Amplitudes at Weak Coupling as Polytopes in AdS_5

Details Amplitudes at Weak Coupling as Polytopes in AdS_5 Amplitudes at Weak Coupling as Polytopes in AdS_5

2010-04-20

arxiv.org/abs/1004.3498v1

J.Phys.A44:135401,2011

Physics

High Energy Physics

High Energy Physics - Theory

16 pages, 5 figures

Scientific paper

10.1088/1751-8113/44/13/135401

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.

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