Topological Field Theory Coupled with 2D Quantum Gravity

拓扑场论与二维量子引力相结合

• 批准号: 08640364
• 负责人: AOYAMA Shogo
• 金额: $ 1.28万
• 依托单位: Shizuoka University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 1996
• 资助国家: 日本
• 起止时间: 1996 至 1997
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-08640364/
• 关键词: Prticle Theory; 2D Quantum Gravity; topological field theory

Continuing the research project in 1996, I tried to phenomelogically understand the Seiberg-Witten solution of quantum chromo dynamics (QCD) by means of a two-dimensional topological field theory. Namely I expected that moduli of QCD vacuum depend on the renormalization group flow and satisfy a kind of renormalization group equation. I conjectured that sucth a renormalization group equation has an integrable structure and the moduli of QCD vacuum could be described by a two dimensional topological field theory with a Landau-Ginzburg potential. I concretely checked this conjecture by computer calculation. The results for the gauge group SU (3) are given below, depending on the representation of Higgs fields.1. The fundamental or six-dimensional representation :The moduli of QCD vacuum is described by a topological field theory with a Landau-Ginzburg potential which is a polynomial of the third degree.2. The eigth-dimensional representation :This case is also described by a topological field theory, but its Landaw-Ginzburg potential is irrational function with branch-cuts. One should remark that the fusion algebra is the same as in the case 1 and therefore the topological field theory with this potential is equivalent to the one discussed in the case 1.However it is an important future subject to know an equation to determine the renormalization group flow.3. The ten-dimnsional representation :The Landau-Ginzburg potentail in this case is the same as obtained in the case 1 or the case 2.One may expect the similar phenomena of the Landau-Ginzburg potential even in higher dimensional representation of the Higgs fields. I emphasize that the result in the case 2 in important in the sense that a two-dimensional topological field theory can be described by two different Lndaw-Ginzburg potential.

在1996年继续进行研究项目，我试图通过二维拓扑场理论从现象中理解量子Chromo动力学（QCD）的Seiberg-witten解决方案。也就是说，我期望QCD真空的模量取决于重新归一化的流动，并满足一种重新归一化组方程。我猜想，取消重新归一化组方程具有可集成的结构，QCD真空的模量可以通过具有Landau-Ginzburg潜力的二维拓扑场理论来描述。我通过计算机计算来具体检查了这个猜想。量规组SU（3）的结果如下，具体取决于希格斯字段的表示。1。基本或六维表示：QCD真空的模量由具有Landau-Ginzburg电位的拓扑场理论描述，该理论是第三度的多项式。2。特定维表示：这种情况也由拓扑场理论描述，但其Landaw-Ginzburg的潜力是分支切割的非理性功能。应该指出，融合代数与情况1相同，因此，具有该潜力的拓扑字段理论等于案例中讨论的拓扑场理论1.何时了解一个重要的对象，知道确定重新分配群体流量的方程。3。十维表示：在这种情况下，Landau-Ginzburg Potentail与情况1或情况2相同。我强调的是，在案例2中，重要的是，在某种意义上，可以通过两个不同的lndaw-ginzburg潜力来描述二维拓扑场理论。

项目成果

青山昭五: "Topological Landau-Ginsburg Theory with a Rational Potential and the Dispersionless KP Hierarchy" Communication in Mathematical Physics. 182. 185-219 (1996)

Shogo Aoyama：“具有有理势的拓扑朗道-金斯伯格理论和无色散 KP 层次结构”数学物理通讯 182. 185-219 (1996)