Vertex Algebras in Geometry and Physics

几何和物理中的顶点代数

• 批准号: SAPIN-2020-00039
• 负责人: Creutzig, Thomas
• 金额: $ 3.79万
• 依托单位: University of Alberta
• 依托单位国家: 加拿大
• 项目类别: Subatomic Physics Envelope - Individual
• 财政年份: 2022
• 资助国家: 加拿大
• 起止时间: 2022-01-01 至 2023-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=758336
• 关键词: Vertex; Algebras; Geometry; Physics

A vertex operator algebra (VOA) or chiral algebra is the infinite dimensional symmetry algebra of a two dimensional conformal field theory (CFT). Historically these CFTs have appeared as worldsheet theories of strings and also in the description of two-dimensional critical phenomena. Over the last years the importance of VOAs themselves in the context of higher dimensional supersymmetric gauge theories, quantum gravity and string theories has been realized. For example protected sectors of 4-dimensional N=2 superconformal field theories are described by VOAs and so a current theme is to study important aspects of VOA theory in order to gain a better understanding of higher dimensional theories: I want to solve various important problems in the theory of VOAs that are inspired by physics and use these insights for a better understanding of corresponding physical theories. 1) Representation categories of VOAs appear in 4-dimensional supersymmetric theories as categories of line defects ending on three dimensional topological boundary conditions while the VOAs themselves are attached to the two-dimensional intersection (corner) of boundary conditions. These corner VOAs turn out to be very useful to my research program and so in the first place I want to find good constructions of them. The corner VOAs allow for a large coupling limit in which the VOA degenerates to the semi direct product of a VOA and a Poisson vertex algebra. These limits are closely related to the best known logarithmic CFTs. Jointly with Dimofte and Geer we will use corresponding quantum groups to construct three dimensional topological field theories coupled to flat connections. These will give new and interesting invariants of 3-manifolds and links.  2) The AGT correspondence relates correlation functions of W-algebras to partition functions of N=2 4-dimensional super Yang-Mills theory. In a limit these correlation functions become norms of Whittaker vectors which satisfy the celebrated Nakajima-Yoshioka blowup equations. Jointly with Arakawa and Feigin a magical property of the quantum Hamiltonian reduction functor will be proven. As a corollary certain algebraic blowup equations follow and these can be easily specialized to the geometric blowup equations. However our results will be much more general and so they should lead to new exciting insights.  3) WZW theories of supergroups and their underlying affine superVOAs appear as special corner VOAs in S-duality, are key ingredients in the AdS/CFT correspondence and provide rich examples of logarithmic CFTs. My aim is to develop Wakimoto free field realizations for them and to study many examples that appear in the context of S-duality. Moreover the explicit coset construction of the small and large N=4 superconformal algebras of Feigin, Linshaw and myself will be used to study the representation theory of the superconformal algebras. The main eventual goal is to use these insights for the AdS_3/CFT_2 correspondence.

顶点算子代数（VOA）或手征代数是二维共形场论（CFT）的无穷维对称代数。从历史上看，这些CFTs曾作为弦的世界面理论出现，也出现在二维临界现象的描述中。在过去的几年中，VOA本身在高维超对称规范理论、量子引力和弦理论中的重要性已经被认识到。例如，四维N=2超共形场论的保护扇区由VOA描述，因此当前的主题是研究VOA理论的重要方面，以便更好地理解高维理论：我想解决VOA理论中受到物理启发的各种重要问题，并使用这些见解更好地理解相应的物理理论。 1)在四维超对称理论中，VOA的表示类别是以三维拓扑边界条件为终点的线缺陷类别，而VOA本身则附着在边界条件的二维交点（角）上。这些角VOA对我的研究计划非常有用，所以首先我想找到它们的好结构。角VOA允许一个大的耦合限制，其中的VOA退化到一个VOA和泊松顶点代数的半直积。这些限制与最著名的对数CFTs密切相关。与Dimofte和Geer一起，我们将使用相应的量子群来构建耦合到平坦连接的三维拓扑场论。2）AGT对应将W-代数的相关函数与N=2的四维超Yang-Mills理论的配分函数联系起来。在极限下，这些相关函数成为满足著名的Nakajima-Yoshioka爆破方程的Whittaker向量的范数。与荒川和费金一起，量子哈密顿约化函子的一个神奇的性质将被证明。作为推论，某些代数爆破方程如下，这些可以很容易地专门的几何爆破方程。3）超群的WZW理论及其基础的仿射超VOA在S-对偶中表现为特殊的角VOA，是AdS/CFT对应的关键成分，并提供了对数CFTs的丰富例子。我的目标是为他们开发Wakimoto自由场实现，并研究在S-对偶的背景下出现的许多例子。此外，Feigin，Linshaw和我自己的小和大N=4超共形代数的显式陪集构造将用于研究超共形代数的表示理论。主要的最终目标是将这些见解用于AdS_3/CFT_2对应。