Amplitudes at Weak Coupling as Polytopes in AdS_5

Abstract

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.

16 pages, 5 figures

References in corpus (20)

• One-Loop n-Point Gauge Theory Amplitudes, Unitarity and Collinear Limits
• Gluon scattering amplitudes at strong coupling
• Generalized Unitarity and One-Loop Amplitudes in N=4 Super-Yang-Mills
• Magic identities for conformal four-point integrals
• Conformal Ward identities for Wilson loops and a test of the duality with gluon amplitudes
• Yangian symmetry of scattering amplitudes in N=4 super Yang-Mills theory
• Fermionic T-Duality, Dual Superconformal Symmetry, and the Amplitude/Wilson Loop Connection
• Dual Superconformal Invariance, Momentum Twistors and Grassmannians
• Maximally Supersymmetric Planar Yang-Mills Amplitudes at Five Loops
• Dual Superconformal Symmetry from AdS5 x S5 Superstring Integrability
• A note on dual superconformal symmetry of the N=4 super Yang-Mills S-matrix
• Scattering into the fifth dimension of N=4 super Yang-Mills
• All tree-level amplitudes in N=4 SYM
• A geometrical angle on Feynman integrals
• Multi-loop Feynman integrals and conformal quantum mechanics
• Proof of the Dual Conformal Anomaly of One-Loop Amplitudes in N=4 SYM
• Thermodynamic Bethe Ansatz and Threefold Triangulations
• ADE functional dilogarithm identities and integrable models
• IR finite one-loop box scalar integral with massless internal lines
• One-loop derivation of the Wilson polygon - MHV amplitude duality

Cited by in corpus (27)

• The Amplituhedron
• Local Integrals for Planar Scattering Amplitudes
• Projectors, Shadows, and Conformal Blocks
• The Complete Planar S-matrix of N=4 SYM as a Wilson Loop in Twistor Space
• Superconformal symmetry and two-loop amplitudes in planar N=4 super Yang-Mills
• Scattering Amplitudes
• Generic multiloop methods and application to N=4 super-Yang-Mills
• New differential equations for on-shell loop integrals
• Scattering Amplitudes and Wilson Loops in Twistor Space
• A Note on Polytopes for Scattering Amplitudes
• More loops and legs in Higgs-regulated N=4 SYM amplitudes
• Dualities for Loop Amplitudes of N=6 Chern-Simons Matter Theory
• Rationalizing Loop Integration
• Mellin Amplitudes for Dual Conformal Integrals
• Yangian-invariant fishnet integrals in 2 dimensions as volumes of Calabi-Yau varieties
• Simple loop integrals and amplitudes in N=4 SYM
• Gauge and Gravity Amplitude Relations
• Dual conformal symmetry at loop level; massive regularization
• Manifestly Dual-Conformal Loop Integration
• All-Mass $n$-gon Integrals in $n$ Dimensions
• MHV Diagrams from an All-Line Recursion Relation
• Amplituhedron-like geometries
• One-loop SYM-supergravity relation for five-point amplitudes
• One-loop N = 4 super Yang-Mills scattering amplitudes in d dimensions, relation to open strings and polygonal Wilson loops
• Applications of Subleading-color Amplitudes in N=4 SYM Theory
• Twistor and Polytope Interpretations for Subleading Color One-Loop Amplitudes
• Triangular tessellations of one-loop scattering amplitudes in ${\varphi }^3$ theory