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Moduli spaces in positive geometry (2405.17332v2)

Published 27 May 2024 in math.AG, hep-th, and math.CO

Abstract: These are lecture notes for five lectures given at MPI Leipzig in May 2024. We study the moduli space M_{0,n} of n distinct points on P1 as a positive geometry and a binary geometry. We develop mathematical formalism to study Cachazo-He-Yuan's scattering equations and the associated scalar and Yang-Mills amplitudes. We discuss open superstring amplitudes and relations to tropical geometry.

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References (66)
  1. Grassmannian Geometry of Scattering Amplitudes. Cambridge University Press, 2016.
  2. Unification of Residues and Grassmannian Dualities. JHEP, 01:049, 2011.
  3. Scattering Forms and the Positive Geometry of Kinematics, Color and the Worldsheet. JHEP, 05:096, 2018.
  4. Positive Geometries and Canonical Forms. JHEP, 11:039, 2017.
  5. Scalar-scaffolded gluons and the combinatorial origins of yang-mills theory. arXiv:2401.00041, 2024.
  6. All loop scattering as a counting problem. arXiv:2309.15913, 2023.
  7. Cluster configuration spaces of finite type. SIGMA Symmetry Integrability Geom. Methods Appl., 17:Paper No. 092, 41, 2021.
  8. Stringy canonical forms. J. High Energy Phys., (2):Paper No. 069, 59, 2021.
  9. Binary geometries, generalized particles and strings, and cluster algebras. Phys. Rev. D, 107(6):Paper No. 066015, 8, 2023.
  10. Non-perturbative geometries for planar 𝒩=4𝒩4\mathcal{N}=4caligraphic_N = 4 SYM amplitudes. J. High Energy Phys., (3):Paper No. 065, 14, 2021.
  11. Positive configuration space. Comm. Math. Phys., 384(2):909–954, 2021.
  12. The Amplituhedron. JHEP, 10:030, 2014.
  13. Integration rules for loop scattering equations. J. High Energy Phys., (11):080, front matter+19, 2015.
  14. Wondertopes. arXiv:2403.04610, 2024.
  15. Ruth Britto. Constructing scattering amplitudes. https://www.maths.tcd.ie/~britto/lecture-notes.pdf.
  16. Francis C. S. Brown. Multiple zeta values and periods of moduli spaces 𝔐0,nsubscript𝔐0𝑛\mathfrak{M}_{0,n}fraktur_M start_POSTSUBSCRIPT 0 , italic_n end_POSTSUBSCRIPT. Annales Sci. Ecole Norm. Sup., 42:371, 2009.
  17. Planar kinematics: Cyclic fixed points, mirror superpotential, k-dimensional catalan numbers, and root polytopes. arXiv:2010.09708, 2020.
  18. Scattering Equations: From Projective Spaces to Tropical Grassmannians. JHEP, 06:039, 2019.
  19. Color-dressed generalized biadjoint scalar amplitudes: Local planarity. arXiv:2212.11243, 2022.
  20. Scattering in Three Dimensions from Rational Maps. JHEP, 10:141, 2013.
  21. Scattering equations and Kawai-Lewellen-Tye orthogonality. Phys. Rev., D90(6):065001, 2014.
  22. Scattering of Massless Particles in Arbitrary Dimensions. Phys. Rev. Lett., 113(17):171601, 2014.
  23. Scattering of Massless Particles: Scalars, Gluons and Gravitons. JHEP, 07:033, 2014.
  24. The Momentum Amplituhedron. JHEP, 08:042, 2019.
  25. Proof of the formula of Cachazo, He and Yuan for Yang-Mills tree amplitudes in arbitrary dimension. J. High Energy Phys., (5):010, front matter+23, 2014.
  26. Lance J. Dixon. Calculating scattering amplitudes efficiently. arXiv:hep-ph/9601359.
  27. Scattering amplitudes in gauge theory and gravity. Cambridge University Press, Cambridge, 2015.
  28. Regularity theorem for totally nonnegative flag varieties. J. Amer. Math. Soc., 35(2):513–579, 2022.
  29. The totally nonnegative Grassmannian is a ball. Adv. Math., 397:Paper No. 108123, 23, 2022.
  30. Parity duality for the amplituhedron. Compos. Math., 156(11):2207–2262, 2020.
  31. The momentum amplituhedron of SYM and ABJM from twistor-string maps. J. High Energy Phys., (2):Paper No. 148, 32, 2022.
  32. Stringy canonical forms and binary geometries from associahedra, cyclohedra and generalized permutohedra. J. High Energy Phys., (10):054, 35, 2020.
  33. Likelihood geometry. In Combinatorial algebraic geometry, volume 2108 of Lecture Notes in Math., pages 63–117. Springer, Cham, 2014.
  34. June Huh. The maximum likelihood degree of a very affine variety. Compos. Math., 149(8):1245–1266, 2013.
  35. Notes on Scattering Amplitudes as Differential Forms. JHEP, 10:054, 2018.
  36. Steven N. Karp. Sign variation, the Grassmannian, and total positivity. J. Combin. Theory Ser. A, 145:308–339, 2017.
  37. Sean Keel. Intersection theory of moduli space of stable n𝑛nitalic_n-pointed curves of genus zero. Trans. Amer. Math. Soc., 330(2):545–574, 1992.
  38. Positroid varieties: juggling and geometry. Compos. Math., 149(10):1710–1752, 2013.
  39. Manifestly crossing invariant parametrization of n meson amplitude. Nucl. Phys., B12:517–536, 1969.
  40. Polar homology and holomorphic bundles. volume 359, pages 1413–1427. 2001. Topological methods in the physical sciences (London, 2000).
  41. Thomas Lam. in preparation.
  42. Thomas Lam. Amplituhedron cells and Stanley symmetric functions. Comm. Math. Phys., 343(3):1025–1037, 2016.
  43. Thomas Lam. Totally nonnegative Grassmannian and Grassmann polytopes. pages 51–152, 2016.
  44. Thomas Lam. An invitation to positive geometries. arXiv:2208.05407, 2022.
  45. Four lectures on euler integrals. arXiv:2306.13578, 2023.
  46. Sebastian Mizera. Scattering amplitudes from intersection theory. Phys. Rev. Lett., 120(14):141602, 6, 2018.
  47. Sebastian Mizera. Aspects of scattering amplitudes and moduli space localization. Springer Theses. Springer, Cham, [2020] ©2020. Doctoral thesis accepted by the University of Waterloo, Canada.
  48. Ambitwistor strings and the scattering equations. Journal of High Energy Physics, 2014(7):48, 2014.
  49. Introduction to tropical geometry, volume 161 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 2015.
  50. Mellin transforms of multivariate rational functions. J. Geom. Anal., 23(1):24–46, 2013.
  51. The number of critical points of a product of powers of linear functions. Invent. Math., 120(1):1–14, 1995.
  52. Rahul Pandharipande. The canonical class of M¯0,n⁢(ℙr,d)subscript¯𝑀0𝑛superscriptℙ𝑟𝑑\overline{M}_{0,n}(\mathbb{P}^{r},d)over¯ start_ARG italic_M end_ARG start_POSTSUBSCRIPT 0 , italic_n end_POSTSUBSCRIPT ( blackboard_P start_POSTSUPERSCRIPT italic_r end_POSTSUPERSCRIPT , italic_d ) and enumerative geometry. Internat. Math. Res. Notices, (4):173–186, 1997.
  53. Alexander Postnikov. Total positivity, Grassmannians, and networks. 2006.
  54. Claudio Procesi. Lie groups. Universitext. Springer, New York, 2007. An approach through invariants and representations.
  55. Maxwell Rosenlicht. Some rationality questions on algebraic groups. Ann. Mat. Pura Appl. (4), 43:25–50, 1957.
  56. On the tree level S matrix of Yang-Mills theory. Phys. Rev., D70:026009, 2004.
  57. Pierre Samuel. À propos du théorème des unités. Bull. Sci. Math. (2), 90:89–96, 1966.
  58. Rob Silversmith. Cross-ratio degrees and triangulations. arXiv:2310.07377, 2023.
  59. O. Schlotterer and S. Stieberger. Motivic multiple zeta values and superstring amplitudes. J. Phys. A, 46(47):475401, 37, 2013.
  60. Likelihood equations and scattering amplitudes. Algebr. Stat., 12(2):167–186, 2021.
  61. The tropical totally positive grassmannian. Journal of Algebraic Combinatorics, 22(2):189–210, 2005.
  62. The positive Dressian equals the positive tropical Grassmannian. Trans. Amer. Math. Soc. Ser. B, 8:330–353, 2021.
  63. Jenia Tevelev. Compactifications of subvarieties of tori. Amer. J. Math., 129(4):1087–1104, 2007.
  64. A. Varchenko. Critical points of the product of powers of linear functions and families of bases of singular vectors. Compositio Math., 97(3):385–401, 1995.
  65. Gabriele Veneziano. Construction of a crossing-symmetric, Regge-behaved amplitude for linearly rising trajectories. Il Nuovo Cimento A (1965-1970), 57(1):190–197, 1968.
  66. Edward Witten. Perturbative gauge theory as a string theory in twistor space. Commun. Math. Phys., 252:189–258, 2004.
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Authors (1)

  1. Thomas Lam
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