Abstract
We address the appearance of algebraic singularities in the symbol alphabet of scattering amplitudes in the context of planar mathcal{N} = 4 super Yang-Mills theory. We argue that connections between cluster algebras and tropical geometry provide a natural language for postulating a finite alphabet for scattering amplitudes beyond six and seven points where the corresponding Grassmannian cluster algebras are finite. As well as generating natural finite sets of letters, the tropical fans we discuss provide letters containing square roots. Remarkably, the minimal fan we consider provides all the square root letters recently discovered in an explicit two-loop eight-point NMHV calculation.
Highlights
Euclidean sheet, seems to be implied by them)
We address the appearance of algebraic singularities in the symbol alphabet of scattering amplitudes in the context of planar N = 4 super Yang-Mills theory
We argue that connections between cluster algebras and tropical geometry provide a natural language for postulating a finite alphabet for scattering amplitudes beyond six and seven points where the corresponding Grassmannian cluster algebras are finite
Summary
Seems to be implied by them). An important point about the property of cluster adjacency is that it extends the role of the cluster algebra beyond the union of the A-coordinates it generates; it provides a role for the way the clusters themselves appear in the algebra. We can use the g-vectors of the cluster algebra as candidate rays of any fan F (S) defined by tropical evaluation of a finite set S of cluster A-coordinates. As we already outlined in [38], if we consider the Speyer-Williams fan where we take S to be the set of all minors we find that 356 g-vectors of the cluster algebra are rays of the fan.
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