CAREER: Conformal Symmetry in Theoretical Physics

这个职业奖是一个综合的研究，教育和推广计划，重点是量子场论与共形对称性（CFTs）。主要目标是：1）进一步了解共形对称性如何约束CFTs中的相关函数。虽然2点和3点函数的形式是由共形对称固定的，但4点函数并不完全确定，并且通常可以扩展为称为共形块的函数的基。知道如何显式地执行这些展开对于在任何具有共形对称性的场论中制定基本的一致性条件（称为交叉对称性）是极其重要的。PI将开发共形块的理论，重点放在情况（例如，具有奇数维自旋和CFTs的算子的相关函数），其中这些分解知之甚少。2)详细分析和数值研究交叉对称性和么正性（共形引导）在CFTs中的含义。这些研究将产生一般严格的物理观测值（运营商尺寸）的界限，并开发技术来预测其值在各种各样的物理相关的理论。通过进行这些研究，PI正在寻求绘制出CFTs的空间，识别新的CFTs并了解已知的CFTs，对实验可实现的凝聚态系统进行定量预测，探索标准模型之外的物理学思想的有效性，并测试和制定与AdS/CFT对应和量子引力相关的新理论体系。共形不变性是尺度不变性的推广，它只要求相应曲线之间的角度不改变。 共形场论（CFT）位于理论物理的核心，连接物理学部门内代表的所有物理学领域以及NSF其他部门的其他计划。 它们在量子场论的空间中无处不在;它们全息地描述了（将体积中的属性与该体积表面中编码的信息连接起来）量子引力的一致性理论;它们在各种各样的现实世界凝聚态系统中实现，例如铁磁性的临界三维伊辛模型，根据相邻原子的自旋之间的耦合;它们甚至可能在标准模型之外的物理学中发挥重要作用，标准模型是用夸克、轻子和规范场精确描述已知基本粒子的概念。有广泛的物理相关的CFTs类传统的场论技术，如微扰理论和大N展开不能使用。递归共形引导方法是目前欠发达的，似乎是一个非常有前途的方式，使定量预测的物理观测量，否则棘手的theors.Broader影响：教育和推广的组成部分，这个计划将介绍学生在所有教育水平的对称性在物理学中的力量。1)PI正在开发一系列实践演示，说明物理学中的对称性原理，通过耶鲁大学的“数学早晨”系列讲座和PI主办的“理论物理日”，向种族多样化和高度贫困学区的初中和高中学生提供。2)PI正在开发一个关于“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Pathways Science Collaborative Hands-On Learning and Research（SCHOLAR）计划开发和教授两个不同的为期3天的模块，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积极地让耶鲁大学的本科生参与他的研究，并让他们参与外展活动。5)PI正在开发一个新的研究生课程“共形场论”，介绍耶鲁大学的学生现代研究课题，让他们有机会发展自己的口头表达技巧，并制作一套将公开传播的讲义。

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.