Local geometric Langlands correspondence and affine Kac-Moody algebras

By E. Frenkel, D. Gaitsgory. 2005

144 citations recorded by the source corpus.

Let \( \mathfrak{g} \) be a simple Lie algebra over ℂ and G a connected algebraic group with Lie algebra \( \mathfrak{g} \). The affine Kac-Moody algebra \( \hat {\mathfrak{g}} \) is the universal central extension of the formal loop agebra \( \mathfrak{g} \)((t)). Representations of \( \hat {\mathfrak{g}} \) have a parameter, an invariant bilinear form on \( \mathfrak{g} \), which is called the level. Representations corresponding to the bilinear form which is equal to minus one half of the Killing form are called representations of critical level. Such representations can be realized in spaces of global sections of twisted D-modules on the quotient of the loop group G((t)) by its “open compact” subgroup K, such as G[[t]] or the Iwahori subgroup I.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/29818b5fb073ee4b2a9de395ed9f9d5654d0beed

How researchers see this paper

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In Algebra and Number Theory

1. E. Witten
2. D. Nadler
3. David Ben-Zvi
4. Sergei Gukov
5. B. Gross
6. R. Bezrukavnikov
7. E. Frenkel
8. A. Kapustin

In Geometry and Topology

1. E. Witten
2. Sergei Gukov
3. David Ben-Zvi
4. D. Nadler
5. A. Kapustin
6. E. Frenkel
7. R. Bezrukavnikov
8. D. Gaiotto

In Mathematical Physics

1. E. Witten
2. Sergei Gukov
3. D. Nadler
4. A. Kapustin
5. David Ben-Zvi
6. R. Bezrukavnikov
7. D. Gaiotto
8. B. Gross

Citations in the corpus

• Dynkin Automorphism Actions on Gaudin Algebras (2023, unknown): “In this survey, we will give the descriptions of all our constructions for C -points only, and the precise treatment of R -points can be found in [FG06, 1.1-1.5].”
• Cohomology of tilting modules over quantum groups and t-structures on derived categories of coherent sheaves (2004, background): “N appears in some other contexts, e.g., it is related to the t-structures appearing in [BMR], and, conjecturally, 2 to modules over the affine Lie algebra on the critical level studied by Frenkel and Gaitsgory, cf. [ FG ].”
• A rigid irregular connection on the projective line (2009, methodology): “According to the proposal of [FG], the sought-after category Cσ should be equivalent to ĝcrit -modχ (these categories should therefore be equivalent to each other for all χ ∈ p−1(σ)).”
• Lectures on the Langlands program and conformal field theory (2005, methodology): “We explain this briefly following [ 41 ] where we refer the reader for more details.”
• Betti Geometric Langlands (2016, background): “7), this would be an A-space in the sense of [FG].”
• Electric-Magnetic Duality And The Geometric Langlands Program (2006, background): “[5,9, 27-30 ]. The last of these references applies conformal field theory to the ramified case.”
• Gauge Theory, Ramification, And The Geometric Langlands Program (2006, methodology): “Using conformal field theory, a proposal has been made [ 6 ] for a unified approach to”