Recently,planar collections of Feynman diagrams were proposed by Borges and one of the authors as the natural generalization of Feynman diagrams for the computation of k=3 biadjoint amplitudes.Planar collections are o...Recently,planar collections of Feynman diagrams were proposed by Borges and one of the authors as the natural generalization of Feynman diagrams for the computation of k=3 biadjoint amplitudes.Planar collections are one-dimensional arrays of metric trees satisfying an induced planarity and compatibility condition.In this work,we introduce planar matrices of Feynman diagrams as the objects that compute k=4 biadjoint amplitudes.These are symmetric matrices of metric trees satisfying compatibility conditions.We introduce two notions of combinatorial bootstrap techniques for finding collections from Feynman diagrams and matrices from collections.As applications of the first,we find all 693,13612 and 346710 collections for(k,n)=(3,7),(3,8)and(3,9),respectively.As applications of the second kind,we find all90608 and 30659424 planar matrices that compute(k,n)=(4,8)and(4,9)biadjoint amplitudes,respectively.As an example of the evaluation of matrices of Feynman diagrams,we present the complete form of the(4,8)and(4,9)biadjoint amplitudes.We also start a study of higher-dimensional arrays of Feynman diagrams,including the combinatorial version of the duality between(k,n)and(n-k,n)objects.展开更多
Planar arrays of tree diagrams were introduced as a generalization of Feynman diagrams that enable the computation of biadjoint amplitudes m_(n)(^(k))for k>2.In this follow-up work,we investigate the poles of m_(n)...Planar arrays of tree diagrams were introduced as a generalization of Feynman diagrams that enable the computation of biadjoint amplitudes m_(n)(^(k))for k>2.In this follow-up work,we investigate the poles of m_(n)(^(k))from the perspective of such arrays.For general k,we characterize the underlying polytope as a Flag Complex and propose a computation of the amplitude-based solely on the knowledge of the poles,whose number is drastically less than the number of the full arrays.As an example,we first provide all the poles for the cases(k,n)=(3,7),(3,8),(3,9),(3,10),(4,8)and(4,9)in terms of their planar arrays of degenerate Feynman diagrams.We then implement simple compatibility criteria together with an addition operation between arrays and recover the full collections/arrays for such cases.Along the way,we implement hard and soft kinematical limits,which provide a map between the poles in kinematic space and their combinatoric arrays.We use the operation to give a proof of a previously conjectured combinatorial duality for arrays in(k,n)and(n-k,n).We also outline the relation to boundary maps of the hypersimplex Δ_(k,n) and rays in the tropical Grassmannian Tr(k,n).展开更多
基金supported in part by the Government of Canada through the Department of Innovation,Science and Economic Development Canadaby the Province of Ontario through the Ministry of Economic Development,Job Creation and Trade。
文摘Recently,planar collections of Feynman diagrams were proposed by Borges and one of the authors as the natural generalization of Feynman diagrams for the computation of k=3 biadjoint amplitudes.Planar collections are one-dimensional arrays of metric trees satisfying an induced planarity and compatibility condition.In this work,we introduce planar matrices of Feynman diagrams as the objects that compute k=4 biadjoint amplitudes.These are symmetric matrices of metric trees satisfying compatibility conditions.We introduce two notions of combinatorial bootstrap techniques for finding collections from Feynman diagrams and matrices from collections.As applications of the first,we find all 693,13612 and 346710 collections for(k,n)=(3,7),(3,8)and(3,9),respectively.As applications of the second kind,we find all90608 and 30659424 planar matrices that compute(k,n)=(4,8)and(4,9)biadjoint amplitudes,respectively.As an example of the evaluation of matrices of Feynman diagrams,we present the complete form of the(4,8)and(4,9)biadjoint amplitudes.We also start a study of higher-dimensional arrays of Feynman diagrams,including the combinatorial version of the duality between(k,n)and(n-k,n)objects.
基金supported in part by the Government of Canada through the Department of Innovation, Science and Economic Development Canadaby the Province of Ontario through the Ministry of Economic Development, Job Creation and Trade
文摘Planar arrays of tree diagrams were introduced as a generalization of Feynman diagrams that enable the computation of biadjoint amplitudes m_(n)(^(k))for k>2.In this follow-up work,we investigate the poles of m_(n)(^(k))from the perspective of such arrays.For general k,we characterize the underlying polytope as a Flag Complex and propose a computation of the amplitude-based solely on the knowledge of the poles,whose number is drastically less than the number of the full arrays.As an example,we first provide all the poles for the cases(k,n)=(3,7),(3,8),(3,9),(3,10),(4,8)and(4,9)in terms of their planar arrays of degenerate Feynman diagrams.We then implement simple compatibility criteria together with an addition operation between arrays and recover the full collections/arrays for such cases.Along the way,we implement hard and soft kinematical limits,which provide a map between the poles in kinematic space and their combinatoric arrays.We use the operation to give a proof of a previously conjectured combinatorial duality for arrays in(k,n)and(n-k,n).We also outline the relation to boundary maps of the hypersimplex Δ_(k,n) and rays in the tropical Grassmannian Tr(k,n).
