arXiv:1609.01923v1 [hep-th] 7 Sep 2016 arXiv:xxxx.xxxxx On-shell diagrams and the geometry of planar N<4SYM theories Paolo Benincasa, David Gordo †Instituto de F´ısica Te´orica, Universidad Aut´onoma de Madrid / CSIC Calle Nicolas Cabrera 13, Cantoblanco 28049, Madrid, Spain paolo.benincasa@csic.es, david.gordo@csic.es Abstract We continue the discussion of the decorated on-shell diagrammatics for planarN<4 Super- symmetric 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 sin- gularities, 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) Pl¨ucker coordinates involving at least two non-adjacent columns, or new relations among Pl¨ucker 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 onGr(2,4) andGr(3,6) on which the residue theorem allows to obtain a plethora of identities among them. September 2016
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