.00008 | DOI | MR | Zbl

[19] Cachazo F., Rojas J.M., “Notes on biadjoint amplitudes, ${\rm Trop}\, G(3,7)$ and ${\rm X}(3,7)$ scattering equations”, J. High Energy Phys., 2020:4 (2020), 176, 13 pp., arXiv: 1906.05979 | DOI | MR | Zbl

[20] Cachazo F., Umbert B., Zhang Y., “Singular solutions in soft limits”, J. High Energy Phys., 2020:5 (2020), 148, 32 pp., arXiv: 1911.02594 | DOI | MR

[21] Chicherin D., Henn J.M., Papathanasiou G., “Cluster algebras for Feynman integrals”, Phys. Rev. Lett., 126 (2021), 091603, 7 pp., arXiv: 2012.12285 | DOI | MR

[22] Deligne P., Mumford D., “The irreducibility of the space of curves of given genus”, Inst. Hautes Études Sci. Publ. Math., 36 (1969), 75–109 | DOI | MR | Zbl

[23] Drummond J., Foster J., Gürdoğan O., “Cluster adjacency beyond MHV”, J. High Energy Phys., 2019:3 (2019), 086, 43 pp., arXiv: 1810.08149 | DOI | MR

[24] Drummond J., Foster J., Gürdoğan O., Kalousios C., “Tropical Grassmannians, cluster algebras and scattering amplitudes”, J. High Energy Phys., 2020:4 (2020), 146, 22 pp., arXiv: 1907.01053 | DOI | MR | Zbl

[25] Drummond J., Foster J., Gürdoğan O., Kalousios C., “Algebraic singularities of scattering amplitudes from tropical geometry”, J. High Energy Phys., 2021:4 (2021), 002, 18 pp., arXiv: 1912.08217 | DOI | MR

[26] Drummond J., Foster J., Gürdoğan O., Kalousios C., “Tropical fans, scattering equations and amplitudes”, J. High Energy Phys., 2021:11 (2021), 071, 26 pp., arXiv: 2002.04624 | DOI | MR

[27] Early N., Planar kinematic invariants, matroid subdivisions and generalized Feynman diagrams, arXiv: 1912.13513

[28] Early N., “From weakly separated collections to matroid subdivisions”, Comb. Theory, 2:2 (2022), 2, 35 pp., arXiv: 1910.11522 | DOI | MR

[29] Fock V.V., Goncharov A.B., “Cluster ensembles, quantization and the dilogarithm”, Ann. Sci. Éc. Norm. Supér, 42 (2009), 865–930, arXiv: math.AG/0311245 | DOI | MR | Zbl

[30] Fomin S., Shapiro M., Thurston D., “Cluster algebras and triangulated surfaces. I Cluster complexes”, Acta Math., 201 (2008), 83–146, arXiv: math.RA/0608367 | DOI | MR | Zbl

[31] Fomin S., Zelevinsky A., “Cluster algebras. II Finite type classification”, Invent. Math., 154 (2003), 63–121, arXiv: math.RA/0208229 | DOI | MR | Zbl

[32] Fomin S., Zelevinsky A., “$Y$-systems and generalized associahedra”, Ann. of Math., 158 (2003), 977–1018, arXiv: hep-th/0111053 | DOI | MR | Zbl

[33] Gekhtman M., Shapiro M., Vainshtein A., “Cluster algebras and Weil–Petersson forms”, Duke Math. J., 127 (2005), 291–311, arXiv: math.QA/0309138 | DOI | MR | Zbl

[34] Guevara A., Zhang Y., Planar matrices and arrays of Feynman diagrams: poles for higher $k$, arXiv: 2007.15679

[35] He S., Li Z., Yang Q., “Notes on cluster algebras and some all-loop Feynman integrals”, J. High Energy Phys., 2021:6 (2021), 119, 26 pp., arXiv: 2103.02796 | DOI | MR

[36] He S., Li Z., Yang Q., “Truncated cluster algebras and Feynman integrals with algebraic letters”, J. High Energy Phys., 2021:12 (2021), 110, 28 pp., arXiv: 2106.09314 | DOI | MR

[37] He S., Li Z., Yang Q., “Comments on all-loop constraints for scattering amplitudes and Feynman integrals”, J. High Energy Phys., 2022:1 (2022), 073, 23 pp., arXiv: 2108.07959 | DOI | MR

[38] He S., Ren L., Zhang Y., “Notes on polytopes, amplitudes and boundary configurations for Grassmannian string integrals”, J. High Energy Phys., 2020:4 (2020), 140, 37 pp., arXiv: 2001.09603 | DOI | MR

[39] Henke N., Papathanasiou G., How tropical are seven- and eight-particle amplitudes?, J. High Energy Phys., 2020:8 (2020), 005, 49 pp., arXiv: 1912.08254 | DOI | MR

[40] Henke N., Papathanasiou G., “Singularities of eight- and nine-particle amplitudes from cluster algebras and tropical geometry”, J. High Energy Phys., 2021:10 (2021), 007, 60 pp., arXiv: 2106.01392 | DOI | MR

[41] Kalousios C., “Massless scattering at special kinematics as Jacobi polynomials”, J. Phys. A, 47 (2014), 215402, 8 pp., arXiv: 1312.7743 | DOI | MR | Zbl

[42] Knudsen F.F., “The projectivity of the moduli space of stable curves. II The stacks $M_{g,n}$”, Math. Scand., 52 (1983), 161–199 | DOI | MR | Zbl

[43] Koba Z., Nielsen H.B., “Manifestly crossing invariant parametrization of $n$ meson amplitude”, Nuclear Phys. B, 12 (1969), 517–536 | DOI

[44] Koba Z., Nielsen H.B., “Reaction amplitude for $n$ mesons: a generalization of the Veneziano–Bardakci–Ruegg–Virasora model”, Nuclear Phys. B, 10 (1969), 633–655 | DOI

[45] Łukowski T., Parisi M., Williams L.K., “The positive tropical Grassmannian, the hypersimplex, and the $m=2$ amplituhedron”, Int. Math. Res. Not. (to appear) , arXiv: 2002.06164 | DOI

[46] Orlik P., Solomon L., “Combinatorics and topology of complements of hyperplanes”, Invent. Math., 56 (1980), 167–189 | DOI | MR | Zbl

[47] Ren L., Spradlin M., Volovich A., “Symbol alphabets from tensor diagrams”, J. High Energy Phys., 2021:12 (2021), 079, 41 pp., arXiv: 2106.01405 | DOI | MR

[48] Skorobogatov A.N., “On the number of representations of matroids over finite fields”, Des. Codes Cryptogr., 9 (1996), 215–226 | DOI | MR | Zbl

[49] Sturmfels B., Telen S., “Likelihood equations and scattering amplitudes”, Algebr. Stat., 12 (2021), 167–186, arXiv: 2012.05041 | DOI | MR

[50] Volkov A.Yu., “On the periodicity conjecture for $Y$-systems”, Comm. Math. Phys., 276 (2007), 509–517 | DOI | MR | Zbl

[51] Wang Y., Zhao P., $Y$-systems, $z$-variables, and cluster alphabets, in preparation

[52] Zamolodchikov A.B., “On the thermodynamic Bethe ansatz equations for reflectionless $ADE$ scattering theories”, Phys. Lett. B, 253 (1991), 391–394 | DOI | MR