New Algebraic Structures inspired by Gauge/Gravity Dualities

在过去的二十年里，数学和物理相互影响，变得高度纠缠在一起。数学物理总是产生各种各样的新概念和新问题，成为纯数学研究的重要课题。规范、引力和弦理论的发展使这些学科之间的联系比以往任何时候都更加紧密。一个重要的驱动力是量子群和规范/引力对偶的发现。在这里，起主导作用的是所谓的ADS/CFT对偶性及其基本的可积结构。一个影响深远的概念是边界及其相应的边界条件的影响。它们在几乎所有数学物理模型中都是不可避免的，并且具有根本性的重要性。在量子群理论中引入边界导致了一类全新的所谓反射代数。这样的代数被证明出现在许多数学模型中，并处于它们的可积结构的核心。此外，这些代数也被证明在ADS/CFT中起着重要的作用。然而，描述这类代数的连贯框架尚不清楚，反射代数的许多性质仍然是一个悬而未决的问题。本研究的目的是在ADS/CFT的可积结构的启发下，发展新的代数方法和代数公理理论与量子群理论之间的学科联系，特别是通过揭示边界和不同边界构型的影响。这项研究是通过应用诸如量子仿射和杨氏代数等代数对象来找到优雅的、精确的解来描述由规范/引力对偶产生并受到其启发的模型。

During the last twenty years mathematics and physics have significantly influenced each other and became highly entangled. Mathematical physics was always producing a wide variety of new concepts and problems that became important subjects of the pure mathematical research. The growth of gauge, gravity and string theories have made the relation between these subjects closer than ever before. An important driving force was the discovery of quantum groups and of the gauge/gravity dualities. Here the leading role was played by the the so-called AdS/CFT duality and the underlying integrable structure of it.A far-reaching concept is the effect of boundaries and the corresponding boundary conditions. They are unavoidable in almost all models of mathematical physics and are of the fundamental importance. The introduction of boundaries into the theory of quantum groups leads to a whole new class of the so-called reflection algebras. Such algebras were shown to appear in numerous mathematical models and are at the core of the integrable structure of them. Furthermore, these algebras were also shown to play a prominent role in the AdS/CFT. However a coherent framework for describing such algebras is not known, and many properties of the reflection algebras are still an open question.The goal of this research is to develop new algebraic methods and intradisciplinary connections between the axiomatic theory of algebras and the theory of quantum groups inspired by the integrable structure of the AdS/CFT, in particular by shedding more light on the effects of boundaries and different boundary configurations. The research is driven by applying algebraic objects such as the quantum affine and Yangian algebras to find elegant, exact solutions describing the models that arise from and are inspired by the gauge/gravity dualities.