Vertex algebras and Hodge structures
In this short note we discuss some natural inter-relations between Hodge structures and vertex algebras of conformal field theory. Some part of this on a correspondence between Higgs bundles and opers already is known in the literature as non-abelian Hodge theorem due to C. Simpson. The same kind of correspondence has been well studied over flag manifolds of semisimple Lie groups known as Beilinson-Bernstein localization. Our goal is to explain how the data of a variation of Hodge structure as a solution of a regular holonomic system is matched with similar system of vertex algebra modules arising in conformal field theory. The result of the discussion is an analogue of the Bernstein correspondence over a local manifold. We associate to flat connections of mixed Hodge structures a generalized version of Harish-Chandra modules called Wakimoto modules and a generalized Harish-Chandra homomorphism. Therefore the map of correspondence is a more developed form of Harish-Chandra isomorphism. This text mainly proposes to motivates some ideas of representations of vertex algebras into Hodge theory. We have brought the basic ideas in the two fields close to each other. We enclose with an explanation of geometric Langlands correspondence as a generalization of the discussion.