The amplituhedron and the one-loop Grassmannian measure

Self Cite

197

471

Recent years have seen a surprising connection between the physics of scattering amplitudes and a class of mathematical objects -the positive Grassmannian, positive loop Grassmannians, tree and loop Amplituhedra -which have been loosely referred to as "positive geometries". The connection between the geometry and physics is provided by a unique differential form canonically determined by the property of having logarithmic singularities (only) on all the boundaries of the space, with residues on each boundary given by the canonical form on that boundary. The structures seen in the physical setting of the Amplituhedron are both rigid and rich enough to motivate an investigation of the notions of "positive geometries" and their associated "canonical forms" as objects of study in their own right, in a more general mathematical setting. In this paper we take the first steps in this direction. We begin by giving a precise definition of positive geometries and canonical forms, and introduce two general methods for finding forms for more complicated positive geometries from simpler ones -via "triangulation" on the one hand, and "push-forward" maps between geometries on the other. We present numerous examples of positive geometries in projective spaces, Grassmannians, and toric, cluster and flag varieties, both for the simplest "simplex-like" geometries and the richer "polytope-like" ones. We also illustrate a number of strategies for computing canonical forms for large classes of positive geometries, ranging from a direct determination exploiting knowledge of zeros and poles, to the use of the general triangulation and push-forward methods, to the representation of the form as volume integrals over dual geometries and contour integrals over auxiliary spaces. These methods yield interesting representations for the canonical forms of wide classes of positive geometries, ranging from the simplest Amplituhedra to new expressions for the volume of arbitrary convex polytopes.

Section: A 1-loop Grassmannianmentioning

confidence: 99%

Positive geometries and canonical forms

Arkani-Hamed¹,

Bai²,

Lam³

2017

Self Cite

197

471

“…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].…”

Section: Jhep01(2018)016mentioning

confidence: 99%

Unwinding the amplituhedron in binary

Arkani-Hamed

Thomas

Trnka

2018

139

347

We present new, fundamentally combinatorial and topological characterizations of the amplituhedron. Upon projecting external data through the amplituhedron, the resulting configuration of points has a specified (and maximal) generalized "winding number". Equivalently, the amplituhedron can be fully described in binary: canonical projections of the geometry down to one dimension have a specified (and maximal) number of "sign flips" of the projected data. The locality and unitarity of scattering amplitudes are easily derived as elementary consequences of this binary code. Minimal winding defines a natural "dual" of the amplituhedron. This picture gives us an avatar of the amplituhedron purely in the configuration space of points in vector space (momentum-twistor space in the physics), a new interpretation of the canonical amplituhedron form, and a direct bosonic understanding of the scattering super-amplitude in planar N = 4 SYM as a differential form on the space of physical kinematical data.

“…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…”

Section: Introduction and Overviewmentioning

confidence: 99%

Prescriptive unitarity

Bourjaily

Herrmann

Trnka

2017

116

We introduce a prescriptive approach to generalized unitarity, resulting in a strictly-diagonal basis of loop integrands with coefficients given by specifically-tailored residues in field theory. We illustrate the power of this strategy in the case of planar, maximally supersymmetric Yang-Mills theory (SYM), where we construct closed-form representations of all (n-point N k MHV) scattering amplitudes through three loops. The prescriptive approach contrasts with the ordinary description of unitarity-based methods by avoiding any need for linear algebra to determine integrand coefficients. We describe this approach in general terms as it should have applications to many quantum field theories, including those without planarity, supersymmetry, or massless spectra defined in any number of dimensions.