[Paper Review] A quantization of the Hitchin hamiltonian system and the Beilinson-Drinfeld isomorphism

Ken‐ichi Sugiyama|ArXiv.org|Aug 22, 2007

Advanced Algebra and Geometry8 references3 citations

TL;DR

This paper generalizes the Beilinson-Drinfeld quantization of the Hitchin system to principal $SL_2(\mathbb{C})$-bundles with parabolic structures (D-flats) on a smooth projective curve $X$ of genus $g > 1$. It constructs an isomorphism between the global sections of $\Omega^{\otimes 2}(D)$ and twisted differential operators on the moduli stack of such bundles, establishing a geometric ramified nonabelian class field theory via holonomic D-modules that are conjectured to be Hecke eigensheaves.

ABSTRACT

We will study the Hitchin's hamiltonian system for a modular stack of principal SL_2(C) bundle on a smooth projective curve which has a parabolic reduction at certain points. As an application we will obtain a generalization of the Beilinson-Drinfeld isomorphism, which is a quantization of the Hitchin's hamiltonian system.

Motivation & Objective

• To extend the Beilinson-Drinfeld isomorphism to the case of ramified local systems on $SL_2(\mathbb{C})$-bundles with parabolic reductions at marked points.
• To construct a quantization of the Hitchin system on the cotangent bundle of the moduli stack of $SL_2(\mathbb{C})$-bundles with $D$-flags.
• To establish a geometric ramified nonabelian class field theory assigning holonomic D-modules to elements of $H^0(X, \Omega^{\otimes 2}(D))$.

Proposed method

• Define the moduli stack $\mathrm{Bun}_{G,X}^{D\text{-}fl}$ of $SL_2(\mathbb{C})$-bundles with $D$-flags at $N$ marked points $z_1, \dots, z_N$.
• Construct a sheaf of twisted differential operators $\mathcal{D}'_{D\text{-}fl,\lambda}$ on $\mathrm{Bun}_{G,X}^{D\text{-}fl}$ for $\lambda \in \mathbb{Z}^N$.
• Prove an isomorphism $\Gamma(H^0(X, \Omega^{\otimes 2}(D)), \mathcal{O}) \simeq D'_{D\text{-}fl,\lambda}$, generalizing the Beilinson-Drinfeld isomorphism.
• Use a localization functor $\Delta_{D\text{-}fl}$ to associate D-modules on $\mathrm{Bun}_{G,X}^{D\text{-}fl}$ to representations of the affine Lie algebra $\hat{\mathfrak{g}}_N$.
• Define $\lambda$-admissible and global local sections of $\Omega^{\otimes 2}(2D)$ to parametrize D-modules via central character actions.
• Establish that the resulting D-modules are holonomic and nonzero precisely when the section is both global and $\lambda$-admissible.

Experimental results

Research questions

• RQ1 Can the Beilinson-Drinfeld isomorphism be generalized to the case of ramified local systems on $SL_2(\mathbb{C})$-bundles with parabolic structures?
• RQ2 Is there a quantization of the Hitchin system on the moduli stack of $SL_2(\mathbb{C})$-bundles with $D$-flags that extends the classical Hitchin integrable system?
• RQ3 Do the D-modules constructed via the generalized isomorphism form a Hecke eigensheaf structure, as conjectured?

Key findings

• The paper establishes a $\mathbb{C}$-algebra isomorphism $\Gamma(H^0(X, \Omega^{\otimes 2}(D)), \mathcal{O}) \simeq D'_{D\text{-}fl,\lambda}$, proving a non-abelian generalization of the Beilinson-Drinfeld isomorphism.
• The D-module $\mathcal{M}_q$ associated to $q \in H^0(X, \Omega^{\otimes 2}(D))$ is holonomic, as shown in Theorem 5.3.
• The localization functor $\Delta_{D\text{-}fl}$ sends $\lambda$-admissible and global $q$-data to nonzero D-modules, as stated in Theorem 6.1.
• The D-module $\mathcal{D}'_{D\text{-}fl,\lambda} \otimes_{\otimes \Gamma(H^0(D_i, \Omega^{\otimes 2}(2\mathbb{O})), \mathcal{O})} \mathbb{C}$ is isomorphic to $\Delta_{D\text{-}fl}(M_{-2,\lambda}^q)$, confirming consistency of the construction.
• The correspondence assigning $\mathcal{M}_q$ to $q$ defines a generalized Beilinson-Drinfeld correspondence, extending geometric class field theory to the ramified case.
• The construction provides a framework for studying Hecke eigensheaves in the ramified setting, with the conjecture that the D-modules are regular holonomic and Hecke eigensheaves.