On-shell diagrams and the geometry of planar N < 4 SYM theories

Abstract

We continue the discussion of the decorated on-shell diagrammatics for planar N < 4 Supersymmetric Yang-Mills theories started in arXiv:1510.03642. In particular, we focus on its relation with the structure of varieties on the Grassmannian. The decoration of the on-shell diagrams, which physically keeps tracks of the helicity of the coherent states propagating along their edges, defines new on-shell functions on the Grassmannian and can introduce novel higher-order singularities, which graphically are reflected into the presence of helicity loops in the diagrams. These new structures turn out to have similar features as in the non-planar case: the related higher-codimension varieties are identified by either the vanishing of one (or more) Plucker coordinates involving at least two non-adjacent columns, or new relations among Plucker coordinates. A distinctive feature is that the functions living on these higher-codimenson varieties can be thought of distributionally as having support on derivative delta-functions. After a general discussion, we explore in some detail the structures of the on-shell functions on Gr(2,4) and Gr(3,6) on which the residue theorem allows to obtain a plethora of identities among them.

arXiv comment34 pages, 65 figures

References in corpus (29)

• Direct Proof Of Tree-Level Recursion Relation In Yang-Mills Theory
• New Recursion Relations for Tree Amplitudes of Gluons
• The Amplituhedron
• Magic identities for conformal four-point integrals
• The All-Loop Integrand For Scattering Amplitudes in Planar N=4 SYM
• Yangian symmetry of scattering amplitudes in N=4 super Yang-Mills theory
• A Duality For The S Matrix
• Dual Superconformal Invariance, Momentum Twistors and Grassmannians
• The Grassmannian Origin Of Dual Superconformal Invariance
• Into the Amplituhedron
• Taming Tree Amplitudes In General Relativity
• Positive Amplitudes In The Amplituhedron
• Evidence for a Nonplanar Amplituhedron
• On-Shell Recursion Relations for Generic Theories
• ABJM amplitudes and the positive orthogonal grassmannian
• On-shell superamplitudes in N<4 SYM
• Gravity On-shell Diagrams
• The Amplituhedron from Momentum Twistor Diagrams
• Towards the Amplituhedron Volume
• Unraveling L_{n,k}: Grassmannian Kinematics
• Stratifying On-Shell Cluster Varieties: the Geometry of Non-Planar On-Shell Diagrams
• The Amplituhedron and the One-loop Grassmannian Measure
• On the Tree-Level Structure of Scattering Amplitudes of Massless Particles
• On-Shell Diagrams for N = 8 Supergravity Amplitudes
• New structures in scattering amplitudes: a review
• Grassmannians for scattering amplitudes in 4d $\mathcal{N}=4$ SYM and 3d ABJM
• The Geometry of On-Shell Diagrams
• Positroid Stratification of Orthogonal Grassmannian and ABJM Amplitudes
• Nonplanar On-shell Diagrams and Leading Singularities of Scattering Amplitudes

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