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Logarithms and Polytopes
Michael Enciso1408.0932 ME (2014) + To Appear
QCD Meets Gravity 2016
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Motivation
● Better understand geometry of dual amplituhedron.
● Gain some insight into how polylogarithms arise in amplitudes.
● Give intrinsic, triangulation-independent definition of polytopes in complex projective space.
Arkani-Hamed, Trnka (2013)
Outline
● Describe a new way of computing volumes of polytopes.
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Basic Of Projective Geometry
Points in one space determine hyperplanes in the other:
Intersections always exist in the expected way.
E.g:
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Volumes of 2D Simplices
Vertices
Lines
(Complex or real) Projectively well-defined area in terms of the LINES defining the triangle.
Arkani-Hamed, Bourjaily, Cachazo, Hodges, Trnka (2010)
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Even Simpler: Volume of 1D Simplex
Gives projectively well-defined definition of length in terms of dual points.
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Volumes Of Simplices in D-Dimensions
D=4 is relevant for amplitudes:
With P defining the plane at infinity and Z’s real, these give genuine volumes of simplices.
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Volumes Of General Polytopes
General polytopes can be defined by sums of simplices—i.e., via a triangulation:
Area of quadrilateral:
=
=However, we also have =
(Infinitely many) different triangulations correspond to the same polytope.
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NMHV Tree Amplitudes in Planar N=4 SYM
It turns out:
Encode the external kinematics
As well as...
Two different expressions come from two different BCFW shifts.
Modern understanding: two different triangulations of the same polytope.
For completeness:
Identities can be understood using GRTs in the Grassmannian
Mason, Skinner (2009)
Arkani-Hamed, Bourjaily, Cachazo, Goncharov, Postnikov, Trnka (2012)
And others
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Revisiting 1D
Allow x’s to be complex from the outset.
Logarithms appear naturally in calculation of (complex) lengths.
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1D Continued
Again define:
Contour naturally defined by integrand.
As expected.
Can express lengths as contour integrals of logarithms, where contour is canonically defined by the integrand itself.
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2D Revisited: QuadrilateralsMotivated by the 1D case, we consider:
Contour naturally defined by integrand
So A is just the area of the quadrilateral bounded by the lines 1, 2, 3, and 4 =
becomes manifest from integrand.
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3D “Quadrilateral”
All of these identities become manifest.
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Can we “quadrilateralate” a general polytope?
Yes
Thus in principle we can use integrals like these to represent any volume.
Each contour defined by integrand
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Why would we want to?
There is a nice organizing principle
One can show:
Key Identity
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General Polygons
General polygon can be defined by a list of vertices:
Consider the sum over vertices:
Using Key Identity
All possible triangulations can be obtained from sum of F’s. This is a triangulation-independent expression of the volume.
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General Polygons
The types of polygons this applies to are completely general—not even necessarily “connected”.
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Equivalence of Triangulations
Spurious vertices
The area is expressed uniquely in terms of F objects.
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Applications to NMHV amplitudes
Key Identity
The amplitude is uniquely expressed in terms of the F’s, precisely one for each vertex of the polytope.
For general n: A sum over vertices
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Summary● Introduced “F” objects, used to uniquely express volumes of polytopes.
● Polytopes defined, and volumes given, in a triangulation-independent way.
● Non-trivial identities obtained algebraically using the “Key Identities.”
● These objects are connected with the Cech cohomology of the underlying space.
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Future Work
● Extend this work to k>1 in N^kMHV○ Find the objects that uniquely express the amplitude, and understand
(cohomological?) origins of the identities generated by these objects.
● Explore the relationship between the F objects and the Grassmannian picture.
○ Connection with gravity?
● Explore loop-level structure.
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Thank you