Abstract
We take up the study of integrable structures behind non-planar contributions to scattering amplitudes in $$ \mathcal{N}=4 $$ super Yang-Mills theory. Focusing on leading singularities, we derive the action of the Yangian generators on color-ordered subsets of the external states. Each subset corresponds to a single boundary of the non-planar on-shell diagram. While Yangian invariance is broken, we find that higher-level Yangian generators still annihilate the non-planar on-shell diagram. For a given diagram, the number of these generators is governed by the degree of non-planarity. Furthermore, we present additional identities involving integrable transfer matrices. In particular, for diagrams on a cylinder we obtain a conservation rule similar to the Yangian invariance condition of planar on-shell diagrams. To exemplify our results, we consider a five-point MHV on-shell function on a cylinder.
Highlights
Every on-shell diagram encodes an expression in terms of a Graßmannian contour integral of the form [11, 17,18,19]
While Yangian invariance is broken, we find that higher-level Yangian generators still annihilate the non-planar on-shell diagram
We show that identities similar to the Yangian invariance of the planar on-shell diagram Ap in (2.2) hold for the non-planar diagram Anp
Summary
Let us consider an arbitrary non-planar on-shell diagram Anp, with nnp external particles and MHV degree knp. ∆CC MR(1 − u)Ap. To obtain the action of the Yangian generators on the boundary B we expand the monodromy in terms of the spectral parameter u. To obtain the action of the Yangian generators on the boundary B we expand the monodromy in terms of the spectral parameter u We find that the remaining higher levels of the Yangian generators that act on the boundary B annihilate the non-planar on-shell diagram Anp, and generate unbroken symmetries, M[Bi] ab Anp = 0 , i = kp + 1, . The number of external states nB fixes the number of levels of the Yangian generators and kp = knp + ncut can be regarded as a measure of non-planarity, as each additional boundary or handle requires further internal lines to be cut. The preceding discussion shows that the actual symmetries are determined by the minimal way to cut the diagram, and that one should consider all possible embeddings of the diagram to identify as many symmetries as possible
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