论文标题
弦理论中的模块图形式和散射幅度
Modular Graph Forms and Scattering Amplitudes in String Theory
论文作者
论文摘要
在本论文中，我们使用称为模块化图形的特殊类别的功能，研究了十维Minkowski背景下，在十维Minkowski背景下，在一环级别（即在属属中）隔离字符串散射幅度的低能膨胀。这些允许对低能扩张进行系统评估，并满足许多非平凡的代数和差异关系。我们详细研究了这些关系，从而导致了大量模块化图形形式的基础分解，这些图形大大降低了振幅中积分的扩展的复杂性。本论文的结果之一是Mathematica软件包，该软件包自动化了这些简化。我们使用这些技术来计算单循环水平上杂质弦中四个胶子的散射幅度的领先低能顺序。 此外，我们研究了一个生成函数，该函数猜想包含所有扰动闭合理论的圆环积分。我们以众多艾森斯坦系列的迭代积分来编写此生成功能，并使用此方法比以前可能对模块图形式的空间进行更严格的表征。 对于树级的字符串振幅，多个Zeta值的单值映射图映射了开放式弦振幅到封闭弦振幅。合适的单循环概括的定义，即所谓的椭圆形单值地图，是一个活跃的研究领域，我们使用圆环积分的生成函数为该主题提供了新的观点。 该论文的原始版本于2020年6月提交给柏林洪堡大学，可在DOI 10.18452/21829下获得。本文与此版本相比包含较小的更新，反映了文献中的进一步发展，特别是关于构建椭圆形单值地图的构建。
In this thesis, we investigate the low-energy expansion of scattering amplitudes of closed strings at one-loop level (i.e. at genus one) in a ten-dimensional Minkowski background using a special class of functions called modular graph forms. These allow for a systematic evaluation of the low-energy expansion and satisfy many non-trivial algebraic and differential relations. We study these relations in detail, leading to basis decompositions for a large number of modular graph forms which greatly reduce the complexity of the expansions of the integrals appearing in the amplitude. One of the results of this thesis is a Mathematica package which automatizes these simplifications. We use these techniques to compute the leading low-energy orders of the scattering amplitude of four gluons in the heterotic string at one-loop level. Furthermore, we study a generating function which conjecturally contains the torus integrals of all perturbative closed-string theories. We write this generating function in terms of iterated integrals of holomorphic Eisenstein series and use this approach to arrive at a more rigorous characterization of the space of modular graph forms than was possible before. For tree-level string amplitudes, the single-valued map of multiple zeta values maps open-string amplitudes to closed-string amplitudes. The definition of a suitable one-loop generalization, a so-called elliptic single-valued map, is an active area of research and we provide a new perspective on this topic using our generating function of torus integrals. The original version of this thesis, as submitted in June 2020 to the Humboldt University Berlin, is available under the DOI 10.18452/21829. The present text contains minor updates compared to this version, reflecting further developments in the literature, in particular concerning the construction of an elliptic single-valued map.
