Superstring world-sheet on the lattice

弦理论是一种理论物理模型，对于它来说，自然界中的基本实体不是点状的，而是一维的弦状物体-弦。这个模型最初引起了物理学家和数学家的注意，作为一种统一描述自然界基本力的可能方法，但它从未被实验证明。尽管如此，它是理论物理的主要研究领域，因为它对数学中的新结构和粒子物理中的新思想具有强大的触发作用。特别是20年前，弦理论被认为是一种工具，用于理解规范理论的一些普遍特征，这些规范理论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，电磁学理论是规范理论)。这个突破性的想法被称为规范/弦对偶性，它是这样一个猜想：如果弦在某个弯曲的时空中运动，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双重的(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，研究任何规范理论量都变得困难或不可能。重要的是，根据规范/弦对偶性，规范理论的这种所谓的强耦合机制与弦理论的研究是直接一致的，反之亦然。因此，如果对偶性是真的，使用弦理论，人们可以很容易地获得通过其他明显方式无法获得的物理信息。更重要的是，最近的进展表明，某些特定的规范理论(及其弦对偶)可能是精确可积的，即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成真了：现在有可能用一个单一的公式来描述“夸克-反夸克”势(尽管是在非常特殊的规范理论中)作为耦合的函数！为了验证这些猜想(模型的可积性，规范/弦对偶本身)，并尝试将全耦合分析扩展到一般的不可积例子，人们应该对这两个模型中的任何一个进行研究，以一种潜在地能够在不需要假设的情况下探索其强耦合机制的方式。一种已知的继续进行的方法是将定义模型的时空离散到格子上。从一个具有无限多个自由度的系统(与其在连续体中的原始性质相关)，该模型变成一个具有ND个自由度(其中N是一维格子的长度，d是时空维数)的统计系统，然后可以用被称为蒙特卡罗方法的数值算法来研究它。这个项目是关于在规范/弦对偶性相关的弦理论模型中使用蒙特卡罗方法。

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.