Topological Field Theory Coupled with 2D Quantum Gravity

1996年，我继续这个研究项目，尝试用二维拓扑场论的方法从现象学上理解量子色动力学（QCD）的Seiberg-Witten解。即我期望QCD真空的模依赖于重整化群流并满足一类重整化群方程。推测该重整化群方程具有可积结构，QCD真空的模可以用具有朗道-金兹堡势的二维拓扑场论来描述。我用计算机计算具体地验证了这个猜想。根据希格斯场的表示，规范群SU(3)的结果如下所示。基本的或六维的表示：QCD真空的模是用三次多项式朗道-金兹堡势的拓扑场论来描述的。八维表示：这种情况也可以用拓扑场论来描述，但它的朗道-金兹堡势是带有分支的非理性函数。需要注意的是，融合代数与情形1中相同，因此具有这种势的拓扑场论与情形1中讨论的拓扑场论是等价的。然而，如何确定重整化群流的方程是今后的一个重要课题。十维表示：这种情况下的朗道-金兹堡势与情况1或情况2中得到的相同。人们可能期望在更高维度的希格斯场表示中也会出现类似的朗道-金兹堡势现象。我要强调的是，在情形2中的结果很重要，因为二维拓扑场论可以用两个不同的伦道-金兹堡势来描述。

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.