By William Fulton
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Additional info for Categorical Framework for the Study of Singular Spaces (Memoirs of the American Mathematical Society)
The picture is obviously not cyclically invariant—as a rotation would exchange its black and white vertices. 57) The merge and square moves can be used to show the physical equivalence of many seemingly different on-shell diagrams. 59) ⇔ 1 25 2 3 5 4 5 4 5 4 Here, each step down involves one or more square moves, and each step up involves one or more mergers. 61) 1 4 These forms are completely equivalent, but suggest very different physical interpretations. 60), clearly exposes its origin as a forward-limit—arising through the gluing of two of the external legs of a six-particle tree-amplitude.
As each vertex carries two auxiliary degrees of freedom, and each GL(1) from the internal lines can be used to remove one of them, the ‘dimension’ associated with an on-shell graph is simply: dim(C) = 2nV − nI . 42) We should mention that this can be counted in a more direct way from the graph as follows. Because each on-shell graph is trivalent, we have 3nV =2nI + n so that dim(C)=2nV − nI =nI − nV + n; and restricting our attention to planar graphs, Euler’s formula tells us that (nF − n) − nI + nV = 1 (where nF is the number of faces of the graph including the n faces of the boundary).
At first sight, it certainly seems as if a “combinatorial S-matrix” would be far too simple an object to capture anything remotely resembling the richness of physical scattering amplitudes. However, we will soon discover that this is not the case: on-shell diagrams are fully determined by permutations, and hence the entire S-matrix of N =4 can be described combinatorially! Recall that something very much like this happens for integrable theories in (1 + 1) dimensions [84, 85]. Consider for instance the permutation given by 1 2 3 4 5 6 ↓ ↓ ↓ ↓ ↓ ↓ .
Categorical Framework for the Study of Singular Spaces (Memoirs of the American Mathematical Society) by William Fulton
Categories: Algebraic Geometry