Integrable models and deformations of vertex algebras via symmetric functions

数学结构或物理定律被称为尺度不变性，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而保持不变。这种现象可以在各种数学和物理环境中观察到。在数学中，分形形成了一个最好的例子——不管分形曲线的一部分被放大，人们总能找到一个自相似的结构。在物理学中，这种现象发生在统计力学中所谓的相临界点。水的临界点（374摄氏度，218倍标准大气压）就是一个例子。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，例如粒子物理学的标准模型，当人们将注意力限制在无质量的粒子，如光子（光的量子）上时，人们也会遇到尺度不变性。在大多数情况下，尺度不变性是称为保形不变性的更大对称性的一部分，保形不变性是描述物理系统的数学方程的不变性，它与保持角度而不需要保持长度的变换有关。用来描述这种系统的数学模型称为共形场论。数学和物理学都对它们非常感兴趣，因为它们具有显著的对称性，这往往使它们成为极小的精确可解模型集的成员，从而能够更深入地了解物理现象。人们也有兴趣了解当一个长度尺度突然被引入时会发生什么，例如，一个粒子获得非零静止质量——这一事件一定发生在大爆炸之后的宇宙中的某个时刻。一些物理模型保留了大量的对称性，尽管保形不变性本身丢失了，因此仍然可以精确地解决。这样的模型称为可积模型。这些可积模型和共形场论为更复杂的世界模型提供了高度非平凡的理想化。因此，他们的研究可以教会我们很多关于自然的基本属性。因此，推进对这些理论的理解不仅是一个有趣的数学问题，也是一个在理论物理学和前沿数学研究之间建立进一步桥梁的机会。从长远来看，这些进展将为完全理解凝聚态物理中的普适类、量子场论和超弦理论中的对偶性提供关键的一步。这是一个连接数学物理和纯数学的跨学科项目。研究的主要对象将是共形场论和可积模型，可由著名的代数海森堡代数或自由玻色子代数构造。虽然共形场论和可积模型可能是非常不同的，但海森堡代数的存在导致它们共享许多数学特征。这个项目的主要目的是连接这两种类型的理论（这样来自一方的见解可以用来尽可能多地从另一方学习），给所有这些理论一个统一的结构，并阐明它们的更深层次的结构。

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.