Superstring world-sheet on the lattice

弦理论是理论物理学的一个模型，在这个模型中，自然界的基本实体不是点状的，而是一维的弦状物体——弦。这个模型最初吸引了物理学家和数学家的注意，因为它可能是统一自然界基本力描述的一种方法，但它从未被实验证明。尽管如此，它仍然是理论物理研究的一个主要领域，因为它在数学中的新结构和粒子物理中的新思想中发挥了强大的触发作用。特别是，二十年前，弦理论被认为是理解规范理论的一些普遍特征的工具，这些特征代表了描述我们周围世界及其相互作用的当前标准方式（例如电磁理论就是规范理论）。这个突破性的想法，被称为规范/弦对偶性，是一种假设，如果弦在特定的弯曲时空中运动，那么它们的动力学与特定规范理论中某个物理量的动力学是对偶的（具有相同的行为）。现在，在描述相互作用强度的某个参数（耦合）很大的情况下，对任何规范理论量的研究都变得困难或不可能。至关重要的是，根据规范/弦对偶性，这种所谓的规范理论的强耦合状态与弦理论研究的直接状态是一致的，反之亦然。因此，如果对偶是真的，那么使用弦理论就可以很容易地获得用其他明显方法无法获得的物理信息。更重要的是，最近的进展表明，一些特定的规范理论（以及它们的弦对偶）可能是完全可积的，也就是说，有可能预测某些特征物理观测值在所有耦合值下的行为！如果这是真的，那么每个理论粒子物理学家的梦想之一就实现了：拥有一个单一的“夸克-反夸克”势的公式（尽管是在一个非常特定的规范理论中）作为耦合的函数现在是可能的！为了验证这些猜想（模型的可积性，规范/弦对偶性本身）以及尝试将全耦合分析扩展到一般的，不可积的例子，人们应该以一种不需要假设就能探索其强耦合状态的方式对其中任何一个模型进行研究。一种已知的方法是将模型定义在晶格上的时空离散化。从一个具有无限多个自由度的系统（与连续统中的原始性质有关），模型成为一个具有Nd个自由度的统计系统（其中N是一维晶格的长度，d是时空维度的数量），然后可以用称为蒙特卡罗方法的数值算法进行研究。这个项目是关于在规范/弦对偶中使用蒙特卡罗方法的弦理论模型。

String theory is a model of theoretical physics for which the fundamental entities in nature are not point-like, but one-dimensional string-like objects - the strings. This model has primarily attracted the attention of physicists and mathematicians as a possible way to unify the description of the fundamental forces in nature, but it has never been proven by experiments. Despite this, it is a primary area of research in theoretical physics in its role of powerful trigger for novel structures in mathematics and new ideas in particle physics. In particular, twenty years ago string theory has been indicated as a tool to understand some universal features of gauge theories, which represent the current, standard way to describe the world around us and its interactions (e.g. the theory of electromagnetism is a gauge theory). This breakthrough idea, known as gauge/string duality, is the conjecture that if strings move through a certain curved spacetime then their dynamics is dual to (has the same behavior of) the dynamics of a certain physical quantity in a specific gauge theory. Now, the study of any gauge theory quantity becomes hard or impossible in the regime in which a certain parameter describing the strength of the interaction - the coupling - is big. Crucially, such so-called strong coupling regime of the gauge theory coincides according to the gauge/string duality with the one in which the study of string theory is straightforward, and vice versa. Therefore, if the duality is true, using string theory one can have easy access to physical information that cannot be obtained in other obvious ways. What is more, recent progress has suggested that some specific gauge theories (and their string dual) may be exactly integrable, namely it may be possible to predict the behavior of some characteristic physical observables at all values of the coupling! If true, one of the dreams of every theoretical particle physicist comes true: having a single formula for the "quark-antiquark" potential (albeit in a very specific gauge theory) as a function of the coupling is now possible!To verify these conjectures (the integrability of the models, the gauge/string duality itself) as well as to try extending the all-coupling analysis to general, non-integrable examples, one should perform a study of either one of the models in a way that is potentially able to explore its strong coupling regime with no assumptions needed. A known way to proceed is to discretize the spacetime in which the model is defined onto a lattice. From being a system with an infinite number of degrees of freedom (related to its original nature in the continuum), the model becomes a statistical system with Nd degrees of freedom (where N is the length of the lattice in one dimension, d is the number of spacetime dimensions) which may then be studied with numerical algorithms known as Monte Carlo methods. This project is about the use of Monte Carlo methods for string theory models relevant in the gauge/string duality.