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Positive Geometries and Canonical Forms

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Positive Geometries and Canonical Forms Prepared for submission to JHEP Positive Geometries and Canonical Forms Nima Arkani-Hamed,a Yuntao Bai,b Thomas Lamc aSchool of Natural Sciences, Institute for Advanced Study, Princeton, NJ 08540, USA bDepartment of Physics, Princeton University, Princeton, NJ 08544, USA cDepartment of Mathematics, University of Michigan, 530 Church St, Ann Arbor, MI 48109, USA Abstract: Recent years have seen a surprising connection between the physics of scat- tering 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 sin- gularities (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 geome- tries 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 arXiv:1703.04541v2 [hep-th] 30 Jul 2017 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. Contents 1 Introduction1 2 Positive geometries5 2.1 Positive geometries and their canonical forms5 2.2 Pseudo-positive geometries6 2.3 Reversing orientation, disjoint unions and direct products7 2.4 One-dimensional positive geometries7 3 Triangulations of positive geometries8 3.1 Triangulations of pseudo-positive geometries8 3.2 Signed triangulations8 3.3 The Grothendieck group of pseudo-positive geometries in X 9 3.4 Physical versus spurious poles 10 4 Morphisms of positive geometries 11 5 Generalized simplices 12 5.1 The standard simplex 12 5.2 Projective simplices 13 5.3 Generalized simplices on the projective plane 15 5.3.1 An example of a non-normal geometry 18 5.4 Generalized simplices in higher-dimensional projective spaces 19 5.5 Grassmannians 21 5.5.1 Grassmannians and positroid varieties 21 5.5.2 Positive Grassmannians and positroid cells 21 5.6 Toric varieties and their positive parts 22 5.6.1 Projective toric varieties 22 5.6.2 The canonical form of a toric variety 23 5.7 Cluster varieties and their positive parts 24 5.8 Flag varieties and total positivity 25 6 Generalized polytopes 26 6.1 Projective polytopes 26 6.1.1 Projective and Euclidean polytopes 26 6.1.2 Cyclic polytopes 27 6.1.3 Dual polytopes 28 6.2 Generalized polytopes on the projective plane 29 6.3 A naive positive part of partial flag varieties 30 6.4 L-loop Grassmannians 31 6.5 Grassmann, loop and flag polytopes 33 6.6 Amplituhedra and scattering amplitudes 35 { i { 7 Canonical forms 36 7.1 Direct construction from poles and zeros 36 7.1.1 Cyclic polytopes 37 7.1.2 Generalized polytopes on the projective plane 40 7.2 Triangulations 41 7.2.1 Projective polytopes 41 7.2.2 Generalized polytopes on the projective plane 43 7.2.3 Amplituhedra and BCFW recursion 43 7.2.4 The tree Amplituhedron for m = 1; 2 45 7.2.5 A 1-loop Grassmannian 49 7.2.6 An example of a Grassmann polytope 51 7.3 Push-forwards 52 7.3.1 Projective simplices 53 7.3.2 Algebraic moment map and an algebraic analogue of the Duistermaat-Heckman measure 56 7.3.3 Projective polytopes from Newton polytopes 58 7.3.4 Recursive properties of the Newton polytope map 61 7.3.5 Newton polytopes from constraints 63 7.3.6 Generalized polytopes on the projective plane 66 7.3.7 Amplituhedra 68 7.4 Integral representations 70 7.4.1 Dual polytopes 70 7.4.2 Laplace transforms 73 7.4.3 Dual Amplituhedra 75 7.4.4 Dual Grassmannian volumes 76 7.4.5 Wilson loops and surfaces 78 7.4.6 Projective space contours part I 82 7.4.7 Projective space contours part II 88 7.4.8 Grassmannian contours 90 8 Integration of canonical forms 91 8.1 Canonical integrals 91 8.2 Duality of canonical integrals and the Aomoto form 92 9 Positively convex geometries 93 10 Beyond \rational" positive geometries 95 11 Outlook 101 A Assumptions on positive geometries 102 A.1 Assumptions on X≥0 and definition of boundary components 102 A.2 Assumptions on X 103 A.3 The residue operator 104 { ii { B Near-equivalence of three notions of signed triangulation 104 C Rational differential forms on projective spaces and Grassmannians 105 C.1 Forms on projective spaces 105 C.2 Forms on Grassmannians 106 C.3 Forms on L-loop Grassmannians 108 D Cones and projective polytopes 108 E Monomial parametrizations of polytopes 109 F The global residue theorem 114 G The canonical form of a toric variety 115 H Canonical form of a polytope via toric varieties 116 I Oriented matroids 119 J The Tarski-Seidenberg theorem 120 1 Introduction Recent years have revealed an unexpected and fascinating new interplay between physics and geometry in the study of gauge theory scattering amplitudes. In the context of planar N = 4 super Yang-Mills theory, we now have a complete formulation of the physics of scattering amplitudes in terms of the geometry of the \Amplituhedron" [1{4], which is a Grassmannian generalization of polygons and polytopes. Neither space-time nor Hilbert space make any appearance in this formulation { the associated physics of locality and unitarity arise as consequences of the geometry. This new connection between physics and mathematics involves a number of interesting mathematical ideas. The notion of \positivity" plays a crucial role. In its simplest form n−1 we consider the interior of a simplex in projective space P as points with homogeneous n−1 co-ordinates (x0; : : : ; xn−1) with all xa > 0. We can call this the \positive part" P>0 of projective space; thinking of projective space as the space of 1-planes in n dimensions, we can also call this the \positive part" of the Grassmannian of 1-planes in n dimensions, G>0(1; n). This notion generalizes from G>0(1; n) to the \positive part" of the Grassmannian of k-planes in n dimensions, G>0(k; n)[5,6]. The Amplituhedron is a further extension of this idea, roughly generalizing the positive Grassmannian in the same way that convex plane polygons generalize triangles. These spaces have loosely been referred to as \positive geometries" in the physics literature; like polygons and polytopes they have an \interior", with boundaries or facets of all dimensionalities. Another crucial idea, which gives a { 1 { 1.0 2.0 0.8 1.5 0.6 y y 1.0 0.4 0.2 0.5 0.0 0.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.5 1.0 1.5 2.0 x x dxdy dxdy(12−x−4y) (a) xy(1−x−y) (b) x(2y−x)(3−x−y)(2−y) 1.0 1.0 0.8 0.8 y 0.6 y 0.6 0.4 0.4 0.2 0.2 0.0 0.0 -1.0 -0.5 0.0 0.5 1.0 -1.0 -0.5 0.0 0.5 1.0 x x p p (3 11=5)dxdy 2 3(1+2y)dxdy (c) (d) p p (1−x2−y2)(y−(1=10)) (1−x2−y2)( 3y+x)( 3y−x) 1.0 0.5 y 0.0 -0.5 -1.0 -1.0 -0.5 0.0 0.5 1.0 x (e) 0 dxdy Figure 1: Canonical forms of (a) a triangle, (b) a quadrilateral, (c) a segment of the unit disk with y ≥ 1=10, (d) a sector of the unit disk with central angle 2π=3 symmetric about the y-axis, and (e) the unit disk. The form is identically zero for the unit disk because there are no zero-dimensional boundaries. For each of the other figures, the form has simple poles along each boundary component, all leading residues are ±1 at zero-dimensional boundaries and zero elsewhere, and the form is positively oriented on the interior. dictionary for converting these geometries into physical scattering amplitudes, is a certain (complex) meromorphic differential form that is canonically associated with the positive geometry. This form is fixed by the requirement of having simple poles on (and only on) all the boundaries of the geometry, with its residue along each being given by the canonical { 2 { form of the boundary. The calculation of scattering amplitudes is then reduced to the natural mathematical question of determining this canonical form.
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