Vertex algebras and Hodge structures (Beilinson-Bernstein correspondence)

Mohammad Reza Rahmati and Zacarias Malacara Hernandez
¹Centro de Investigaciones en Óptica, A. C., Loma del Bosque 115,
Lomas del Campestre, 37150, León, Guanajuato, México

Dedicated to Prof Daniel Malacara-Hernández

This article surveys the connections between Hodge structures and vertex algebras in conformal field theory and is expository text. These structures appear in various contexts of theoretical and mathematical physics. Hodge structure is a generalized complex structure which appears on solution space of integrable systems. Vertex algebras, as highest weight representations of infinite-dimensional Lie algebras, provide a frame work that intersects with both mathematical and physical theories. We explain the connection between Hodge structures and theory of highest weight modules over affine algebras by a generalization of the Beilinson-Bernstein correspondence. The Beilinson-Bernstein localization draws parallels between the variation of Hodge structures and highest weight modules over flag manifolds of semisimple Lie groups. This framework has profound implications for quantum field theory, where the structure of vertex algebras influences the understanding of symmetries and interactions in conformal field theories. We also consider a broader version of the Bernstein correspondence within the context of the geometric Langlands correspondence over local manifolds. Vertex algebras and representations of a fine algebras provide a powerful frame work for describing symmetry and state evolution in physics, optics, and information technology. They under-pin conformal field theories in quantum physics, model photon correlations, entanglement in quantum optics, and enhance quantum information processing. These structures offer a unified approach to analyzing complex systems across these fields. © Anita Publications. All rights reserved.
Doi: 10.54955/AJP.33.12.2024.843-864
Keywords: Vertex operator algebras, Hodgestructure, Highest weight modules, D-modules, Beilinson-Bernstein correspondence, Geometric Langlands correspondence.

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