Integrable models and deformations of vertex algebras via symmetric functions

数学结构或物理定律称为尺度不变量，如果它们不依赖于长度尺度，也就是说，它们通过重新缩放参数保持不变。这种现象在各种数学和物理环境中都能观察到。在数学中，分形图形成了一个最好的例子--无论一段分形线的放大倍数如何，人们总能找到一个自相似的结构。在物理学中，这一现象在统计力学中出现在所谓的相临界点。一个例子是水的临界点(374摄氏度，218倍标准大气压)。正是在这个临界点上，水的气态和液态之间不再有任何区别。在量子场论中，例如粒子物理的标准模型，当人们将注意力限制在无质量的粒子上时，例如光子(光的量子)，也会遇到尺度不变性。在大多数情况下，标度不变性是一个更大的对称性的一部分，称为共形不变性--描述物理系统的数学方程相对于保持角度而不需要保持长度的变换的不变性。用来描述这类系统的数学模型称为共形场理论。它们对数学和物理学都很感兴趣，因为它们具有惊人的对称性，这往往使它们成为极小的精确可解模型集的成员，从而使人们能够更深入地了解物理现象。人们还有兴趣了解当突然引入长度标尺时会发生什么，例如，一个获得非零静止质量的粒子--这一事件肯定发生在大爆炸后我们宇宙中的某个时候。一些物理模型保留了大量的对称性，尽管共形不变性本身消失了，因此仍然可以精确地求解。这样的模型称为可积模型。这些可积模型和保形场理论提供了更复杂的世界模型的高度非平凡的理想化。因此，他们的研究可以教给我们许多关于自然的基本属性。因此，促进对这些理论的理解不仅本身是一个有趣的数学问题，也是在理论物理和尖端数学研究之间建立进一步桥梁的机会。从长远来看，这些进展将为全面理解凝聚态物理中的普适性类以及量子场论和超弦理论中的对偶性提供关键一步。这是一个连接数学物理和纯数学的跨学科项目。主要的研究对象将是共形场理论和可由著名的称为海森堡代数或自由玻色子代数的代数构造的可积模型。虽然共形场理论和可积模型可能非常不同，但海森伯格代数的存在使它们具有许多共同的数学特征。这个项目的主要目的是将这两种类型的理论联系起来(这样一方的见解可以用来尽可能多地从另一方学习)，给出所有这样的理论的统一结构，并阐明它们的更深层次结构。

Mathematical structures or physical laws are called scale invariant, if they do not depend on length scales, that is they are left invariant by rescaling parameters. This phenomenon is observed in various mathematical and physical settings. In mathematics, fractals form a prime example - regardless of the magnification of a section of a fractal curve, one always finds a self-similar structure. In physics, the phenomenon occurs in statistical mechanics at so-called phase critical points. An example is water at its critical point (at 374 C and 218 times standard atmospheric pressure). It is at this critical point where there ceases to be any distinction between the gaseous and liquid states of water. In quantum field theory, for example the Standard Model of Particle Physics, one also encounters scale invariance, when one restricts one's attention to massless particles, such as photons (the quanta of light). In most cases scale invariance is part of a larger symmetry known as conformal invariance - invariance of the mathematical equations describing a physical system with respect to transformations which preserve angles yet need not preserve lengths. The mathematical models used to describe such systems are called conformal field theories. They are of great interest to both mathematics and physics due to their remarkable amount of symmetry, which often elevates them to membership of the exceedingly small set of exactly solvable models, thereby enabling deeper insights into physical phenomena. One is also interested in understanding what happens when a length scale is suddenly introduced, for example, by a particle acquiring nonzero rest mass - an event which must have occurred at some point in our universe after the big bang. Some physical models retain large amounts of symmetry, despite conformal invariance itself being lost, and can thus still be solved exactly. Such models are called integrable. These integrable models and conformal field theories offer highly non-trivial idealisations of more complicated models of the world. Thus their study can teach us much about the fundamental properties of nature. Advancing the understanding of such theories is thus not just an interesting mathematical problem in its own right, it is also an opportunity to build further bridges between theoretical physics and cutting-edge mathematical research. In the long run, such advances will provide a key step towards a complete understanding of universality classes in condensed matter physics, and dualities in quantum field theory and superstring theory. This is an intradisciplinary project bridging mathematical physics and pure mathematics. The main objects of study will be the conformal field theories and integrable models constructable from a famous algebra called the Heisenberg algebra or free boson algebra. Though conformal field theories and integrable models can be very different, the presence of the Heisenberg algebra leads to them sharing a number of mathematical features. The main aims of this project are to bridge these two types of theories (so that insights from one side can be used to learn as much as possible from the other side), to give a uniform construction of all such theories, and to elucidate their deeper structures.