Triangulations and Canonical Forms of Amplituhedra: A Fiber-Based Approach Beyond Polytopes

Abstract

Any totally positive (k + m) × n matrix induces a map π + from the positive Grassmannian Gr + (k, n) to the Grassmannian Gr(k, k + m), whose image is the amplituhedron A n,k,m and is endowed with a top-degree form called the canonical form (A n,k,m ). This construction was introduced by Arkani-Hamed and Trnka (J High Energy Phys 2014(10):30, 2014), where they showed that (A n,k,4 ) encodes scattering amplitudes in N = 4 super Yang-Mills theory. One way to compute (A n,k,m ) is to subdivide A n,k,m into so-called generalized triangles and sum over their associated canonical forms. Hence, the physical computation of scattering amplitudes is reduced to finding the triangulations of A n,k,4 . However, while triangulations of polytopes are fully captured by their secondary and fiber polytopes (Billera and Sturmfels in Ann Math 527-549, 1992; Gelfand et al. in Discriminants, resultants, and multidimensional determinants, Birkahuser, Boston, 1994), the study of triangulations of objects beyond polytopes is still underdeveloped. In this work, we initiate the geometric study of subdivisions of A n,k,m in order to establish the notion of secondary amplituhedron. For this purpose, we first extend the projection π + to a rational map π : Gr(k, n)Gr(k, k +m) and provide a concrete birational parametrization of the fibers of π . We then use this to explicitly describe a rational top-degree form ω n,k,m (with simple poles) on the fibers and compute (A n,k,m ) as a summation of certain residues of ω n,k,m . As main application of our approach, we develop a well-structured notion of secondary amplituhedra for conjugate to polytopes, i.e. when n − k − 1 = m (even). We show that, in this case, each fiber of π is parametrized by a projective space and its volume form ω n,k,m has only poles on a hyperplane arrangement. Using such linear structures, for amplituhedra which are cyclic polytopes or conjugate to polytopes, we show that the Jeffrey-Kirwan residue computes (A n,k,m ) from the fiber volume form ω n,k,m . In particular, we give conceptual proofs of the statements of Ferro et al. (J Phys A Math Theor 52(4):045201, 2018). Finally, we propose a more general framework of fiber positive geometries and analyze new families of examples such as fiber polytopes and Grassmann polytopes.

Positive Geometries and Differential Forms with Non-Logarithmic Singularities I

“…The form is not the canonical form of the fiber, but it is in covariant pairing with it. In [25], these type of forms will also be defined in the context of objects which are more general than polytopes, such as the amplituhedra.…”

“…Integration contours capturing all the (regular) 2 triangulations of a given polytope can be defined via a method [21] which relies on the Jeffrey-Kirwan Residue [20,24]. We will give a brief review and refer to [21] and the upcoming work [25] for further details. In Section (4) we will present an explicit example and explain its connection with this work.…”

Internal boundaries of the loop amplituhedron

“…Each 4 loop all-in-onepoint cut is given by 3 intersections and 2 sidings. Denoting the intersection between lines L i and L j by (i, j) and a sliding between the sets of lines I and J as (I, J), we explore the cut {(1, 2), (1, 3), (1, 4) ; (12,13), (123, 14)} ,…”

“…This type of structure appears in the boundaries of the loop amplituhedron, implying that the loop amplituhedron is not a 'positive geometry' in the sense usually understood [10]. 2 There has been tremendous progress in understanding the geometry of the tree level amplituhedron and its canonical form [8,[12][13][14][15]. The loop amplituhedron and its triangulations on the other hand are much less well understood.…”

Towards the Gravituhedron: New Expressions for NMHV Gravity Amplitudes

“…The discovery of the Amplituhedron was based on earlier polytope picture for NMHV tree-level amplitudes [4,5], loop level recursion relations [6] and the positive Grassmannian [7][8][9][10][11][12]. The geometry of Amplituhedron has been studied extensively in last years, including the boundary structure [13][14][15][16][17][18][19], triangulations [20,21], Yangian invariance [22,23] and more rigorous mathematical understanding of the geometry [24][25][26][27][28][29]. The geometric picture has been used to provide some all-loop results which could not be obtained using standard methods [30][31][32].…”