CAREER: Factorization Algebras in Quantum Field Theory

几个世纪以来，数学和物理一直保持着持续的、不断扩大的对话，通过交换问题和见解丰富了这两门学科。在过去的几十年里，这种对话导致了一次卓有成效的交流，中心是高等代数(如范畴理论和同伦代数)在量子场论(QFT)中的作用。它改变和扩大了我们对量子力学的看法，因此它不仅在粒子理论的更深层次，而且在凝聚态物理的具体发现中，特别是在物质的拓扑态和相态方面，都发挥着积极的作用。这种交换的一个最新创新是因式分解代数，它在物理学中自然出现，作为场论(经典和量子)的观测值，但最初出现在数学中。这个项目探索了这个工具在四维规范理论设置中的力量。为了实现这项研究的目标，首席研究员将与数学家和物理学家合作。这种跨学科的努力是这一研究领域的一个鼓舞人心和鼓舞人心的方面，但有效的沟通往往是困难的，因为尽管数学和物理有着长期的关系，但每个社区都有自己的特权和话语模式。因此，该项目的一个关键组成部分是为各级研究人员提供机会，使他们能够流利地与这两个学科对话，此外，还可以建立直接的个人桥梁。在研究生和博士后层面，该项目将为数学家和理论物理学家举办年度暑期班，重点关注共同感兴趣的主题。此外，在每个学年，它还将推出高质量的在线大师班，由专家就这类主题进行讨论，并提供课堂讲稿和练习。最后，该项目将支持来自马萨诸塞大学和附近五所大学的本科生的暑期研究，解决数学和物理之间的问题。更详细地说，这个项目围绕两个焦点运行。第一个是研究复杂曲面上的全纯理论，特别是与全纯G-丛的模有关的理论。PI将寻求仿射李代数、丛的模和手征CFT之间的关系的类比，这涉及黎曼曲面上的全纯理论。长期目标是Seiberg对偶的全纯版本(源于N=1的超对称规范理论)，它与镜像对称有着自然的相似之处。第一个项目使用了一个由PI与合作者共同开发的工具包，用于构建全纯场论的BV量子化，然后分析它们的可观测因式分解代数。第二个焦点涉及具有边界和角的流形上的拓扑理论，其中长期目标是关于顶点代数和带状范畴如何从Kapustin-Witten规范理论产生的Frenkel-Gaiotto猜想的严格证明，Kapustin-Witten规范理论在几何Langland对应的物理方法中扮演着关键角色。第二个项目将使用全局方法(通过派生的代数几何，开发具有边界和角的流形的AKSZ程序，并由此得到更高的范畴形变量化)和微扰方法(建立在有边界流形上的场论的BV/因式分解包的扩展基础上，从而产生可观测的分层因式分解代数)。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

Mathematics and physics have sustained an ongoing, ever-expanding dialogue for centuries, enriching both subjects by the swapping of problems and insights. In the last few decades, this conversation has led to a fruitful exchange centered on the role of higher algebra (such as category theory and homotopical algebra) in quantum field theory (QFT). It has transformed and enlarged our view of QFT, and thus it plays an active role not only in the farther reaches of particle theory, but also in the concrete discoveries of condensed matter physics, notably in topological states and phases of matter. A recent innovation from this exchange is factorization algebras, which appear naturally in physics as the observables of field theories (both classical and quantum) but originally appeared in mathematics. This project explores the power of this tool in the setting of 4-dimensional gauge theories. To pursue goals of this research, the Principal Investigator will collaborate with both mathematicians and physicists. This kind of interdisciplinary effort is an inspiring and motivating aspect of this area of research, but effective communication is often difficult because, despite the long-running relationship of mathematics and physics, each community has its own prerogatives and modes of discourse. A key component of this project is thus to offer chances for researchers at all levels to become fluent in speaking to both disciplines and, moreover, to build direct personal bridges. At the graduate and postdoctoral level, the project will run annual summer schools for both mathematicians and theoretical physicists, focused on topics of mutual interest. In addition, each academic year, it will produce high-quality, online masterclasses by experts about such topics, with lecture notes and exercises. Finally, the project will support summer research for undergraduates, tackling problems between mathematics and physics, from the University of Massachusetts and nearby Five Colleges.In more detail, this project orbits around two foci. The first is the study of holomorphic theories on complex surfaces, particularly those related to moduli of holomorphic G-bundles. The PI will pursue analogs of relationships between affine Lie algebras, moduli of bundles, and chiral CFT, which involve holomorphic theories on Riemann surfaces. The long-term target is a holomorphic version of Seiberg duality (arising from N = 1 supersymmetric gauge theories), which bears natural analogies to mirror symmetry. This first project uses a toolkit, developed by the PI with collaborators, for constructing BV quantization of holomorphic field theories and then analyzing their factorization algebras of observables. The second focus involves topological theories on manifolds with boundaries and corners, where the long-term target is a rigorous demonstration of the Frenkel-Gaiotto conjecture about how vertex algebras and ribbon categories arise from the Kapustin-Witten gauge theories, which play a key role in the physical approach to the geometric Langlands correspondence. This second project will approach the conjecture using both global methods (by developing, via derived algebraic geometry, the AKSZ procedure with manifolds with boundaries and corners, and thence a higher categorical deformation quantization) and perturbative methods (building upon an extension of the BV/factorization package for field theories on manifolds with boundary, which yields stratified factorization algebras of observables).This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.