Abstract
We continue the discussion of the decorated on-shell diagrammatics for planar mathcal{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.
Highlights
For the cases of interest, which typically deal with massless particles, such building blocks are provided by the three-particle amplitudes, which are fully determined by Poincare invariance [12]
We continue the discussion of the decorated on-shell diagrammatics for planar N < 4 Supersymmetric Yang-Mills theories started in [1]
The on-shell diagram formulation of N = 4 Supersymmetric Yang-Mills (SYM) is intimately related to novel mathematical structures such as the Grassmannian Gr(k, n) [16,17,18,19,20], whose positivity preserving diffeomorphisms correspond to the Yangian symmetry of the theory [9], and the permutations, which define equivalence classes for the diagrams [9, 21]
Summary
When we build complicated on-shell processes by gluing several three-particle amplitudes, not all of them turns out to be inequivalent. It is important to stress that this operation is possible if and only if the p-valent node has all the helicity arrows with the same direction These are the only cases given the relation outlined before between helicity flows and singularity structure of an on-shell diagram. If instead the states with the same helicity are not adjacent, the helicity flow structure is sensibly different, with one of the two configuration allowing for both the multiplet to propagate in the internal lines generating helicity loops: the square move is an equivalence relation if and only if the on-shell box shows two states with the same helicity direction as adjacent. There is a further operation which maps a given diagram into another diagram with one face less singling out one degree of freedom It can be performed whenever a diagram has, as a sub-diagram, a black node and a white node connected through two edges forming a bubble. We will quickly review the basic definition of the Grassmannian and its properties and we will discuss the Grassmannian formulation for amplitudes in N < 4 SYM theories
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