[Paper Review] Quantization of Hitchin's equations for Higgs Bundles I

Mario García-Fernández, Julien Keller+1 more|arXiv (Cornell University)|Jan 19, 2016

Geometry and complex manifolds18 references3 citations

TL;DR

This paper introduces a quantization framework for Higgs bundles by defining balanced metrics as finite-dimensional approximations to solutions of the Hitchin equation. Using Geometric Invariant Theory and moment map techniques, it establishes that balanced metrics converge to solutions of the Hitchin equation in the quantum limit, and proves that their existence is equivalent to Gieseker stability of the Higgs bundle.

ABSTRACT

We provide an algebraic framework for quantization of Hermitian metrics that are solutions of the Hitchin equation for Higgs bundles over a projective manifold. Using Geometric Invariant Theory, we introduce a notion of balanced metrics in this context. We show that balanced metrics converge at the quantum limit towards the solution of the Hitchin equation. We relate the existence of balanced metrics to the Gieseker stability of the Higgs bundle.

Motivation & Objective

• To develop an algebraic quantization framework for Hermitian metrics solving the Hitchin equation on Higgs bundles over projective manifolds.
• To define a notion of balanced metrics in the context of Higgs bundles, generalizing the concept from Hermitian-Einstein metrics.
• To establish a correspondence between the existence of balanced metrics and the Gieseker stability of the Higgs bundle.
• To show that balanced metrics converge to solutions of the Hitchin equation in the quantum limit (large k limit).
• To relate the moment map formalism and Bergman function to the stability condition via Geometric Invariant Theory.

Proposed method

• Introduce a Kähler structure on a space Z parameterizing holomorphic maps from the base manifold to a Grassmannian and sections of endomorphism bundles twisted by the canonical bundle.
• Define a Kähler form Ωₖ on Z using the Fubini-Study metric on the Grassmannian and a correction term involving the Bergman function and a parameter λ.
• Use the SU_N-action on the space of holomorphic maps and define a moment map for this action with respect to Ωₖ.
• Define a balanced metric as a Hermitian metric h on E such that the induced map uₛ is a zero of the moment map for the SU_N-action.
• Re-express the balanced condition intrinsically via the Bergman function Bₖ(h), linking it to the L²-orthonormal basis of H⁰(E⊗Lᵏ).
• Use a 1-parameter C*-action to analyze the weight of the moment map, leading to a stability criterion involving the trace of the curvature term and the Higgs field.

Experimental results

Research questions

• RQ1 Can a finite-dimensional approximation scheme be constructed for solutions of the Hitchin equation on Higgs bundles?
• RQ2 Is there a notion of balanced metric for Higgs bundles that generalizes the classical balanced metrics for vector bundles?
• RQ3 Does the existence of balanced metrics at level k correspond to a stability condition analogous to Gieseker stability?
• RQ4 How does the quantum limit (k→∞) of balanced metrics relate to the solution of the Hitchin equation?
• RQ5 Can the moment map formalism be extended to include the Higgs field in the stability criterion?

Key findings

• Balanced metrics exist if and only if the Higgs bundle is Gieseker stable, establishing a direct link between algebraic stability and the existence of finite-dimensional approximations.
• The balanced metric condition is equivalent to the vanishing of the moment map for the SU_N-action on the space of holomorphic maps to the Grassmannian, with a corrected Kähler form involving the Bergman function.
• In the large k limit, balanced metrics converge to solutions of the Hitchin equation, providing a quantization procedure for the Hermitian metric on the Higgs bundle.
• The weight of the moment map under a C*-action yields a stability inequality: the difference in normalized dimensions of global sections of subbundles is bounded below by a non-negative term involving the Higgs field’s action.
• The Higgs field contributes a negative term to the weight, which vanishes if and only if the Higgs field preserves the subbundle, indicating that splitting of the Higgs field is obstructed by instability.
• The key inequality derived from the moment map weight computation is: (h⁰(E⊗Lᵏ)/rk(E)) − (h⁰(F⊗Lᵏ)/rk(F)) − (1/V rk(E) rk(F)) ||αβ/√(1+α|φ|²) π_F φ (Id−π_F)||²_{L²} ≥ 0, which holds for all large k and is strict if (E,φ) is simple.