Algebraic and geometric aspects of string dualities

弦对偶性的代数和几何方面

基本信息

批准号：
13640264
负责人：
KATO Akishi
金额：
$ 2.11万
依托单位：
The University of Tokyo
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2001
资助国家：
日本
起止时间：
2001 至 2004
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-13640264/
关键词：
quantum field theory string theory supersymmetric gauge theory duality renormalization group derived category rational elliptic surface moduli space E-string 弦双対性位相的場の理論

项目摘要

The renormalization of quantum field theories are commonly phased as "the procedure to extract finite results from divergent quntities by subtracting infinity." This obscures why one can get a reliable answer avoiding arbitrariness. Cannes and Kreimer proposed a Hopf algebra point of view to the renormalization. In their paper, however, it was not completely clear which part is assumption and which part is logical consequence. As a part of our project of understanding dualities algebraically, I started to clarify the relationship between Hopf algebra and theory of renormalization. In particular, I studied the renormalizability and the Birkhoff decomposition in detail using a toy model.Discovery of string dualities enabled us to "geometrize" the various phenomena in physics, and provided a tool to analyze the dynamics beyond the reach of perturbative method. In particular, new type of fixed points were discovered in six dimensional N=(1,0) supersymmetric gauge theories with E8 global symmetry. F-theoretic interpretation of this critical point is the vanishing locus of codimension one rational elliptic surface in Calabi-Yau threefold. Conjecturally elementary excitations called "E-strings" will play an important role in the critical point. As has been successful in Seiberg-Witten theory, renormalization group flow can be seen as the deformation family of elliptic curves fibered over a moduli space. The duality group of the quantum field theory acts not only as covering transformations associated with the fibration, but also the autoequivalence of the derived categories on the surface. I investigated the relationship between the geometry of rational elliptic surfaces and the dynamics of the supersymmetric gauge theories, especially the partition functions of the topological gauge theories (E-strings). This work was done with Professors H.Awata (Nagoya), S.Kondo (Nagoya), Y.Saito (Tokyo), Y.Shimizu (ICU) and A.Tsuchiya (Nagoya).

量子场理论的重新归一化通常被逐步逐步逐步，因为“通过减去无穷大从不同的Q提取有限结果的程序”。这掩盖了为什么人们可以得到可靠的答案以避免任意性。戛纳和克雷默（Cannes）和克雷默（Kreimer）提出了霍普夫代数的观点，以实现重新归一化。但是，在他们的论文中，尚不完全清楚哪个部分是假设，哪一部分是合乎逻辑的结果。作为我们理解二元性的项目的一部分，我开始阐明Hopf代数与重新归一化理论之间的关系。特别是，我使用玩具模型详细研究了可授权性和Birkhoff分解。弦乐二元性的分配使我们能够“几何化”物理学中的各种现象，并提供了一种工具来分析超出扰动方法触及的动态。特别是，具有E8全局对称性的六个维度n =（1,0）超对称量规理论，发现了新型的固定点。对这个临界点的f理论解释是Colabi-yau三倍的Condimension One Rational Elliptic表面的消失源。猜想的基本激发称为“电子弦”将在临界点起重要作用。正如Seiberg-Witten理论中成功的那样，重新归一化组流量可以看作是在模量空间上纤维纤维的变形家族。量子场理论的二元性群不仅涵盖了与纤维相关的转换，而且还涵盖了表面上派生类别的自动等效性。我调查了有理椭圆形表面的几何形状与超对称仪表理论的动力学，尤其是拓扑规理论（E弦）的分区函数之间的关系。这项工作是由H.Awata（Nagoya），S.Kondo（Nagoya），Y.Saito（东京），Y.Shimizu（ICU）和A.Tsuchiya（Nagoya）完成的。