[Paper Review] Singular Hermitian-Einstein monopoles on the product of a circle and a Riemann surface

Benoit Charbonneau, Jacques Hurtubise|ArXiv.org|Dec 1, 2008

Algebraic Geometry and Number Theory18 references4 citations

TL;DR

This paper establishes a correspondence between singular unitary Hermitian-Einstein monopoles on $S^1 \times \Sigma$ and stable holomorphic pairs $(\mathcal{E}, \rho)$ on a Riemann surface $\Sigma$, where $\mathcal{E}$ is a holomorphic vector bundle and $\rho$ is a meromorphic automorphism. The key result is a bijective correspondence between the moduli space of such monopoles and a moduli space of stable pairs, with singularities of the monopole corresponding to poles of the meromorphic section $\rho$, and the complex geometry of the moduli space is analyzed via dimension computations from both gauge-theoretic and complex-geometric perspectives.

ABSTRACT

In this paper, the moduli space of singular unitary Hermitian--Einstein monopoles on the product of a circle and a Riemann surface is shown to correspond to a moduli space of stable pairs on the Riemann surface. These pairs consist of a holomorphic vector bundle on the surface and a meromorphic automorphism of the bundle. The singularities of this automorphism correspond to the singularities of the singular monopole. We then consider the complex geometry of the moduli space; in particular, we compute dimensions, both from the complex geometric and the gauge theoretic point of view.

Motivation & Objective

• To establish a correspondence between singular unitary Hermitian-Einstein monopoles on $S^1 \times \Sigma$ and stable holomorphic pairs on $\Sigma$.
• To characterize the moduli space of such monopoles using complex geometry and gauge theory.
• To define and compute the dimension of the moduli space from both gauge-theoretic and complex-geometric viewpoints.
• To analyze the role of singularities in the monopole, showing they correspond to poles of a meromorphic automorphism on the bundle.
• To extend the correspondence to circle bundles that become trivial over finite covers of $\Sigma$, using parallel transport and deck transformations.

Proposed method

• The authors use the generalized Bogomolny equation with a constant central term on $Y = S^1 \times \Sigma$, incorporating Dirac-type singularities at fixed points $p_i = (t_i, z_i)$.
• They define singular monopoles with asymptotic behavior near singularities determined by integer sequences $\vec{k}_i$, where the Higgs field $\phi$ behaves like $\frac{\sqrt{-1}}{2R_i} \mathrm{diag}(k_{i1}, \dots, k_{in})$.
• The construction lifts the monopole to the universal cover $\widetilde{Y} \to Y$, where a flat section $S$ trivializes the bundle, enabling the use of parallel transport to define a holomorphic structure on $\Sigma$.
• The monopole's $S^1$-invariance and the flat connection allow the construction of a holomorphic vector bundle $\mathcal{E}$ on $\Sigma$ and a meromorphic automorphism $\rho$ via parallel transport and deck transformations.
• The authors define a $(\vec{t}, U)$-degree $\delta_{\vec{t},U}(\mathcal{E}, \rho)$ to generalize stability conditions for non-trivial circle bundles.
• They prove a bijective correspondence between irreducible HE-monopole moduli and stable pairs $(\mathcal{E}, \rho)$ with specified singular type $\mathbf{K}$, using invariance under $S^1 \times \mathfrak{D}$ and descent via deck group actions.

Experimental results

Research questions

• RQ1 How do singular Hermitian-Einstein monopoles on $S^1 \times \Sigma$ relate to holomorphic structures on $\Sigma$?
• RQ2 What is the precise correspondence between the moduli space of singular monopoles and a moduli space of stable holomorphic pairs?
• RQ3 How can the complex geometry of the monopole moduli space be analyzed, particularly in terms of dimension?
• RQ4 What is the role of the circle bundle's topology in defining stability and degree for the associated holomorphic pairs?
• RQ5 How do the singularities of the monopole manifest in the meromorphic automorphism of the holomorphic bundle?

Key findings

• The moduli space of irreducible $\mathrm{U}(n)$ Hermitian-Einstein monopoles on a circle bundle $Y \to \Sigma$ that becomes trivial over a finite cover is in bijective correspondence with the moduli space of $(\vec{t}, U)$-stable holomorphic pairs $(\mathcal{E}, \rho)$.
• The holomorphic bundle $\mathcal{E}$ on $\Sigma$ has rank $n$ and degree $k_0$, and the meromorphic automorphism $\rho$ has poles at $z_j$ with local form $F_j(z) \mathrm{diag}(z^{k_{jl}}) G_j(z)$, where $F_j, G_j$ are holomorphic and invertible.
• The determinant of $\rho$ has divisor $\sum_j \mathrm{tr}(\vec{k}_j) z_j$, linking the monopole's singularity data to the meromorphic section's divisor.
• The dimension of the moduli space is computed from both gauge-theoretic and complex-geometric perspectives, confirming consistency between the two approaches.
• The $(\vec{t}, U)$-degree $\delta_{\vec{t},U}(\mathcal{E}, \rho)$ is invariant under translation of the section $S$ by $\widetilde{M}_t$, ensuring stability is well-defined under the circle action.
• For circle bundles not trivializing over finite covers, the stability condition is defined as a limit of average degrees over increasingly large fundamental domains.