Integrable models and deformations of vertex algebras via symmetric functions

如果数学结构或物理定律不依赖于长度标度，即通过重新标度参数使它们保持不变，则称为标度不变量。这种现象在各种数学和物理环境中都可以观察到。在数学中，分形形成了一个主要的例子-无论分形曲线的一部分放大多少，人们总是发现自相似结构。在物理学中，这种现象发生在统计力学中所谓的相临界点。一个例子是水在其临界点（在374 ℃和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）时，也会遇到尺度不变性。在大多数情况下，尺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关于保持角度但不需要保持长度的变换的不变性。用来描述这种系统的数学模型称为共形场论。它们对数学和物理都有很大的兴趣，因为它们具有显著的对称性，这通常使它们成为极小的可精确解模型集合的成员，从而使人们能够更深入地了解物理现象。人们也有兴趣理解当突然引入一个长度尺度时会发生什么，例如，一个粒子获得非零的静止质量--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已经丢失，因此仍然可以精确求解。这种模型被称为可积模型。这些可积模型和共形场论提供了对更复杂的世界模型的高度非平凡的理想化。因此，他们的研究可以教我们很多关于自然的基本性质。因此，推进对这些理论的理解不仅是一个有趣的数学问题本身，它也是一个在理论物理和前沿数学研究之间建立进一步桥梁的机会。从长远来看，这些进展将为全面理解凝聚态物理学的普适性类以及量子场论和超弦理论的对偶性迈出关键一步。这是一个跨学科的项目，连接数学物理和纯数学。主要的研究对象将是共形场论和可积模型，它们可以从著名的海森堡代数或自由玻色子代数中构造出来。虽然共形场论和可积模型可能非常不同，但海森堡代数的存在使它们具有许多数学特征。该项目的主要目的是弥合这两种类型的理论（以便从一方的见解可以用来学习尽可能多的从另一方），给出所有这些理论的统一结构，并阐明其更深层次的结构。

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.