Vertex Algebras in Geometry and Physics

顶点算符代数(VOA)或手征代数是二维共形场论(CFT)的无限维对称代数。从历史上看，这些CFT作为弦的世界表理论出现，也出现在对二维临界现象的描述中。在过去的几年里，VOAs本身在高维超对称规范理论、量子引力和弦理论的背景下的重要性已经被认识到。例如，VOAS描述了4维N=2超共形场论的受保护扇区，因此当前的一个主题是研究VOA理论的重要方面，以便更好地理解更高维的理论： 我想解决VOAS理论中受到物理学启发的各种重要问题，并利用这些见解更好地理解相应的物理理论。 1)VOAs的表示范畴在四维超对称理论中表现为以三维拓扑边界条件结束的线缺陷范畴，而VOAs本身依附于边界条件的二维交点(角点)。这些拐角的VOA对我的研究项目非常有用，所以首先我想找到它们的好结构。角VOA允许较大的耦合极限，其中VOA退化为VOA和泊松顶点代数的半直积。这些极限与最著名的对数CFT密切相关。我们将与Dimofte和Geer一起使用相应的量子群来构造耦合到平面连接的三维拓扑场理论。这将给出三维流形和链环的新的有趣的不变量。 2)AGT对应将W-代数的关联函数与N=2 4维超Yang-Mills理论的配分函数联系起来。在一个极限内，这些关联函数成为满足著名的中岛-吉冈爆破方程的惠特克向量的范数。我们将与Arakawa和Feigin一起证明量子哈密顿约化函子的一个神奇性质。作为推论，随后出现了某些代数爆破方程，这些方程可以很容易地专门化为几何爆破方程。然而，我们的结果将更加普遍，因此它们应该会带来新的令人兴奋的见解。 3)WZW超群理论及其下的仿射超VOA在S对偶中表现为特殊的角VOA，是ADS/CFT对应的关键成分，并提供了丰富的对数CFT例子。我的目的是为他们发展Wakimoto自由场实现，并研究出现在S对偶背景下的许多例子。此外，还将利用Feigin、Linshaw和我的N=4超共形代数的小的和大的N=4的显式陪集构造来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通信。

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.