The $m=1$ Amplituhedron and Cyclic Hyperplane Arrangements

Abstract

Abstract. The (tree) amplituhedron A n,k,m is the image in the Grassmannian Gr k,k+m of the totally nonnegative part of Gr k,n , under a (map induced by a) linear map which is totally positive. It was introduced by Arkani-Hamed and Trnka in 2013 in order to give a geometric basis for the computation of scattering amplitudes in N = 4 supersymmetric Yang-Mills theory. When k + m = n, the amplituhedron is isomorphic to the totally nonnegative Grassmannian, and when k = 1, the amplituhedron is a cyclic polytope. While the case m = 4 is most relevant to physics, the amplituhedron is an interesting mathematical object for any m. In this paper we study it in the case m = 1. We start by taking an orthogonal point of view and define a related "B-amplituhedron" B n,k,m , which we show is isomorphic to A n,k,m . We use this reformulation to describe the amplituhedron in terms of sign variation. We then give a cell decomposition of the amplituhedron A n,k,1 using the images of a collection of distinguished cells of the totally nonnegative Grassmannian. We also show that A n,k,1 can be identified with the complex of bounded faces of a cyclic hyperplane arrangement, and describe how its cells fit together. We deduce that A n,k,1 is homeomorphic to a ball.

Unwinding the amplituhedron in binary

“…This is equivalent to the characterization of the m = 1 amplituhedron recently given in [15]. Let's now look at m = 2.…”

“…This physics and mathematics has been explored from a variety of perspectives in the past few years (see e.g. [6][7][8][9][10][11][12][13][14][15][16]), and a systematic mathematical exploration of the notion of "positive geometries" has recently been initiated in [17].…”

Prescriptive unitarity

“…Most of these discoveries have been fueled by direct computation -pushing the limits of our theoretical reach (often for toy models) to uncover unanticipated, simplifying structures in the formulae that result, and using these insights to build more powerful tools. The lessons learned through such investigations include the (BCFW) on-shell recursion relations at tree-and loop-level, [7,8] and [9]; the discovery of a hidden dual conformal invariance [10][11][12] as well as the duality to Wilson loops and correlation functions [13][14][15][16][17][18][19][20]; the connection to Grassmannian geometry [21][22][23][24][25][26][27] and the amplituhedron [28][29][30][31][32][33][34][35][36][37][38][39]; various bootstrap methods [40][41][42][43][44][45][46][47][48][49][50][51]; the twistor…”

Multi-loop positivity of the planar N $$ \mathcal{N} $$ = 4 SYM six-point amplitude

“…[5][6][7][8][9][10] for recent progress). The idea is to rewrite the kinematical and helicity variables in terms of bosonized momentum twistors Z serving as vertices of a geometric object -the Amplituhedron -whose volume is equal to the integrand of scattering amplitudes in planar N = 4 SYM.…”

“…This attempt immediately runs into trouble, because the limiting behavior of Y , 10) like that of any one-loop function, does not have enough logarithms of v or w to compete with the four powers of logs in the problematic terms in R…”