Edward Frenkel's Research on the Langlands Program
Bridging representation theory, algebraic geometry, and mathematical physics
Edward Frenkel has published significant research on the Langlands program, with particular focus on the geometric Langlands correspondence. His work establishes deep connections between representation theory, algebraic geometry, and mathematical physics, often emphasizing the role of Lie groups and algebras in unifying these disparate mathematical domains.
Geometric Langlands Program
Frenkel's most influential contributions lie in the geometric Langlands correspondence, a geometric reformulation of the classical Langlands program. This framework relates holomorphic D-modules on moduli stacks of G-bundles on a Riemann surface to sheaves on the moduli stack of Langlands dual group ^L G-local systems.
In collaboration with Dennis Gaitsgory and Kari Vilonen, Frenkel helped formalize key aspects of the geometric Langlands conjecture, establishing foundational results for GL(n) and other reductive groups. Their work provided rigorous mathematical underpinnings for this deep correspondence between algebraic geometry and representation theory.
Affine Lie Algebras and Vertex Algebras
Frenkel has extensively studied affine Kac-Moody algebras and vertex operator algebras, which emerge naturally in the geometric Langlands context. His research explores how representation theory of infinite-dimensional Lie algebras interfaces with the Langlands program.
His influential book "Vertex Algebras and Algebraic Curves," co-authored with David Ben-Zvi, has become a standard reference that systematically links vertex algebras to the geometric Langlands program, providing essential tools for researchers in this interdisciplinary field.
Opers and Gaudin Models
Frenkel introduced the concept of G-opers, a generalization of differential operators that serves as a bridge between differential equations and Langlands dual groups. This framework has proven instrumental in understanding the geometric Langlands correspondence.
His investigations into integrable systems, particularly the Gaudin model, revealed how spectral curves of Gaudin Hamiltonians correspond to opers. This connection has opened new pathways for understanding the relationship between integrable systems and the Langlands program.
Quantum Langlands Duality
Frenkel has made substantial contributions to q-deformations of the Langlands correspondence, exploring connections to quantum groups and quantum geometric Langlands. This work frequently intersects with topological field theory and mirror symmetry.
His research in this area demonstrates how quantization procedures can extend and enrich the classical Langlands correspondence, revealing deeper structures that connect seemingly unrelated mathematical domains.
Number Theory and Automorphic Forms
While Frenkel's primary focus is geometric, he has also contributed to the classical Langlands program. His work on "Langlands duality for loop groups" explores how automorphic forms on loop groups relate to their dual groups.
This research demonstrates the profound connections between the geometric and classical aspects of the Langlands program, showing how insights from one domain can illuminate problems in the other.
Select Publications
Geometric Langlands Correspondence and Affine Kac-Moody Algebras
Co-authored with D. Gaitsgory. A foundational paper linking affine Lie algebras to the geometric Langlands program, establishing important connections between representation theory and algebraic geometry.
Lectures on the Langlands Program and Conformal Field Theory
A comprehensive survey connecting the Langlands program to physics, particularly exploring relationships with conformal field theory and quantum physics.
Opers and the Quantum Langlands Correspondence
Explores q-deformations of opers and their role in quantum Langlands, extending the geometric Langlands correspondence to quantum groups and deformed algebras.
Langlands Duality for Representations of Quantum Groups
Co-authored with D. Hernandez. Extends Langlands duality to quantum affine algebras, demonstrating how quantum groups fit into the Langlands correspondence framework.
Affine Kac-Moody Algebras and the Geometric Langlands Program
Discusses how Wakimoto modules and other representation-theoretic tools appear in Langlands, providing technical machinery for advancing the geometric Langlands program.
Key Collaborators
Frenkel frequently collaborates with leading mathematicians across prestigious institutions:
Interdisciplinary Impact
Frenkel's work consistently emphasizes connections to physics, particularly 2D conformal field theory and S-duality. He has articulated how the geometric Langlands correspondence represents a mathematical realization of S-duality in quantum field theory.
Through his expository writings, most notably his book "Love and Math," and numerous public lectures, Frenkel has played a pivotal role in popularizing the Langlands program among non-specialists, presenting it as a "grand unified theory of mathematics" that connects diverse mathematical disciplines.
Conclusion
Edward Frenkel stands as a central figure in the geometric Langlands program, with contributions spanning representation theory, algebraic geometry, and mathematical physics. His research has illuminated deep structural aspects of Langlands duality, particularly through the innovative application of vertex algebras, opers, and quantum deformations.
While his work is primarily geometric in nature, it carries significant implications for the classical Langlands program and number theory. Frenkel's ability to bridge disparate mathematical domains and communicate complex ideas to diverse audiences has established him as both a leading researcher and influential ambassador for modern mathematics.
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