219 research outputs found
The Amplituhedron
Perturbative scattering amplitudes in gauge theories have remarkable
simplicity and hidden infinite dimensional symmetries that are completely
obscured in the conventional formulation of field theory using Feynman
diagrams. This suggests the existence of a new understanding for scattering
amplitudes where locality and unitarity do not play a central role but are
derived consequences from a different starting point. In this note we provide
such an understanding for N=4 SYM scattering amplitudes in the planar limit,
which we identify as ``the volume" of a new mathematical object--the
Amplituhedron--generalizing the positive Grassmannian. Locality and unitarity
emerge hand-in-hand from positive geometry.Comment: 36 pages, 14 figure
Unwinding the Amplituhedron in Binary
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.Comment: 42 pages, 13 figure
Positive configuration space
We define and study the totally nonnegative part of the Chow quotient of the
Grassmannian, or more simply the nonnegative configuration space. This space
has a natural stratification by positive Chow cells, and we show that
nonnegative configuration space is homeomorphic to a polytope as a stratified
space. We establish bijections between positive Chow cells and the following
sets: (a) regular subdivisions of the hypersimplex into positroid polytopes,
(b) the set of cones in the positive tropical Grassmannian, and (c) the set of
cones in the positive Dressian. Our work is motivated by connections to super
Yang-Mills scattering amplitudes, which will be discussed in a sequel.Comment: 46 pages; citations adde
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