Vertex Algebras in Geometry and Physics

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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.