Gauge Theory and Quantum Integrability

1988年，Sklyanin写下了控制有边界量子系统可积性的方程，后来被称为反射方程。该方程在统计力学中的边界精确可解模型和二维可积QFT中的粒子离边界的因子化散射中起着核心作用。从那时起，许多解决方案的反射方程已书面下来，其相关的代数结构已被探讨。尽管如此，它的解决方案的一个简单的分类已被证明是难以捉摸的，许多关于它的结构的研究是不透明的。利用科斯特洛、维滕和山崎在从规范理论构建量子可积系统时提出的思想，我已经在许多简单情况下生成了反射方程的解，现在可以全面应用该技术。此外，这种方法使透明的起源，许多代数结构存在于量子可积系统的边界，并给他们一个自然的解释方面的配置奇异威尔逊线在一个特定的量子场论。在未来，我打算进一步探讨科斯特洛的理论在一些不同的背景。特别是我相信有一个连接这个理论和描述的经典可积系统的对称性减少的自对偶杨米尔斯方程。希望是以类似的方式，量子可积系统作为自对偶超对称杨-米尔斯QFT的对称约化而出现，其本身在扭量空间上具有自然的解释。

In 1988 Sklyanin wrote down the equation governing the integrability of quantum systems with boundary, which was subsequently referred to as the reflection equation. This equation plays a central role in exactly solvable models with boundary in statistical mechanics, and in the factorised scattering of particles off a boundary in 2 dimensional integrable QFT. Since then many solutions to the reflection equation have been written down, and its associated algebraic structures have been explored. Despite this a simple classification of its solutions has proved elusive, and much of the research concerning its structure is opaque. Using ideas developed by Costello, Witten, and Yamazaki in their construction of quantum integrable systems from gauge theory I have generated solutions to the reflection equation in a number of simple cases and am now in a position to apply the technique in full generality. Furthermore, this approach makes transparent the origin of many of the algebraic structures present in quantum integrable systems with boundary and gives them a natural interpretation in terms of configurations of exotic Wilson lines in a particular quantum field theory. In the future it is my intention to further explore Costello's theory in a number of different contexts. In particular I believe there exists a connection between this theory and the description of classical integrable systems as symmetry reductions of the self-dual Yang-Mills equations. The hope is that in an analogous way quantum integrable systems arise as symmetry reductions of the self-dual supersymmetric Yang-Mills QFT, which itself has a natural interpretation on twistor space.