Abstract
Stringy canonical forms are a class of integrals that provide α′-deformations of the canonical form of any polytopes. For generalized associahedra of finite-type cluster algebras, there exist completely rigid stringy integrals, whose configuration spaces are the so-called binary geometries, and for classical types are associated with (generalized) scattering of particles and strings. In this paper, we propose a large class of rigid stringy canonical forms for another class of polytopes, generalized permutohedra, which also include associahedra and cyclohedra as special cases (type An and Bn generalized associahedra). Remarkably, we find that the configuration spaces of such integrals are also binary geometries, which were suspected to exist for generalized associahedra only. For any generalized permutohedron that can be written as Minkowski sum of coordinate simplices, we show that its rigid stringy integral factorizes into products of lower integrals for massless poles at finite α′, and the configuration space is binary although the u equations take a more general form than those “perfect” ones for cluster cases. Moreover, we provide an infinite class of examples obtained by degenerations of type An and Bn integrals, which have perfect u equations as well. Our results provide yet another family of generalizations of the usual string integral and moduli space, whose physical interpretations remain to be explored.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
N. Arkani-Hamed, S. He and T. Lam, Stringy canonical forms, arXiv:1912.08707 [INSPIRE].
L. Nilsson and M. Passare, Mellin transforms of multivariate rational functions, J. Geom. Anal. 23 (2011) 24.
C. Berkesch, J. Forsgård and M. Passare, Euler-Mellin integrals and A-hypergeometric functions, Michigan Math. J. 63 (2014) 101.
E. Panzer, Hepp’s bound for Feynman graphs and matroids, arXiv:1908.09820 [INSPIRE].
F.C.S. Brown, Multiple zeta values and periods of moduli spaces M0,n (R), Annales Sci. Ecole Norm. Sup. 42 (2009) 371 [math/0606419] [INSPIRE].
K. Aomoto and M. Kita, Theory of hypergeometric functions, Springer, Japan (2011).
S. Mizera, Scattering amplitudes from intersection theory, Phys. Rev. Lett. 120 (2018) 141602 [arXiv:1711.00469] [INSPIRE].
P. Mastrolia and S. Mizera, Feynman integrals and intersection theory, JHEP 02 (2019) 139 [arXiv:1810.03818] [INSPIRE].
S. Mizera and A. Pokraka, From infinity to four dimensions: higher residue pairings and Feynman integrals, JHEP 02 (2020) 159 [arXiv:1910.11852] [INSPIRE].
F. Brown and C. Dupont, Single-valued integration and double copy, arXiv:1810.07682 [INSPIRE].
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 [arXiv:1711.09102] [INSPIRE].
F. Cachazo, S. He and E.Y. Yuan, Scattering of massless particles: scalars, gluons and gravitons, JHEP 07 (2014) 033 [arXiv:1309.0885] [INSPIRE].
N. Arkani-Hamed, Y. Bai and T. Lam, Positive geometries and canonical forms, JHEP 11 (2017) 039 [arXiv:1703.04541] [INSPIRE].
F. Cachazo, S. He and E.Y. Yuan, Scattering of massless particles in arbitrary dimensions, Phys. Rev. Lett. 113 (2014) 171601 [arXiv:1307.2199] [INSPIRE].
N. Arkani-Hamed, S. He, T. Lam and H. Thomas, Binary geometries, generalized particles and strings, and cluster algebras, arXiv:1912.11764 [INSPIRE].
N. Arkani-Hamed, S. He and T. Lam, Cluster configuration spaces of finite type, arXiv:2005.11419 [INSPIRE].
G. Salvatori, 1-loop amplitudes from the halohedron, JHEP 12 (2019) 074 [arXiv:1806.01842] [INSPIRE].
N. Arkani-Hamed, S. He, G. Salvatori and H. Thomas, Causal diamonds, cluster polytopes and scattering amplitudes, arXiv:1912.12948 [INSPIRE].
S. Fomin and A. Zelevinsky, Y systems and generalized associahedra, Ann. Math. 158 (2003) 977 [hep-th/0111053] [INSPIRE].
N. Arkani-Hamed, T. Lam and M. Spradlin, Positive configuration space, arXiv:2003.03904 [INSPIRE].
A. Postnikov, Permutohedra, associahedra and beyond, Int. Math. Res. Not. 2009 (2009) 1026 [math.CO/0507163].
V. Bazier-Matte, G. Douville, K. Mousavand, H. Thomas and E. Yıldırım, ABHY associahedra and Newton polytopes of F -polynomials for finite type cluster algebras, arXiv:1808.09986 [INSPIRE].
J.-Y. Shi, The Kazhdan-Lusztig cells in certain affine Weyl groups, Lect. Notes Math. 1179 (1986) 312.
A. Postnikov, V. Reiner and L. Williams, Faces of generalized permutohedra, Documenta Math. 13 (2008) 207 [math.CO/0609184].
Z. Li and C. Zhang, Moduli space of paired punctures, cyclohedra and particle pairs on a circle, JHEP 05 (2019) 029 [arXiv:1812.10727] [INSPIRE].
X. Gao, S. He and Y. Zhang, Labelled tree graphs, Feynman diagrams and disk integrals, JHEP 11 (2017) 144 [arXiv:1708.08701] [INSPIRE].
S. He, G. Yan, C. Zhang and Y. Zhang, Scattering forms, worldsheet forms and amplitudes from subspaces, JHEP 08 (2018) 040 [arXiv:1803.11302] [INSPIRE].
N. Early, Generalized permutohedra, scattering amplitudes, and a cubic three-fold, arXiv:1709.03686 [INSPIRE].
N. Early, Generalized permutohedra in the kinematic space, arXiv:1804.05460 [INSPIRE].
S. He and C. Zhang, Notes on scattering amplitudes as differential forms, JHEP 10 (2018) 054 [arXiv:1807.11051] [INSPIRE].
P. Banerjee, A. Laddha and P. Raman, Stokes polytopes: the positive geometry for 𝜙4 interactions, JHEP 08 (2019) 067 [arXiv:1811.05904] [INSPIRE].
P. Raman, The positive geometry for 𝜙p interactions, JHEP 10 (2019) 271 [arXiv:1906.02985] [INSPIRE].
F. Chapoton, Stokes posets and serpent nests, Discrete Math. Theor. Comput. Sci. 18 (2015) [arXiv:1505.05990]
Y. Baryshnikov, On stokes sets, in New developments in singularity theory, Springer, Dordrecht, The Netherlands (2001), pg. 65.
M. Saito, B. Sturmfels and N. Takayama, Gröbner deformations of hypergeometric differential equations, Springer, Berlin, Heidelberg, Germany (2000).
I. Gelfand, M. Kapranov and A. Zelevinsky, Generalized Euler integrals and A-hypergeometric functions, Adv. Math. 84 (1990) 255.
L. de la Cruz, Feynman integrals as A-hypergeometric functions, JHEP 12 (2019) 123 [arXiv:1907.00507] [INSPIRE].
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
ArXiv ePrint: 2005.07395
Rights and permissions
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.
About this article
Cite this article
He, S., Li, Z., Raman, P. et al. Stringy canonical forms and binary geometries from associahedra, cyclohedra and generalized permutohedra. J. High Energ. Phys. 2020, 54 (2020). https://doi.org/10.1007/JHEP10(2020)054
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP10(2020)054