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.

1996年继续这个研究项目，我试图通过二维拓扑场论从现象学上理解量子色动力学（QCD）的Seiberg-Witten解。也就是说，我期望 QCD 真空的模量取决于重正化群流并满足一种重正化群方程。我推测这样的重正化群方程具有可积结构，并且QCD真空的模可以用具有Landau-Ginzburg势的二维拓扑场论来描述。我通过计算机计算具体验证了这个猜想。下面给出了规范组 SU (3) 的结果，具体取决于希格斯场的表示。1．基本的或六维的表示：QCD真空的模量由具有Landau-Ginzburg势的拓扑场论来描述，Landau-Ginzburg势是三次多项式。 2．八维表示：这种情况也是用拓扑场论来描述的，但它的Landaw-Ginzburg势是带分支切割的无理函数。应该指出的是，融合代数与情况 1 中的相同，因此具有这种势的拓扑场论等效于情况 1 中讨论的拓扑场论。然而，知道确定重正化的方程是未来的一个重要课题群体流动.3.十维表示：这种情况下的Landau-Ginzburg势与情况1或情况2中获得的相同。即使在希格斯场的更高维表示中，人们也可以预期Landau-Ginzburg势的类似现象。我强调案例 2 中的结果很重要，因为二维拓扑场论可以通过两个不同的 Lndaw-Ginzburg 势来描述。

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)