Reflection group relations arising from cluster algebras

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.

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Positivity for cluster algebras

Annals of Mathematics ◽

10.4007/annals.2015.182.1.2 ◽

2015 ◽

pp. 73-125 ◽

Cited By ~ 34

Author(s):

Kyungyong Lee ◽

Ralf Schiffler

Keyword(s):

Cluster Algebras

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Algebraic singularities of scattering amplitudes from tropical geometry

Journal of High Energy Physics ◽

10.1007/jhep04(2021)002 ◽

2021 ◽

Vol 2021 (4) ◽

Author(s):

James Drummond ◽

Jack Foster ◽

Ömer Gürdoğan ◽

Chrysostomos Kalousios

Keyword(s):

Scattering Amplitudes ◽

Natural Language ◽

Tropical Geometry ◽

Cluster Algebras ◽

Finite Alphabet ◽

Square Root ◽

Finite Sets ◽

Yang Mills ◽

Square Roots ◽

Algebraic Singularities

Abstract We address the appearance of algebraic singularities in the symbol alphabet of scattering amplitudes in the context of planar $$ \mathcal{N} $$ N = 4 super Yang-Mills theory. We argue that connections between cluster algebras and tropical geometry provide a natural language for postulating a finite alphabet for scattering amplitudes beyond six and seven points where the corresponding Grassmannian cluster algebras are finite. As well as generating natural finite sets of letters, the tropical fans we discuss provide letters containing square roots. Remarkably, the minimal fan we consider provides all the square root letters recently discovered in an explicit two-loop eight-point NMHV calculation.

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