[Paper Review] Uhlenbeck spaces via affine Lie algebras

Alexander BRAVERMAN, Michael FINKELBERG+1 more|arXiv (Cornell University)|Jan 16, 2003

Algebraic structures and combinatorial models4 citations

TL;DR

This paper constructs Uhlenbeck compactifications of moduli spaces of $G$-bundles on $\mathbb{P}^2$ trivialized along a line, using affine Lie algebras and geometric realizations of Kashiwara's crystals for affine Kac-Moody algebras. The key contribution is a geometric description of the stalks of the intersection cohomology sheaf of the Uhlenbeck space, achieved via a novel link between representation theory and moduli geometry in characteristic zero.

ABSTRACT

Let $G$ be an almost simple simply connected group over $\BC$, and let $\Bun^a_G(\BP^2,\BP^1)$ be the moduli scheme of principal $G$-bundles on the projective plave $\BP^2$, of second Chern class $a$, trivialized along a line $\BP^1\subset \BP^2$. We define the Uhlenbeck compactification $\fU^a_G$ of $\Bun^a_G(\BP^2,\BP^1)$, which classifies, roughly, pairs $(\F_G,D)$, where $D$ is a 0-cycle on $\BA^2=\BP^2-\BP^1$ of degree $b$, and $\F_G$ is a point of $\Bun^{a-b}_G(\BP^2,\BP^1)$, for varying $b$. In addition, we calculate the stalks of the Intersection Cohomology sheaf of $\fU^a_G$. To do that we give a geometric realization of Kashiwara's crystals for affine Kac-Moody algebras.

Motivation & Objective

• To define and construct Uhlenbeck compactifications $\mathfrak{U}^a_G$ of moduli spaces of $G$-bundles on $\mathbb{P}^2$ trivialized along a line $\mathbb{P}^1$, generalizing the Donaldson-Nakajima framework beyond $SL_n$.
• To provide a geometric realization of Kashiwara's crystals for affine Kac-Moody algebras using Uhlenbeck spaces.
• To compute the stalks of the intersection cohomology sheaf of $\mathfrak{U}^a_G$ via representation-theoretic techniques.
• To establish invariance of the construction under different modular descriptions, up to nilpotents, in characteristic zero.

Proposed method

• Constructs $\mathfrak{U}^a_G$ as a union over $b \in \mathbb{N}$ of $\operatorname{Bun}^{a-b}_G(\mathbb{P}^2, \mathbb{P}^1) \times \operatorname{Sym}^b(\mathbb{A}^2)$, parametrizing pairs $({\mathcal{F}}_G, D)$ with $D$ a 0-cycle of degree $b$.
• Uses the canonical filtration $F^n$ on the universal enveloping algebra $U({\mathfrak{n}}^{-})$ to analyze the structure of highest weight vectors in the affine Verma module $\mathcal{V}_{\check{\rho}}$.
• Applies Joseph's lemma to bound the minimal degree of a vector in $F^n$ that generates a weight space of weight $w{\check{\rho}}$, showing $m_w \geq \frac{1}{2}\ell(w)$.
• Constructs a Clifford algebra $C({\mathfrak{g}})$ on $\mathfrak{g}$ with a Lie algebra homomorphism $\varphi: \mathfrak{g} \to C({\mathfrak{g}})$, enabling a triangular decomposition $C({\mathfrak{n}}^{-}) \otimes C({\mathfrak{h}}) \otimes C({\mathfrak{n}}) \cong C({\mathfrak{g}})$.
• Realizes the irreducible representation $\mathcal{V}_{\check{\rho}}$ as a quotient of $C({\mathfrak{g}})/I({\mathfrak{g}})$, where $I({\mathfrak{g}})$ is the left ideal generated by $\mathfrak{n}$, and uses this to analyze weight spaces.
• Relates the degree of the vector $\prod_{\check{\beta} \in S(w^{-1})} e_{-\check{\beta}}$ in $C({\mathfrak{g}})/I({\mathfrak{g}})$ to the length $\ell(w)$, proving the key estimate for the stalk computation.

Experimental results

Research questions

• RQ1 How can Uhlenbeck compactifications be constructed for arbitrary almost simple simply connected groups $G$ over $\mathbb{C}$, beyond the $SL_n$ case?
• RQ2 Can the stalks of the intersection cohomology sheaf of the Uhlenbeck space $\mathfrak{U}^a_G$ be computed geometrically using representation-theoretic data?
• RQ3 Is there a geometric realization of Kashiwara's crystals for affine Kac-Moody algebras via Uhlenbeck-type moduli spaces?
• RQ4 What is the role of the canonical filtration on $U({\mathfrak{n}}^{-})$ in controlling the structure of weight spaces in the affine Verma module $\mathcal{V}_{\check{\rho}}$?
• RQ5 How does the Clifford algebra construction on $\mathfrak{g}$ facilitate the analysis of the weight space decomposition in $C({\mathfrak{g}})/I({\mathfrak{g}})$?

Key findings

• The Uhlenbeck space $\mathfrak{U}^a_G$ is constructed as a union over $b \in \mathbb{N}$ of $\operatorname{Bun}^{a-b}_G(\mathbb{P}^2, \mathbb{P}^1) \times \operatorname{Sym}^b(\mathbb{A}^2)$, providing a modular compactification of the moduli of $G$-bundles with trivialization along $\mathbb{P}^1$.
• The stalks of the intersection cohomology sheaf of $\mathfrak{U}^a_G$ are computed via a geometric realization of Kashiwara's crystals for affine Kac-Moody algebras, linking moduli geometry to representation theory.
• The key estimate $m_w \geq \frac{1}{2}\ell(w)$ is established using Joseph's lemma on the canonical filtration of $U({\mathfrak{n}}^{-})$, which bounds the minimal degree of a vector generating the weight space of weight $w{\check{\rho}}$.
• The Clifford algebra $C({\mathfrak{g}})$ admits a triangular decomposition $C({\mathfrak{n}}^{-}) \otimes C({\mathfrak{h}}) \otimes C({\mathfrak{n}}) \cong C({\mathfrak{g}})$, enabling a systematic analysis of the structure of $C({\mathfrak{g}})/I({\mathfrak{g}})$ as a $\mathfrak{g}$-module.
• The irreducible representation $\mathcal{V}_{\check{\rho}}$ is realized as a quotient of $C({\mathfrak{g}})/I({\mathfrak{g}})$, and the weight space of weight $w{\check{\rho}}$ is isomorphic to $\left(\prod_{\check{\beta} \in S(w^{-1})} e_{-\check{\beta}}\right) C({\mathfrak{h}})$, which lies in degree between $\ell(w)$ and $\ell(w) + \dim{\mathfrak{h}}$.
• The construction is invariant under different modular descriptions of $\mathfrak{U}^a_G$ up to nilpotents, and the proof of this invariance relies on the characteristic zero assumption, suggesting potential limitations in positive characteristic.