CAREER: Conformal Symmetry in Theoretical Physics

这一职业奖是为了表彰一个集研究、教育和推广为一体的计划，重点是共形对称量子场理论(CFTS)。我们的主要目标是：1)进一步了解共形对称性如何约束CFT中的关联函数。虽然两点和三点函数的形式是由共形对称性固定的，但四点函数并不是完全确定的，通常可以展开成称为共形块的函数的基。在任何具有共形对称性的场论中，知道如何显式地执行这些展开对于形成基本的一致性条件(称为交叉对称性)是极其重要的。PI将发展保形块理论，专注于对这些分解知之甚少的情况(例如，算符与奇数维自旋和CFT的关联函数)。2)对CFT中交叉对称性和统一性(共形自举)的含义进行了详细的分析和数值研究。这些研究将对物理可见(操作符维度)产生一般严格的界限，并开发出在各种与物理相关的理论中预测它们的值的技术。通过开展这些研究，PI正在寻求绘制CFT的空间图，识别新的CFT并了解已知的CFT，对可实验实现的凝聚态系统进行定量预测，探索标准模型以外的物理概念的有效性，并测试和制定与ADS/CFT对应和量子引力相关的新理论猜想。共形不变性是尺度不变性的一种推广，它只要求相应曲线之间的角度不变。共形场理论(CFT)是理论物理的核心，它将物理系内的所有物理领域以及国家科学基金会其他系中的其他项目联系起来。它们在量子场论的空间中无处不在；它们全息描述(将一个体积中的属性与该体积表面编码的信息联系起来)量子引力的一致理论；它们在各种现实世界的凝聚态系统中实现，例如通过相邻原子自旋之间的耦合实现铁磁性的临界三维伊辛模型；它们甚至可能在标准模型之外的物理学中发挥重要作用，标准模型的概念是用夸克、轻子和规范场的形式准确描述已知基本粒子。物理上相关的CFT有很大的种类，其中不能使用传统的场论技术，如微扰理论和大N展开式。递归共形自举方法目前还不发达，似乎是在这些原本难以处理的理论中对物理可观测进行定量预测的一种非常有前途的方法。更广泛的影响：该项目的教育和推广部分将向所有教育水平的学生介绍物理学中对称性的力量。1)国际学生联合会正在开发一系列实践演示，说明物理学中的对称性原则，通过耶鲁大学的《数学晨报》系列讲座和国际学生联合会主办的“理论物理日”向种族多元化和高度贫困的学区的初中生和高中生提供这些演示。2)国际物理学会正在筹办一场关于“物理学中的对称性”的公开讲座，将在多个地点举行。3)PI正在为耶鲁大学的路径科学合作实践学习和研究(SCARTER)项目开发和教授两个不同的3天模块，一个侧重于空间和时间的对称性，另一个侧重于探索尺度不变性和额外维度。4)PI积极让耶鲁大学本科生参与他的研究，并让他们参与外展活动。5)PI正在开发一门新的研究生级别的课程，内容是“共形场论”，向耶鲁大学的学生介绍现代研究主题，给他们机会发展他们的口头陈述技能，并制作一套将公开传播的讲稿。

This CAREER award is for an integrated program of research, education, and outreach focusing on quantum field theories with conformal symmetry (CFTs). The primary goals are: 1) To further our knowledge of how conformal symmetry constrains correlation functions in CFTs. While the forms of 2-point and 3-point functions are fixed by conformal symmetry, 4-point functions are not completely determined, and can in general be expanded into a basis of functions called conformal blocks. Knowing how to perform these expansions explicitly is extremely important for formulating basic consistency conditions (called crossing symmetry) in any field theory with conformal symmetry. The PI will develop the theory of conformal blocks, focusing on situations (e.g., correlation functions of operators with spin and CFTs in odd dimensions) where these decompositions are poorly understood. 2) To perform detailed analytical and numerical studies of the implications of crossing symmetry and unitarity (the conformal bootstrap) in CFTs. These studies will produce general rigorous bounds on physical observables (operator dimensions) and develop techniques to predict their values in a wide variety of physically relevant theories. By pursuing these studies, the PI is seeking to map out the space of CFTs, identify new CFTs and learn about known ones, make quantitative predictions for experimentally realizable condensed matter systems, explore the validity of ideas in physics beyond the standard model, and test and formulate new theoretical conjectures related to the AdS/CFT correspondence and quantum gravity. Conformal invariance is a generalization of scale invariance that only requires that the angles between corresponding curves do not change. Conformal field theories (CFT) lie at the heart of theoretical physics, linking all the fields of physics represented within the Physics Division as well as other programs in other Divisions in NSF. They are ubiquitous in the space of quantum field theories; they holographically describe (connect properties in a volume to the information encoded in the surface of that volume) consistent theories of quantum gravity; they are realized in a wide variety of real-world condensed matter systems such as the critical three-dimensional Ising model of ferromagnetism in terms of the coupling between spins of neighboring atoms; and they may even play an important role in physics beyond the Standard Model, the concept that accurately describes the known elementary particles in terms of quarks, leptons and gauge fields. There are wide classes of physically relevant CFTs where conventional field theory techniques such as perturbation theory and large-N expansions cannot be used. The recursive conformal bootstrap approach is currently underdeveloped and appears to be an extremely promising way of making quantitative predictions for physical observables in these otherwise intractable theories.Broader Impacts : The education and outreach component of this program will introduce students at all education levels to the power of symmetry in physics. 1) The PI is developing a series of hands-on demonstrations illustrating principles of symmetry in physics, being made available to middle and high school students in racially diverse and high poverty school districts through Yale's "Math Mornings" lecture series and during a "Theoretical Physics Day" hosted by the PI. 2) The PI is developing a public lecture on "Symmetries in Physics" to be given in multiple venues. 3) The PI is developing and teaching two distinct 3-day Modules for Yale's Pathways Science Collaborative Hands-On Learning and Research (SCHOLAR) program, one focusing on symmetries of space and time and one focusing on exploring scale invariance and extra dimensions. 4) The PI is actively involving Yale undergraduates in his research as well as including their participation in outreach activities. 5) The PI is developing a new graduate level course on "Conformal Field Theories", introducing Yale students to modern research topics, giving them opportunities to develop their oral presentation skills, and producing a set of lecture notes that will be disseminated publicly.