Vertex algebras and Hodge structures

TL;DR

This paper explores the connections between Hodge structures, vertex algebras, and geometric Langlands correspondence, providing an expository overview with new insights into their interrelations.

Contribution

It offers a comparative analysis of Hodge structures and vertex algebras, introducing generalized Harish-Chandra modules and homomorphisms, and discusses their role in geometric Langlands correspondence.

Findings

Identifies parallels between Hodge structures and vertex algebras.

Introduces generalized Harish-Chandra modules and homomorphisms.

Provides new insights into the geometric Langlands correspondence.

Abstract

We compare the context of Hodge structures with that of vertex algebras of conformal field theory. Vertex algebras appear as the highest weight representations of infinite dimensional Lie algebras. A correspondence between Higgs bundles and opers already is known as non-abelian Hodge theorem due to C. Simpson. The Beilinson-Bernstein localization (correspondence) also compares the context of variation of Hodge structure with that of highest weight modules over flag manifolds of semisimple Lie groups. A more general analogue of the Bernstein correspondence over a local manifold can also be formulted in the context of geometric Langlands correspondence. We discuss a generalized version of Harish-Chandra modules called Wakimoto modules and a generalized Harish-Chandra homomorphism. This text is mainly an expository discussion with a new insight toward the two concepts. We conclude with an explanation of geometric Langlands correspondence.