Amplitudes at Weak Coupling as Polytopes in AdS_5

Journal article

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.

Authors: Mason, L; Skinner, D

• Publication status: Published
• Journal: J.Phys.A
• Volume: 44
• Issue: 13
• Pages: 135401
• Publication date: 2010-04-20
• DOI: 10.1088/1751-8113/44/13/135401
• EISSN: 1751-8121
• ISSN: 1751-8113
• Keywords: math-ph; math.MP; hep-th