Beilinson–Drinfeld chiral algebras
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| Beilinson–Drinfeld chiral algebras | |
|---|---|
| Name | Beilinson–Drinfeld chiral algebras |
| Field | Algebraic geometry, Representation theory |
| Introduced | 1990s |
| Founders | Alexander Beilinson, Vladimir Drinfeld |
Beilinson–Drinfeld chiral algebras are a foundational algebraic structure introduced by Alexander Beilinson and Vladimir Drinfeld that reformulates aspects of two-dimensional Conformal field theory and Quantum field theory in the language of algebraic geometry on curves such as Riemann surface and Algebraic curve. They provide a geometric framework linking the work of Richard Borcherds on vertex algebra and the developments of Gérard Laumon, Joseph Bernstein, and Dennis Gaitsgory on sheaf-theoretic approaches, and have influenced research at institutions like the Institut des Hautes Études Scientifiques and the Clay Mathematics Institute.
Introduction
Beilinson and Drinfeld formulated their chiral algebras in the context of the Beilinson–Bernstein localization program and the geometric Langlands program associated with Pierre Deligne and Robert Langlands, building on ideas from Edward Witten and Anton Kapustin that connect Topological quantum field theory to algebraic geometry. The theory situates chiral operations on a smooth projective Algebraic curve and relates to constructions by Igor Frenkel, James Lepowsky, and Arne Meurman in the development of physical and mathematical vertex operator algebras at centers such as Columbia University and Princeton University.
Definitions and formalism
A chiral algebra in their sense is defined as a right D-module on a smooth Algebraic curve equipped with a chiral bracket satisfying locality conditions modeled after operator product expansions used in Conformal field theory. This formalism incorporates techniques from Alexander Grothendieck's theory of D-modules, the homological methods of Jean-Louis Verdier and Luc Illusie, and categorical perspectives influenced by Maxim Kontsevich and Jacob Lurie. The axioms echo those of Richard Borcherds's vertex algebra but are expressed via factorization along diagonals in products of curves, implementing ideas reminiscent of Paul Dirac's locality and the operator formalism championed by John von Neumann.
Examples and constructions
Key examples arise from affine Kac–Moody algebras studied by Victor Kac, where one constructs chiral algebras from vacuum modules of Kac–Moody algebras at various levels, drawing on representation-theoretic input from Igor Frenkel and Nikita Nekrasov. Additional constructions originate from the sheaf of chiral differential operators developed in work connected to Alexander Polishchuk and Maxim Zabzine, and from the center at the critical level related to the Feigin–Frenkel isomorphism involving Boris Feigin and Edward Frenkel. Geometric examples include the chiral de Rham complex introduced in contexts explored by Malikov, Shechtman, and Vaintrob, linked to moduli problems studied at Harvard University and Stanford University.
Relationship to vertex algebras and factorization algebras
The relation to vertex algebra theory is tight: chiral algebras are a coordinate-independent, sheafified incarnation of vertex algebras on Riemann surface studied by Richard Borcherds and Igor Frenkel, while factorization algebras abstract locality properties in a manner paralleling work by Kevin Costello and Owen Gwilliam on perturbative Quantum field theory. Connections to the geometric Langlands correspondence draw on methods by Edward Frenkel and Dennis Gaitsgory and intersect constructions appearing in the categorical frameworks formulated by Jacob Lurie and Vladimir Drinfeld's subsequent work on higher categories.
Applications in geometry and mathematical physics
Beilinson–Drinfeld chiral algebras underpin key advances in the geometric Langlands program associated with Robert Langlands and facilitate descriptions of correlation functions in conformal and topological field theories studied by Edward Witten and Graeme Segal. They furnish algebraic tools for analyzing moduli spaces such as the moduli of G-bundles over curves investigated by Laumon and Ngô Bâc Tường (Ngô), and for understanding dualities related to Montonen–Olive duality and mirror symmetry researched by Shing-Tung Yau and Maxim Kontsevich. In mathematical physics, chiral algebras contribute to precise formulations of chiral conformal blocks appearing in works at Institute for Advanced Study and Perimeter Institute.
Technical developments and variants
Subsequent technical developments include factorization algebras on higher-dimensional manifolds by Kevin Costello and Owen Gwilliam, categorified and derived enhancements influenced by Jacob Lurie and Dennis Gaitsgory, and quantum deformations studied in relation to Drinfeld associator and Vladimir Drinfeld's quantum group theory. Variants such as chiral homology, twisted chiral algebras, and sheaves of chiral differential operators involve inputs from Pierre Deligne's work on categories, Grothendieck-style cohomology theories, and representation-theoretic structures developed by I. M. Gelfand and collaborators, with ongoing research at centers including University of Cambridge, University of Oxford, and California Institute of Technology.