[Paper Review] Uniformization of semistable bundles on elliptic curves

Penghui Li, David Nadler|arXiv (Cornell University)|Oct 29, 2015

Advanced Algebra and Geometry33 references3 citations

TL;DR

This paper introduces a complex analytic uniformization of the stack of semistable $G$-bundles on an elliptic curve $E$, using twisted conjugacy classes in the holomorphic loop group of a reductive group $G$. It generalizes classical uniformizations like $E \simeq \mathbb{C}^*/q^\mathbb{Z}$ to non-abelian settings, constructs a uniformization of $G$ via the exponential map, and applies these results to sheaves with nilpotent singular support and the Betti geometric Langlands program in genus 1.

ABSTRACT

Let $G$ be a connected reductive complex algebraic group, and $E$ a complex elliptic curve. Let $G_E$ denote the connected component of the trivial bundle in the stack of semistable $G$-bundles on $E$. We introduce a complex analytic uniformization of $G_E$ by adjoint quotients of reductive subgroups of the loop group of $G$. This can be viewed as a nonabelian version of the classical complex analytic uniformization $ E \simeq \mathbb{C}^*/q^{\mathbb{Z}}$. We similarly construct a complex analytic uniformization of $G$ itself via the exponential map, providing a nonabelian version of the standard isomorphism $\mathbb{C}^* \simeq \mathbb{C}/\mathbb{Z}$, and a complex analytic uniformization of $G_E$ generalizing the standard presentation $E = \mathbb{C}/(\mathbb{Z} \oplus \mathbb{Z} τ)$. Finally, we apply these results to the study of sheaves with nilpotent singular support. As an application to Betti geometric Langlands conjecture in genus 1, we define a functor from $Sh_\mathcal{N}(G_E)$ (the semistable part of the automorphic category) to ${IndCoh}_{\check{\mathcal{N}}}({Locsys}_{\check G} (E))$ (the spectral category).

Motivation & Objective

• To develop a complex analytic uniformization of the stack $G_E$ of semistable $G$-bundles on an elliptic curve $E$, generalizing classical uniformizations.
• To construct a non-abelian analog of $\mathbb{C}^* \simeq \mathbb{C}/\mathbb{Z}$ via the exponential map for $G$.
• To extend the standard uniformization $E \simeq \mathbb{C}/(\mathbb{Z} \oplus \mathbb{Z}\tau)$ to a non-abelian setting using loop group subvarieties.
• To apply the uniformization to sheaves with nilpotent singular support and to define a functor in the Betti geometric Langlands program.
• To clarify the dependence of restriction functors on parabolic subgroups and their topological nature in genus 1.

Proposed method

• Use twisted conjugacy classes in the holomorphic loop group $L_{\text{hol}}G$ to parametrize $G$-bundles on $E$, generalizing the classical $E \simeq \mathbb{C}^*/q^\mathbb{Z}$ uniformization.
• Construct a uniformization of $G$ itself via the exponential map, providing a non-abelian version of $\mathbb{C}^* \simeq \mathbb{C}/\mathbb{Z}$.
• Define analytic charts on the stack $G_E$ using reductive subgroups of the loop group, with a Lie-theoretic choice of charts for simply-connected and general reductive $G$.
• Apply étale charts and 1-shifted symplectic structures to construct a global uniformization via gluing of local models.
• Use the uniformization to define a functor from $\text{Sh}_{\mathcal{N}}(G_E)$ to $\text{IndCoh}_{\check{\mathcal{N}}}(\text{Locsys}_{\check{G}}(E))$, linking automorphic and spectral categories.
• Leverage the theory of twisted conjugation and gauge theory on $S^1$ and $E$ to relate holomorphic and topological structures.

Experimental results

Research questions

• RQ1 Can the stack of semistable $G$-bundles on an elliptic curve be uniformly parametrized by finite-dimensional subvarieties of the loop group?
• RQ2 How can the classical uniformization $E \simeq \mathbb{C}^*/q^\mathbb{Z}$ be generalized to a non-abelian setting using loop group structures?
• RQ3 What is the role of twisted conjugacy in the holomorphic loop group in parametrizing $G$-bundles on $E$?
• RQ4 How does the uniformization relate to the geometry of sheaves with nilpotent singular support?
• RQ5 What is the dependence of restriction functors on parabolic subgroups in the context of the Betti geometric Langlands conjecture in genus 1?

Key findings

• The stack $G_E$ of semistable $G$-bundles on $E$ admits a complex analytic uniformization by twisted conjugacy classes in the holomorphic loop group $L_{\text{hol}}G$.
• A non-abelian uniformization of $G$ is constructed via the exponential map, generalizing $\mathbb{C}^* \simeq \mathbb{C}/\mathbb{Z}$.
• The uniformization of $G_E$ generalizes the standard presentation $E \simeq \mathbb{C}/(\mathbb{Z} \oplus \mathbb{Z}\tau)$ to a non-abelian setting using reductive subgroups of the loop group.
• The uniformization induces an isomorphism between the analytic stack $T_E^{\text{reg}}/W$ and the open substack $G_E^{\text{reg}}$ of regular semistable bundles.
• The construction yields a functor from $\text{Sh}_{\mathcal{N}}(G_E)$ to $\text{IndCoh}_{\check{\mathcal{N}}}(\text{Locsys}_{\check{G}}(E))$, providing a key step in the Betti geometric Langlands program.
• The moduli stack $\text{Bun}^{0,ss}_{G,x}(E)$ of semistable $G$-bundles with trivialization at a point $x$ is representable by a complex manifold, confirming analytic smoothness.