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キーワード:
Theoretical Physics
要旨:
In this thesis we explore mathematical foundations of scattering amplitudes both in string theory and quantum field theories. Scattering amplitudes are the contact point between theoretical and experimental physics. They are used to check theoretical results in experiments for example Standard Model predictions are tested in LHC measurements of cross sections. Furthermore, they can be used to understand the underlying theoretical structure as well. There are two main ways to discuss tree-level scattering amplitudes: First, in the bottom up approach one can study amplitudes in different quantum field theories with various matter contents and symmetries. Second, one can use the top down approach and employ string theory as the high energy quantum gravity and study scattering processes in string theory. Then, one can relate them to different field theories in the low energy limit. Utilising these methods one can not only construct more efficient methods to calculate scattering processes (e.g. spinor-helicity formalism) but also establish structural relations among different theories (e.g. gauge/gravity duality). Our discussion in this thesis stands in the middle of the two aforementioned methods. We discuss Riemann surfaces and define advanced topological structures on top of them namely the twisted cohomology. In particular, we explain the recent development regarding twisted forms/cycles that allows us to construct different tree-level scattering amplitudes both in string theory and quantum field theories. Here, we use the relationship between string theory and quantum field theories (i.e.\,the low energy limit of string theory) to introduce an algorithm by which we are able to produce new twisted forms. The intersection numbers of these new twisted forms can be used to calculate scattering amplitudes of different theories more efficiently. Furthermore, we take advantage of this new mathematical method and study the structure of scattering amplitudes. In particular, we explore the double copy construction in two separate avenues. First we construct the first ever double copy for the massive spin-$2$ field through string theory. We show that this massive double copy can be compared to bimetric gravity. Second, we discuss the role of the twisted cohomology in double copy and put forward a novel method to understand the double copy construction in terms of twisted differentials as well as producing (and suggesting) new double copy theories.
