By Nima Arkani-Hamed, Jacob Bourjaily, Freddy Cachazo, Alexander Goncharov, Alexander Postnikov, Jaroslav Trnka
Outlining a progressive reformulation of the principles of perturbative quantum box conception, this e-book is a self-contained and authoritative research of the applying of this new formula to the case of planar, maximally supersymmetric Yang-Mills idea. The e-book starts by means of deriving connections among scattering amplitudes and Grassmannian geometry from first rules prior to introducing novel actual and mathematical principles in a scientific demeanour available to either physicists and mathematicians. the primary avid gamers during this procedure are on-shell services that are heavily on the topic of convinced sub-strata of Grassmannian manifolds referred to as positroids - by way of which the type of on-shell capabilities and their family turns into combinatorially happen. this is often an important creation to the geometry and combinatorics of the positroid stratification of the Grassmannian and an excellent textual content for complex scholars and researchers operating within the components of box idea, excessive power physics, and the wider fields of mathematical physics.
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Extra resources for Grassmannian Geometry of Scattering Amplitudes
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 ↓ ↓ ↓ ↓ ↓ ↓ .
Grassmannian Geometry of Scattering Amplitudes by Nima Arkani-Hamed, Jacob Bourjaily, Freddy Cachazo, Alexander Goncharov, Alexander Postnikov, Jaroslav Trnka
Categories: Algebraic Geometry