标题： 随机矩阵、边界与膜 显示英文标题

标题： Random Matrices, Boundaries and Branes

摘要： 本论文致力于将随机矩阵理论应用于离散和连续随机曲面的研究；在此形式主义中，特别强调了曲面边界及其相关的边界条件。 特别是，利用具有置换对称性的多矩阵积分，我们能够计算在随机平面格点上具有各种边界条件的 Potts 模型的配分函数。 我们进一步研究了该模型相图中的临界点与二维 Liouville 理论耦合到具有全局 $\mathcal{W}$-对称性的共形场论之间的对应关系。 在这种情况下，每种边界条件都可以被解释为一族玻色弦背景中一个膜的描述。 这项研究建议，当给定系统放置在一个有界亏格的随机曲面上时，最初不同的边界条件的谱可能会变得退化，从而有效地留下较小的一组独立的边界条件。 这一有趣且备受争议的特点随后通过考虑双尺度极限进行了进一步分析。对于这个模型，我们可以明确展示这种表面上的退化是如何实际上通过考虑弦微扰理论中看不见的贡献来解决的。 总的来说，这些发展为以前未探索的边界条件提供了新的描述，并为边界和膜的非微扰物理提供了新的见解。 显示英文摘要

摘要： This thesis is devoted to the application of random matrix theory to the study of random surfaces, both discrete and continuous; special emphasis is placed on surface boundaries and the associated boundary conditions in this formalism. In particular, using a multi-matrix integral with permutation symmetry, we are able to calculate the partition function of the Potts model on a random planar lattice with various boundary conditions imposed. We proceed to investigate the correspondence between the critical points in the phase diagram of this model and two-dimensional Liouville theory coupled to conformal field theories with global $\mathcal{W}$-symmetry. In this context, each boundary condition can be interpreted as the description of a brane in a family of bosonic string backgrounds. This investigation suggests that a spectrum of initially distinct boundary conditions of a given system may become degenerate when the latter is placed on a random surface of bounded genus, effectively leaving a smaller set of independent boundary conditions. This curious and much-debated feature is then further scrutinised by considering the double scaling limit of a two-matrix integral. For this model, we can show explicitly how this apparent degeneracy is in fact resolved by accounting for contributions invisible in string perturbation theory. Altogether, these developments provide novel descriptions of hitherto unexplored boundary conditions as well as new insights into the non-perturbative physics of boundaries and branes.