ABJM 圈振幅及其正幾何

Abstract

在本篇論文中，我們討論如何計算高點ABJM圈振幅的方法，並引入了正幾何的架構到這個理論。

在第一部分中，我們以八點振幅為例，計算一圈、兩圈振幅，所使用的方法可以直接推廣到更高點的圈振幅。我們首先利用廣義么正性和發散限制條件，確定了該理論的一圈和兩圈被積函數。隨後，我們計算這些被積函數積分後的結果，得到完整的一圈和兩圈八點振幅。

在第二部分中，我們引進並研究了與ABJM振幅和它的正幾何。我們先在動量空間定義他的樹正幾何，接著在動量扭空間將樹正幾何推廣到圈正幾何。我們引進新的正幾何與$\\cal{N}=$4 超對稱楊-米爾斯理論的正幾何有非常簡單的關係：對所有相鄰扭量加上辛條件，並將所有扭量括號由正號改成負號。

In this thesis, we discuss methods for calculating the higher-point ABJM loop amplitudes and introduce the positive geometry framework into this theory.

In the first part, we take the eight-point amplitude as an example and calculate the one-loop and two-loop amplitudes. The methods used can be directly extended to higher-point loop amplitudes. We first determine the one-loop and two-loop integrands of the theory using generalized unitarity and IR constraints. Then, we compute these integrands to derive complete one- and two-loop eight-point amplitudes.

In the second part, we introduce the positive geometry associated with ABJM amplitudes. We first define its tree-level geometry in momentum space and then extend it to loop-level in momentum twistor space. The new positive geometry has a very simple relationship with one of $\\cal{N}=$4 super Yang-Mills theory: imposing symplectic conditions on all adjacent twistors and changing the sign of all twistor brackets from positive to negative.