Built on

F. Cachazo, N. Early, A. Guevara, and S. Mizera, “Scattering Equations: From Projective Spaces to Tropical Grassmannians,” JHEP
Original
1903
Earlier work this paper cites.
D. Damgaard, L. Ferro, T. Lukowski, and M. Parisi, “The Momentum Amplituhedron,” JHEP
Original
1905
Earlier work this paper cites.
J. Drummond, J. Foster, Ö. Gürdoğan, and C. Kalousios, “Tropical Grassmannians, cluster algebras and scattering amplitudes,” arXiv:1907.01053 [hep-th]
Original
1907
Earlier work this paper cites.
H. Frellesvig, F. Gasparotto, M. K. Mandal, P. Mastrolia, L. Mattiazzi, and S. Mizera, “Vector Space of Feynman Integrals and Multivariate Intersection Numbers,” Phys. Rev. Lett
Original
1907
Earlier work this paper cites.
E. Panzer, “Hepp’s bound for Feynman graphs and matroids,” arXiv:1908.09820 [math-ph]
Original
1908
Earlier work this paper cites.
F. Cachazo, B. Umbert, and Y. Zhang, “Singular Solutions in Soft Limits,” arXiv:1911.02594 [hep-th]
Original
1911
Earlier work this paper cites.
N. Arkani-Hamed, S. He, T. Lam, and H. Thomas, “Binary Geometries, Generalized Particles and Strings, and Cluster Algebras,” arXiv:1912.11764 [hep-th]
Original
1912
Earlier work this paper cites.
N. Arkani-Hamed, S. He, G. Salvatori, and H. Thomas, “Causal Diamonds, Cluster Polytopes and Scattering Amplitudes,” arXiv:1912.12948 [hep-th]
Original
1912
Earlier work this paper cites.
A. Herderschee, S. He, F. Teng, and Y. Zhang, “On Positive Geometry and Scattering Forms for Matter Particles,” arXiv:1912.08307 [hep-th]
Original
1912
Earlier work this paper cites.
G. Veneziano, “Construction of a crossing-symmetric, Regge-behaved amplitude for linearly rising trajectories,” Il Nuovo Cimento A (1965-1970)
1968
Earlier work this paper cites.
Z. Koba and H. B. Nielsen, “Manifestly crossing invariant parametrization of n meson amplitude,” Nucl. Phys
1969
Earlier work this paper cites.
P. Deligne and D. Mumford, “The irreducibility of the space of curves of given genus,” Publications Mathématiques de l’Institut des Hautes Études Scientifiques
1969
Earlier work this paper cites.
K. Aomoto, “Equations aux differences lineaires et les integrales des fonctions multiformes, ii. evanouissement des hypercohomologies et exemples,” Proc. Japan Acad
1974
Earlier work this paper cites.
E. R. Speer, “Ultraviolet and infrared singularity structure of generic feynman amplitudes,” Annales de l I.H.P. Physique theorique
1975
Earlier work this paper cites.
D. N. Bernstein, “The number of roots of a system of equations,” Funkcional. Anal. i Prilozen
1975
Earlier work this paper cites.
A. G. Kouchnirenko, “Polyèdres de Newton et nombres de Milnor,” Inventiones Mathematicae
1976
Earlier work this paper cites.
D. J. Gross and P. F. Mende, “The High-Energy Behavior of String Scattering Amplitudes,” Phys. Lett
1987
Earlier work this paper cites.

Similar

Annals of mathematics studies. Princeton Univ. Press, Princeton, NJ, 1993
W. Fulton, Introduction to toric varieties · 1993
Cited alongside, same era.
M. Kita, “On vanishing of the twisted rational de rham cohomology associated with hypergeometric functions,” Nagoya Mathematical Journal
1994
Cited alongside, same era.
F. Chapoton, S. Fomin, and A. Zelevinsky, “Polytopal realizations of generalized associahedra,” Canadian Mathematical Bulletin
2002
Cited alongside, same era.
S. Fomin and A. Zelevinsky, “Y-systems and generalized associahedra,” Annals of Mathematics
2003
Cited alongside, same era.
R. Roiban, M. Spradlin, and A. Volovich, “On the tree level S matrix of Yang-Mills theory,” Phys. Rev
2004
Cited alongside, same era.

Then

F. Cachazo, S. He, and E. Y. Yuan, “Scattering of Massless Particles in Arbitrary Dimensions,” Phys. Rev. Lett
Original
2014
Later among the works it cites.
F. Cachazo, S. He, and E. Y. Yuan, “Scattering of Massless Particles: Scalars, Gluons and Gravitons,” JHEP
Original
2014
Later among the works it cites.
K. Lee and R. Schiffler, “Positivity for cluster algebras,” Annals of Math
2015
Later among the works it cites.
Cambridge University Press, 2016
Original
N. Arkani-Hamed, J. L. Bourjaily, F. Cachazo, A. B. Goncharov, A. Postnikov, and J. Trnka, Grassmannian Geometry of Scattering Amplitudes · 2016
Later among the works it cites.
N. Arkani-Hamed, Y. Bai, and T. Lam, “Positive Geometries and Canonical Forms,” JHEP
Original

Beyond the bibliography

alphaXiv searches the wider corpus for related work and actual follow-ups.

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Reading the bibliography…

2019

Stringy Canonical Forms

Arkani-Hamed, Nima, He, Song, Lam, Thomas

Understand

Canonical forms of positive geometries play an important role in revealing hidden structures of scattering amplitudes, from amplituhedra to associahedra.

In this paper, we introduce "stringy canonical forms", which provide a natural definition and extension of canonical forms for general polytopes, deformed by a parameter $\alpha'$.
They are defined by real or complex integrals regulated with polynomials with exponents, and are meromorphic functions of the exponents, sharing various properties of string amplitudes.
As $\alpha' \to 0$, they reduce to the usual canonical form of a polytope given by the Minkowski sum of the Newton polytopes of the regulating polynomials, or equivalently the volume of the dual of this polytope, naturally determined by tropical functions.

Built on

F. Cachazo, N. Early, A. Guevara, and S. Mizera, “Scattering Equations: From Projective Spaces to Tropical Grassmannians,” JHEP
Original
1903
Earlier work this paper cites.
D. Damgaard, L. Ferro, T. Lukowski, and M. Parisi, “The Momentum Amplituhedron,” JHEP
Original
1905
Earlier work this paper cites.
J. Drummond, J. Foster, Ö. Gürdoğan, and C. Kalousios, “Tropical Grassmannians, cluster algebras and scattering amplitudes,” arXiv:1907.01053 [hep-th]
Original
1907
Earlier work this paper cites.
H. Frellesvig, F. Gasparotto, M. K. Mandal, P. Mastrolia, L. Mattiazzi, and S. Mizera, “Vector Space of Feynman Integrals and Multivariate Intersection Numbers,” Phys. Rev. Lett
Original
1907
Earlier work this paper cites.
E. Panzer, “Hepp’s bound for Feynman graphs and matroids,” arXiv:1908.09820 [math-ph]
Original
1908
Earlier work this paper cites.
F. Cachazo, B. Umbert, and Y. Zhang, “Singular Solutions in Soft Limits,” arXiv:1911.02594 [hep-th]
Original
1911
Earlier work this paper cites.
N. Arkani-Hamed, S. He, T. Lam, and H. Thomas, “Binary Geometries, Generalized Particles and Strings, and Cluster Algebras,” arXiv:1912.11764 [hep-th]
Original
1912
Earlier work this paper cites.
N. Arkani-Hamed, S. He, G. Salvatori, and H. Thomas, “Causal Diamonds, Cluster Polytopes and Scattering Amplitudes,” arXiv:1912.12948 [hep-th]
Original
1912
Earlier work this paper cites.
A. Herderschee, S. He, F. Teng, and Y. Zhang, “On Positive Geometry and Scattering Forms for Matter Particles,” arXiv:1912.08307 [hep-th]
Original
1912
Earlier work this paper cites.
G. Veneziano, “Construction of a crossing-symmetric, Regge-behaved amplitude for linearly rising trajectories,” Il Nuovo Cimento A (1965-1970)
1968
Earlier work this paper cites.
Z. Koba and H. B. Nielsen, “Manifestly crossing invariant parametrization of n meson amplitude,” Nucl. Phys
1969
Earlier work this paper cites.
P. Deligne and D. Mumford, “The irreducibility of the space of curves of given genus,” Publications Mathématiques de l’Institut des Hautes Études Scientifiques
1969
Earlier work this paper cites.
K. Aomoto, “Equations aux differences lineaires et les integrales des fonctions multiformes, ii. evanouissement des hypercohomologies et exemples,” Proc. Japan Acad
1974
Earlier work this paper cites.
E. R. Speer, “Ultraviolet and infrared singularity structure of generic feynman amplitudes,” Annales de l I.H.P. Physique theorique
1975
Earlier work this paper cites.
D. N. Bernstein, “The number of roots of a system of equations,” Funkcional. Anal. i Prilozen
1975
Earlier work this paper cites.
A. G. Kouchnirenko, “Polyèdres de Newton et nombres de Milnor,” Inventiones Mathematicae
1976
Earlier work this paper cites.
D. J. Gross and P. F. Mende, “The High-Energy Behavior of String Scattering Amplitudes,” Phys. Lett
1987
Earlier work this paper cites.

Similar

Annals of mathematics studies. Princeton Univ. Press, Princeton, NJ, 1993
W. Fulton, Introduction to toric varieties · 1993
Cited alongside, same era.
M. Kita, “On vanishing of the twisted rational de rham cohomology associated with hypergeometric functions,” Nagoya Mathematical Journal
1994
Cited alongside, same era.
F. Chapoton, S. Fomin, and A. Zelevinsky, “Polytopal realizations of generalized associahedra,” Canadian Mathematical Bulletin
2002
Cited alongside, same era.
S. Fomin and A. Zelevinsky, “Y-systems and generalized associahedra,” Annals of Mathematics
2003
Cited alongside, same era.
R. Roiban, M. Spradlin, and A. Volovich, “On the tree level S matrix of Yang-Mills theory,” Phys. Rev
2004
Cited alongside, same era.

Then

F. Cachazo, S. He, and E. Y. Yuan, “Scattering of Massless Particles in Arbitrary Dimensions,” Phys. Rev. Lett
Original
2014
Later among the works it cites.
F. Cachazo, S. He, and E. Y. Yuan, “Scattering of Massless Particles: Scalars, Gluons and Gravitons,” JHEP
Original
2014
Later among the works it cites.
K. Lee and R. Schiffler, “Positivity for cluster algebras,” Annals of Math
2015
Later among the works it cites.
Cambridge University Press, 2016
Original
N. Arkani-Hamed, J. L. Bourjaily, F. Cachazo, A. B. Goncharov, A. Postnikov, and J. Trnka, Grassmannian Geometry of Scattering Amplitudes · 2016
Later among the works it cites.
N. Arkani-Hamed, Y. Bai, and T. Lam, “Positive Geometries and Canonical Forms,” JHEP
Original

Beyond the bibliography

alphaXiv searches the wider corpus for related work and actual follow-ups.

Open on alphaXiv

alphaXiv is searching for related work…

D. Speyer and L. Williams, “The tropical totally positive grassmannian,” Journal of Algebraic Combinatorics

2005

Cited alongside, same era.

A. Postnikov, “Permutohedra, associahedra, and beyond,” ArXiv Mathematics e-prints

2005

Cited alongside, same era.

A. Postnikov, “Permutohedra, associahedra, and beyond,” arXiv Mathematics e-prints

2005

Cited alongside, same era.

N. Arkani-Hamed and J. Trnka, “The Amplituhedron,” JHEP

Original

2007

Cited alongside, same era.

F. C. S. Brown, “Multiple zeta values and periods of moduli spaces 𝔐 0 , n \mathfrak{M}_{0,n} ,” Annales Sci. Ecole Norm. Sup

2009

Cited alongside, same era.

N. E. J. Bjerrum-Bohr, P. H. Damgaard, and P. Vanhove, “Minimal Basis for Gauge Theory Amplitudes,” Phys. Rev. Lett

Original

2009

Cited alongside, same era.

L. Nilsson and M. Passare, “Mellin transforms of multivariate rational functions,” Journal of Geometric Analysis

2010

Cited alongside, same era.

2017