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Algebraic Properties of Superconformal Field Theories

超共形场论的代数性质

基本信息

批准号：
2272671
负责人：
金额：
--
依托单位：
University of Oxford
依托单位国家：
英国
项目类别：
Studentship
财政年份：
2019
资助国家：
英国
起止时间：
2019 至无数据
项目状态：
已结题

来源：
https://gtr.ukri.org/projects?ref=studentship-2272671
关键词：
Algebraic Properties Superconformal Field Theories

项目摘要

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.

共形场论（Conformal Field Theories，简称CFTs）是量子场论（Quantum Field Theories，简称QFT）的一个重要分支，它极大地增强了时空对称性。保形场论保留了超对称性，并享有更大的对称群，被称为超保形理论（Superconformal Theories，SCFT）。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微扰方法的限制，而微扰方法在强耦合的情况下会失败。然而，CFT和SCFT的代数性质允许使用强大的非微扰方法。具体地说，共形场论配备了算子乘积展开（OPE），这是局部算子空间中的代数运算。OPE允许本地算子的乘积在主算子及其后代的幂级数中扩展。为了保证相关函数的良好定义性，OPE代数必须是结合的。将结合性的要求作为CFT的关键特征，导致了共形引导的想法，其试图约束定义有效CFT的数据。在二维CFT中，OPE代数受到无限手征子代数的高度约束，OPE代数的其余部分是根据这些手征子代数组织的。在这里，人们发现丰富的代数结构，如Virasoro和Kac-Moody顶点算子代数。在[1]中，证明了这样的手征代数在四维N=2超共形场论中也是上同调的，这导致了对四维物理的一系列重要和令人惊讶的结果的发现，如新的中心电荷界和与模形式的联系。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，这是在黎曼曲面上紧致化的六维超共形理论的低能极限。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]中的手征代数对应研究了S类理论空间的结构。通过将S类理论的手征代数与其UV曲线（二维流形）相关联，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。在[3]中，这种方法被用于严格定义S类理论的手征代数。本项目的目的是利用S类手征代数中存在的新的数学结构来阐明它们的结构，并扩展迄今为止对这些模型的分析。一些具体的目标包括一个澄清的物理解释的结果荒川为非简单的花边李代数，推广这些结果的情况下，S类理论涉及一般扭曲穿刺。这些目标是在超共形场论和顶点算子代数之间的新兴接口的最前沿。