Algebraic Properties of Superconformal Field Theories

该项目属于EPSRC数学物理研究领域，共形场论（conformal Field Theories, CFTs）是量子场论（Quantum Field Theories, qft）的一个重要类别，其时空对称性得到了极大的增强。保持超对称的共形场论享有更大的对称群，被称为超共形理论（SCFTs）。虽然qft在一系列应用中取得了巨大的成功，但它们的分析往往受到依赖摄动方法的限制，这些方法在强耦合的情况下失败。然而，cft和scft的代数性质允许使用强大的非微扰方法。具体来说，共形场理论具有算子积展开（OPE），它是局部算子空间中的一种代数运算。OPE允许将局部算子的乘积扩展为主算子及其子算子的幂级数。为了保证相关函数的定义良好，OPE代数必须是相关联的。将这种结合性要求作为CFT的一个关键特征，导致了共形引导（Conformal Bootstrap）的概念，它试图约束定义有效CFT的数据。在二维cft中，由于存在无限的手性子代数，OPE代数受到高度约束，其余的OPE代数是根据这些子代数组织起来的。在这里可以找到丰富的代数结构，如Virasoro和Kac-Moody顶点算子代数。在[1]中，证明了这些手性代数也在四维N=2超共形场论中出现，这导致了四维物理中一系列重要和令人惊讶的结果的发现，如新的中心电荷边界和与模形式的连接。OPE代数的力量也允许人们分析可能没有拉格朗日描述的理论。例如，有S类理论，它是在黎曼曲面上紧化的六维超共形理论的低能量极限。除了少数例外，这些理论都不能用拉格朗日场论来描述。在[2]中，利用[1]的手性代数对应关系研究了S类理论空间的结构。通过将S类理论的手性代数与其UV曲线（二维流形）联系起来，可以使用广义拓扑量子场论的方法来捕获S类理论的一些结构。这一方法随后在b[3]中被用于严格定义S类理论的手性代数。这个项目的目的是利用S类手性代数中存在的新的数学结构来阐明它们的结构，并扩展迄今为止已经执行的这些模型的分析。一些具体的目标包括澄清Arakawa对非单列李代数结果的物理解释，将这些结果推广到涉及一般扭曲刺的S类理论的情况。这些目标处于超共形场理论和顶点算子代数之间新兴界面的前沿。

This project falls within the EPSRC Mathematical Physics Research AreaConformal Field Theories (CFTs) are an important class of Quantum Field Theories (QFTs) that enjoy a large enhancement of their spacetime symmetries. Conformal field theories which preserve supersymmetry enjoy an even larger group of symmetries and are known as Superconformal Theories (SCFTs). While QFTs have seen great success in a range of applications, their analysis is often limited by a reliance on perturbative methods, which fail in the regime of strong coupling. The algebraic properties of CFTs and SCFTs, however, allow for the use of powerful non-perturbative methods.Specifically, conformal field theories are equipped with an Operator Product Expansion (OPE), which is an algebraic operation in the space of local operators. The OPE allows a product of local operators to be expanded in a power series of primary operators and their descendants. To ensure well-definedness of correlation functions, the OPE algebra must be associative. Enshrining this requirement of associativity as a key feature of a CFT has led to the idea of the Conformal Bootstrap, which seeks to constrain the data that defines a valid CFT.In two-dimensional CFTs, the OPE algebra is highly constrained by the presence of infinite chiral subalgebras in terms of which the rest of the OPE algebra is organized. Here one finds rich algebraic structures such as Virasoro and Kac-Moody vertex operator algebras. In [1], it was shown that such chiral algebras also arise cohomologically in four-dimensional N=2 superconformal field theories, which has led to the discovery of a bevy of important and surprising consequences for the four-dimensional physics such as novel central charge bounds and connections to modular forms.The power of the OPE algebra also allows one to analyse theories which may not have Lagrangian descriptions. For instance, there are the theories of Class S, which are the low-energy limit of a six dimensional superconformal theory compactified on a Riemann Surface. With a few exceptions, these theories are not described by a Lagrangian field theory. In [2], the chiral algebra correspondence of [1] was used to investigate the structure of the space of Class S theories. By associating the chiral algebra of a Class S theory with its UV curve (a two-dimensional manifold), it is possible to use the methods of generalized topological quantum field theory to capture some of the structure of Class S theories. This approach was subsequently used to great effect to define rigorously the chiral algebras of Class S theories in [3].The aim of this project will be to exploit the novel mathematical structures present in chiral algebras of Class S to elucidate their structure and extend the analysis of these models that has been performed to date. Some concrete goals include a clarification of the physical interpretation of the results of Arakawa for non-simply laced Lie algebras, a generalization of those results to the case of class S theories involving general twisted punctures. These targets are at the cutting edge of the emerging interface between superconformal field theory and vertex operator algebras.