Internal and external duality in abstract polytopes

We define an abstract regular polytope to be internally self-dual if its self-duality can be realized as one of its symmetries. This property has many interesting implications on the structure of the polytope, which we present here. Then, we construct many examples of internally self-dual polytopes. In particular, we show that there are internally self-dual regular polyhedra of each type $\{p, p\}$ for $p \geq 3$ and that there are both infinitely many internally self-dual and infinitely many externally self-dual polyhedra of type $\{p, p\}$ for $p$ even. We also show that there are internally self-dual polytopes in each rank, including a new family of polytopes that we construct here

Internal and external duality in abstract polytopes

We define an abstract regular polytope to be internally self-dual if its self-duality can be realized as one of its symmetries. This property has many interesting implications on the structure of the polytope, which we present here. Then, we construct many examples of internally self-dual polytopes. In particular, we show that there are internally self-dual regular polyhedra of each type $\{p, p\}$ for $p \geq 3$ and that there are both infinitely many internally self-dual and infinitely many externally self-dual polyhedra of type $\{p, p\}$ for $p$ even. We also show that there are internally self-dual polytopes in each rank, including a new family of polytopes that we construct here

Brane Tilings and Specular Duality

We study a new duality which pairs 4d N=1 supersymmetric quiver gauge theories. They are represented by brane tilings and are worldvolume theories of D3 branes at Calabi-Yau 3-fold singularities. The new duality identifies theories which have the same combined mesonic and baryonic moduli space, otherwise called the master space. We obtain the associated Hilbert series which encodes both the generators and defining relations of the moduli space. We illustrate our findings with a set of brane tilings that have reflexive toric diagrams.Comment: 42 pages, 16 figures, 5 table

A version of Tutte's polynomial for hypergraphs

Tutte's dichromate T(x,y) is a well known graph invariant. Using the original definition in terms of internal and external activities as our point of departure, we generalize the valuations T(x,1) and T(1,y) to hypergraphs. In the definition, we associate activities to hypertrees, which are generalizations of the indicator function of the edge set of a spanning tree. We prove that hypertrees form a lattice polytope which is the set of bases in a polymatroid. In fact, we extend our invariants to integer polymatroids as well. We also examine hypergraphs that can be represented by planar bipartite graphs, write their hypertree polytopes in the form of a determinant, and prove a duality property that leads to an extension of Tutte's Tree Trinity Theorem.Comment: 49 page

Scattering Amplitudes and Toric Geometry

In this paper we provide a first attempt towards a toric geometric interpretation of scattering amplitudes. In recent investigations it has indeed been proposed that the all-loop integrand of planar N=4 SYM can be represented in terms of well defined finite objects called on-shell diagrams drawn on disks. Furthermore it has been shown that the physical information of on-shell diagrams is encoded in the geometry of auxiliary algebraic varieties called the totally non negative Grassmannians. In this new formulation the infinite dimensional symmetry of the theory is manifest and many results, that are quite tricky to obtain in terms of the standard Lagrangian formulation of the theory, are instead manifest. In this paper, elaborating on previous results, we provide another picture of the scattering amplitudes in terms of toric geometry. In particular we describe in detail the toric varieties associated to an on-shell diagram, how the singularities of the amplitudes are encoded in some subspaces of the toric variety, and how this picture maps onto the Grassmannian description. Eventually we discuss the action of cluster transformations on the toric varieties. The hope is to provide an alternative description of the scattering amplitudes that could contribute in the developing of this very interesting field of research.Comment: 58 pages, 25 figures, typos corrected, a reference added, to be published in JHE

Scattering Forms and the Positive Geometry of Kinematics, Color and the Worldsheet

The search for a theory of the S-Matrix has revealed surprising geometric structures underlying amplitudes ranging from the worldsheet to the amplituhedron, but these are all geometries in auxiliary spaces as opposed to kinematic space where amplitudes live. In this paper, we propose a novel geometric understanding of amplitudes for a large class of theories. The key is to think of amplitudes as differential forms directly on kinematic space. We explore this picture for a wide range of massless theories in general spacetime dimensions. For the bi-adjoint cubic scalar, we establish a direct connection between its "scattering form" and a classic polytope--the associahedron--known to mathematicians since the 1960's. We find an associahedron living naturally in kinematic space, and the tree amplitude is simply the "canonical form" associated with this "positive geometry". Basic physical properties such as locality, unitarity and novel "soft" limits are fully determined by the geometry. Furthermore, the moduli space for the open string worldsheet has also long been recognized as an associahedron. We show that the scattering equations act as a diffeomorphism between this old "worldsheet associahedron" and the new "kinematic associahedron", providing a geometric interpretation and novel derivation of the bi-adjoint CHY formula. We also find "scattering forms" on kinematic space for Yang-Mills and the Non-linear Sigma Model, which are dual to the color-dressed amplitudes despite having no explicit color factors. This is possible due to a remarkable fact--"Color is Kinematics"--whereby kinematic wedge products in the scattering forms satisfy the same Jacobi relations as color factors. Finally, our scattering forms are well-defined on the projectivized kinematic space, a property that provides a geometric origin for color-kinematics duality.Comment: 77 pages, 25 figures; v2, corrected discussion of worldsheet associahedron canonical for

Amplitudes at Weak Coupling as Polytopes in AdS_5

We show that one-loop scalar box functions can be interpreted as volumes of geodesic tetrahedra embedded in a copy of AdS_5 that has dual conformal space-time as boundary. When the tetrahedron is space-like, it lies in a totally geodesic hyperbolic three-space inside AdS_5, with its four vertices on the boundary. It is a classical result that the volume of such a tetrahedron is given by the Bloch-Wigner dilogarithm and this agrees with the standard physics formulae for such box functions. The combinations of box functions that arise in the n-particle one-loop MHV amplitude in N=4 super Yang-Mills correspond to the volume of a three-dimensional polytope without boundary, all of whose vertices are attached to a null polygon (which in other formulations is interpreted as a Wilson loop) at infinity.Comment: 16 pages, 5 figure