The conformal limit for Nakajima quiver varieties

1. Problem Statement and Motivation

The paper addresses the extension of the conformal limit construction, originally developed for Higgs bundles by Gaiotto and rigorously proven by Collier, Li, and others, to the broader class of Nakajima quiver varieties.

• Background: In the context of Higgs bundles on a Riemann surface, the conformal limit describes the behavior of a family of flat connections $\nabla_{R, \bar{h}}$ as a parameter $R \to 0$. This limit maps specific holomorphic Lagrangian submanifolds of the Higgs moduli space ($M_H$) to corresponding submanifolds of the de Rham moduli space ($M_{dR}$).
• The Gap: While the existence of this limit was established for Higgs bundles (which are hyperkähler manifolds with a $\mathbb{C}^\times$-action extending to their twistor space), it was an open question whether a similar construction exists for other hyperkähler manifolds.
• Objective: The authors aim to define and prove the existence of a conformal limit for Nakajima quiver varieties, which are hyperkähler quotients of vector spaces parametrizing framed representations of doubled quivers. They seek to establish that this limit provides a biholomorphism between specific holomorphic Lagrangian submanifolds foliating different fibers of a "Hodge-Nakajima" moduli space.

2. Methodology

The authors adapt the paradigm established in the Higgs bundle literature (specifically [CW19]) to the quiver setting. The methodology proceeds through several key steps:

A. Construction of the Hodge-Nakajima Moduli Space

They define a moduli space $\mathcal{M}_{Hod}^Q$ parametrizing polystable $\lambda$-connections for quiver representations.

• This space is constructed as a union of quiver varieties with varying stability parameters:
  $$\mathcal{M}_{Hod}^Q := \bigcup_{\xi \in \mathbb{C}} M((1-|\xi|^2)\zeta_R, -2i\xi\zeta_R)$$
• They explicitly describe the twistor line (hyperkähler rotation) associated with a point in the quiver variety, showing how the real and complex moment maps transform under the action of $\xi \in \mathbb{C}$.

B. The Bia lynicki-Birula (BB) Slice and Hodge Slice

To handle the non-linearity of the moment map equations, the authors introduce geometric slices:

1. Hodge Slice ($S_p$): Defined at a point $p$ by replacing the linearized complex moment map equation with the full non-linear equation $\mu_C(p+q)=0$, while maintaining the condition that $q$ is orthogonal to the gauge orbit ($l_p^*(q)=0$).
2. BB Slice ($S^+_{p_0}$): Restricted to a $\mathbb{C}^\times$-fixed point $p_0$, this slice is defined by restricting the Hodge slice to the "positive" eigenspaces of the $\mathbb{C}^\times$-action.
   • They prove that $S^+_{p_0}$ is a finite-dimensional subvariety diffeomorphic to the holomorphic Lagrangian submanifold $W^0(p_0)$ (the set of points in the Hodge fiber flowing to $p_0$).

C. The Conformal Limit Construction

The core of the proof involves constructing a family of points $F_{R, \bar{h}}$ and analyzing their limit as $R \to 0$:

1. Scaling: Start with a point $p = p_0 + A$ in the BB slice. Apply the $\mathbb{C}^\times$-action (scaling by $R$) to the "negative" parts of the representation.
2. Gauge Correction: Since the scaled point generally fails to satisfy the real moment map equation, the authors employ a complex gauge transformation $g_{A, R}$ to restore the real moment map condition.
   • A crucial technical lemma (Lemma 4.6) uses the Inverse Function Theorem to prove the existence of a unique gauge transformation $g_{A, R}$ that corrects the error, providing quantitative estimates on the error terms (showing they are of order $O(R^2)$ or higher).
3. Limiting Process: The family is defined as:
   $$\nabla_{R, \bar{h}} = g_{A, \bar{h}} \cdot \text{TwistorAction}(\text{Scaled } p)$$
   The authors prove that as $R \to 0$, this family converges to a well-defined point in the target quiver variety.

3. Key Contributions and Results

Theorem 1.3 (Existence and Biholomorphism)

The primary result establishes that for a fixed point $p_0$ and an element $A$ in the BB slice $S^+_{p_0}$, the conformal limit:
$$CL_{\bar{h}}(p_0 + A) := \lim_{R \to 0} F_{R, \bar{h}}(g_A \cdot (p_0 + A))$$
exists and is given explicitly by a formula involving the fixed point data, the perturbation $A$, and a unique complex gauge transformation $g_{A, \bar{h}}$.

• Significance: This map is a biholomorphism between the holomorphic Lagrangian submanifold $W^0(p_0) \subset M(\zeta_R, 0)$ (the Higgs fiber) and $W^1(p_0) \subset M(0, -2i\zeta_R)$ (the de Rham fiber).

Theorem 5.2

This theorem formalizes the convergence, showing that the limit maps the "upward flow" of the $\mathbb{C}^\times$-action in the Higgs fiber to the corresponding flow in the de Rham fiber.

Simpson's Conjecture for Quiver Varieties (Section 6)

The authors formulate an analogue of Simpson's conjecture, which posits that the holomorphic Lagrangian submanifolds $W^1(p_0)$ are complete submanifolds of the de Rham moduli space.

• Theorem 6.1: They prove a partial result: If $p_0 + A$ is not nilpotent (i.e., it does not lie in the nilpotent cone), then the conformal limit line $CL_{\bar{h}}(p_0 + A)$ leaves every compact subset of the moduli space as the parameter goes to infinity.
• Method: They utilize Lusztig's variant of the Le Bruyn–Procesi theorem, showing that non-nilpotent points have non-vanishing invariants (traces of loops or admissible paths) that blow up under the conformal scaling, forcing the trajectory to escape compact sets.

4. Technical Nuances and Differences from Higgs Bundles

The paper highlights a critical technical difference between the Higgs bundle case and the quiver variety case:

• Gauge Dependence: In the Higgs bundle case, the conformal limit map is relatively direct. In the quiver case, the real moment map is not preserved by the $\mathbb{C}^\times$-action. Consequently, the limit involves a gauge transformation $g_{A, \bar{h}}$ that depends on both the parameter $\bar{h}$ and the initial point $A$. This necessitates the sophisticated use of the Inverse Function Theorem and asymptotic analysis of the metric $h_{A,R}$ to ensure convergence.

5. Significance

1. Generalization of Conformal Limits: The work successfully extends a powerful tool from the theory of Higgs bundles to the broader class of Nakajima quiver varieties, demonstrating that the conformal limit phenomenon is a robust feature of hyperkähler manifolds with $\mathbb{C}^\times$-actions.
2. Coordinate Systems: The conformal limit provides a way to endow the holomorphic Lagrangian submanifolds $W^1(p_0)$ with explicit coordinates, which is essential for studying their geometry and topology.
3. Progress on Completeness: By proving the "escape to infinity" property for non-nilpotent points, the authors provide strong evidence for the completeness of these submanifolds in the quiver setting, mirroring the resolution of Simpson's conjecture in rank 2 for Higgs bundles.
4. Bridge to Real Geometry: The construction connects the geometry of quiver varieties (often related to representation theory and algebraic geometry) with the analytic properties of flat connections, reinforcing the Non-Abelian Hodge correspondence in a discrete setting.

In summary, the paper provides a rigorous geometric construction of the conformal limit for quiver varieties, proving it acts as a biholomorphism between specific Lagrangian strata and offering new insights into the completeness of these strata, thereby deepening the understanding of the interplay between hyperkähler geometry, gauge theory, and representation theory.