[Paper Review] Affine Algebras, Langlands Duality and Bethe Ansatz

Edward Frenkel|ArXiv.org|Jun 5, 1995

Algebraic structures and combinatorial modelsMathematics8 references89 citations

TL;DR

This paper establishes a deep connection between affine Kac-Moody algebras at critical level, geometric Langlands duality, and the Bethe ansatz in integrable systems. It shows that ${\mathfrak{g}}^L$-opers on a curve correspond to $G^L$-local systems and via the Beilinson-Drinfeld localization functor, give rise to ${\mathcal{D}}$-modules on the moduli space of $G$-bundles, realizing the geometric Langlands correspondence. The key result is that completeness of the Bethe ansatz for the $SL_2$ Gaudin model is equivalent to trivial monodromy of the associated projective connection.

ABSTRACT

We review various aspects of representation theory of affine algebras at the critical level, geometric Langlands correspondence, and Bethe ansatz in the Gaudin models. Geometric Langlands correspondence relates D-modules on the moduli space of G-bundles on a complex curve X and flat G^L-bundles on X. Beilinson and Drinfeld construct it by applying a localization functor to representations of affine algebras of critical level. We show that in genus zero the corresponding D-modules are closely related to the diagonalization problem in the Gaudin model associated to G. This allows us to give a new interpretation of the Bethe ansatz and Sklyanin's separation of variables in the Gaudin model in terms of Langlands correspondence.

Motivation & Objective

• To establish a geometric realization of the Langlands correspondence for reductive groups using representation theory of affine Kac-Moody algebras at critical level.
• To clarify the role of ${\mathfrak{g}}^L$-opers as local Langlands parameters for $\widehat{{\mathfrak{g}}}$-modules at critical level.
• To prove completeness of the Bethe ansatz for the $SL_2$ Gaudin model by relating it to monodromy triviality of associated projective connections.
• To explore the correspondence between quantum integrable systems (Gaudin model) and geometric Langlands via localization and $q$-deformations.

Proposed method

• Uses the Beilinson-Drinfeld localization functor to assign ${\mathcal{D}}$-modules on $\mathcal{M}_G(X)$ to $\widehat{{\mathfrak{g}}}$-modules in category $\mathcal{O}^0$.
• Applies the critical level property where $U_{-h^\vee}(\widehat{{\mathfrak{g}}})$ has a large center isomorphic to the classical $\mathcal{W}$-algebra $\mathcal{W}({\mathfrak{g}}^L)$.
• Identifies $\mathcal{W}({\mathfrak{g}}^L)$-functionals with ${\mathfrak{g}}^L$-opers, which parametrize $\widehat{{\mathfrak{g}}}$-modules via character factoring.
• Constructs ${\mathcal{D}}$-modules from ${\mathfrak{g}}^L$-opers and shows they correspond to $G^L$-local systems via monodromy.
• For genus zero, relates the resulting ${\mathcal{D}}$-modules to the Gaudin model's commuting Hamiltonians and their eigenvalue equations.
• Uses the $q$-deformation of the Miura transformation and $q$-difference equations to generalize the separation of variables and spectral theory.

Experimental results

Research questions

• RQ1 How does the critical level structure of affine Kac-Moody algebras relate to the geometric Langlands correspondence?
• RQ2 What is the precise role of ${\mathfrak{g}}^L$-opers in parametrizing $\widehat{{\mathfrak{g}}}$-modules at critical level?
• RQ3 Can the completeness of the Bethe ansatz in the Gaudin model be geometrically characterized via monodromy?
• RQ4 How do $q$-deformations of the Miura transformation and $q$-difference equations generalize the separation of variables in integrable systems?
• RQ5 What is the relationship between the center of the quantum affine algebra at critical level and the classical $\mathcal{W}$-algebra of the Langlands dual?

Key findings

• The center of $U_{-h^\vee}(\widehat{{\mathfrak{g}}})$ is isomorphic to the classical $\mathcal{W}$-algebra $\mathcal{W}({\mathfrak{g}}^L)$, establishing a duality between $\widehat{{\mathfrak{g}}}$-modules and ${\mathfrak{g}}^L$-opers.
• Each regular ${\mathfrak{g}}^L$-oper on a curve $X$ defines a $G^L$-local system and a ${\mathcal{D}}$-module on $\mathcal{M}_G(X)$, realizing the geometric Langlands correspondence.
• For $G=SL_2$, the Bethe ansatz equations are equivalent to the condition that the associated projective connection has trivial monodromy, proving completeness of the Bethe ansatz.
• The Gaudin Hamiltonians arise as commuting differential operators whose eigenvalue problem corresponds to the ${\mathcal{D}}$-module associated to a ${\mathfrak{g}}^L$-oper.
• The $q$-deformed Miura transformation maps the quantum affine algebra's center to $q$-difference operators, generalizing the classical spectral theory.
• The $R$-matrix of $U_q(\widehat{{\mathfrak{g}}})$ at critical level coincides with that of the quantum Toda system, linking quantum groups and integrable systems.