Abstract
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.
Highlights
Guises, but as toy models go, its application to scattering amplitudes is closer to the real world than any other
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
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
Summary
We have motivated the structure of the amplituhedron by mimicking the geometric idea of the “inside” of a convex polygon. It is simple and instructive to see why positivity ensures that the Y = C · Z map is projectively well-defined. We will see this as a by-product of locating the co-dimension one boundaries of the generalized tree amplituhedron. The boundaries are lines (ZiZi+1) as expected This tells us that the map Y = C · Z is projectively well-defined. We can investigate whether the plane (ZiZjZkZl) is a boundary by computing The emergence of boundaries on the plane (ZiZi+1ZjZj+1) is a simple and striking consequence of positivity. We will shortly understand that the location of these boundaries are the “positive origin” of locality from the geometry of the amplituhedron
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