On-shell diagrams and the geometry of planar $$ \mathcal{N}<4 $$ SYM theories

Abstract

Abstract We continue the discussion of the decorated on-shell diagrammatics for planar $$ \mathcal{N}<4 $$ N < 4 Supersymmetric Yang-Mills theories started in [1]. 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) Plücker coordinates involving at least two non-adjacent columns, or new relations among Plücker 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.