Hyperplane Arrangements In Scattering Amplitudes, Asymptotic Solutions Of The Flag And KZ Equations

Scattering amplitude plays an important role in quantum field theory.In perturbation theory,Although great achievement have been gained by using Feynman diagram to calculate scattering amplitude,with the expansion of perturbation,more and more computation is needed,so direct calculation is not effective,and it is necessary to find a mathematical structure to reduce the computation.With the development of the mathematical forms of scattering amplitudes,many hidden symmetries become apparent in some mathematical forms.For example,the amplitude of gravity theory is a double Yang-Mills amplitude.In this paper,we reexpress the bi-adjoint scalar amplitudes by using the dual form between hyperplane arrangement and flag,which is in agreement with the known Feynman diagram form and the scattering equation(Cachazo-He-Yuan)representation.We also find some relations between the asymtotic solutions of flag and Knizhnik-Zamolodchikov(KZ)equation and the scattering amplitudes.In chapter 2,we briefly review the mathematical structures of scattering amplitudes that have been discovered in recent years,and briefly give the related mathematical concepts,as well as the general representation of CHY.The third chapter introduces the Orlik-Solomon algebra generated by hyperplanes and the dual flag complex.The relation between the integral region of Z-amplitude related to string theory amplitude and the flag complex is presented,and the corresponding result of ?'?0 is given through the so-called subdivision.And according to the so-called regularization of flag,it can be seen to be related to the scalar amplitudes in the field theory limit,and the expression is rewritten using the bilinear mapping form of flag.The intersection number of two Twisted differential forms is reexpressed using the standard flag generated by the hyperplanes and the bilinear form between them.The result is the Feynman propagator of scalar amplitudes.According to the equivalence between Grothendick residue and CHY representation,the CHY form of the bi-adjoint scalar amplitudes is expressed by the special flag which is mapped by the critical point of the principal function.Then,the equivalence of the Feynman graph of the scalar amplitudes and CHY representation is found by the substitution relationship between the standard flag and the special flag.In chapter four,we introduce the KZ-equation and its asymptotic solution.It can be seen that the solution project on dual vector space of the first order KZ-equation is related to the z-amplitude,and it can be found that the bilinear form of the asymptotic solution S' is consistent with the representation of CHY.We can see that the dual space of the asymptotic solutions of the KZ-equation is the differential form,that is,the differential form dual to flag.This will find the expression for S'.In chapter five,according to the correspondence between module space and flag,it can be seen that different flags are trivalent graphs of different trivalent factorization of Feynman diagrams.Color-kinematic duality satisfy the Jacobi relationship is actually the residue theorem on moduli space.The space dimension of n point moduli space is(n-3)(due to the action of SL(2,C)group,three selected pointcan be fixed)that is,different points can collide(n-3)times at most,and each collision will produce compactification.After the collision of(n-3)times,a point with the maximum codimension will be generated,and the co-dimension of this point is(n-3).This process is equivalent to forming a flag,with different sequences of collisions between different points corresponding to different flags.At this point,the residue theorem on module space is transformed into a flag with a gap satisfying the identity relation.The sixth chapter summarizes the whole paper and points out the work that can be further studied.