Study of topological condensed matter problems by random matrices

证明了黎曼曲面的模空间的拓扑不变性是由带有外源的高斯随机矩阵模型通过外源的调谐得到的。不变性由一个参数p来分类，该参数表征了外部源的简并性。这种不变性与黎曼曲面上的p-自旋曲线有关。这种拓扑奇性与晶体生长的分叉有关。参数p被扩展到负值(p=-1，-2，...)它代表第k级SL(2，R)/U(1)Wess-Zumino-Witten项。P=-2的情况对应于有相变的么正矩阵模型。用高斯随机矩阵理论考察了强展开和弱展开。在凝聚态问题的相变关系中研究了这种相变。通过降阶为2-矩阵模型，研究了随机矩阵模型的时间相关性，并详细研究了该模型的代数结构。证明了该代数结构是伴随的W-代数。该结构代表了对N=2超对称极小模型的偏离，并研究了随时间变化的Ramond-Ramond项。

The topological invariance of the moduli space of Riemann surface is shown to be obtained from the Gaussian random matrix model with an external source by the tuning of external source. The invariance is classified by a parameter p, which characterizes the degeneracy of the external source. The invariance is related to the p-spin curves on the Riemann surface. This topological singularity is related the bifurcation of the growth of crystals. The parameter p is extended to the negative values(p=-1,-2,...) and it represents level k SL(2, R)/U(1) Wess-Zumino-Witten term. The case of p=-2 corresponds to the unitary matrix model, which has a phase transition. The strong and weak expansion are examined by Gaussian random matrix theory. This transition is studied in the relation of the phase transitions in the condensed matter problems. The time dependence of the random matrix model is studied by the reduction to 2-matrix model and the algebraic structure is investigated in details. The algebraic structure is found to be the associated W-algebra. The structure represents the deviation from N=2 supersymmetric minimal model and the change of the Ramond-Ramond term for the time dependence is studied.