[Paper Review] Singular connections, WZNW action, and moduli of parabolic bundles on the sphere

Claudio Meneses, Leon A. Takhtajan|arXiv (Cornell University)|Jul 24, 2014

Advanced Algebra and Geometry18 references3 citations

TL;DR

This paper constructs a Kähler potential on the regular locus of the moduli space of stable parabolic vector bundles of parabolic degree zero on the Riemann sphere using the regularized Wess-Zumino-Novikov-Witten (WZNW) functional. The potential is shown to be an antiderivative of a (1,0)-form tied to solutions of the Riemann-Hilbert problem, establishing a geometric link between singular connections, conformal field theory, and moduli spaces of parabolic bundles.

ABSTRACT

The moduli space of stable parabolic vector bundles of parabolic degree 0 over the Riemann sphere is considered. The vector bundle analog of the Klein's Hauptmodul is defined and the regular locus, a subset of bundles with minimal Birkhoff-Grothendieck decomposition and generic Bruhat type of the constant term at $\infty$, is introduced. For the restriction of the natural Kaehler metric to the regular locus a potential is constructed as the value of the regularized WZNW functional evaluated on singular Hermitian metrics in the corresponding vector bundles. It is shown that this potential is an antiderivative of a (1,0)-form on the regular locus, associated with a solution of the Riemann-Hilbert problem.

Motivation & Objective

• To define a vector bundle analog of Klein's Hauptmodul for parabolic bundles on the Riemann sphere.
• To introduce and characterize the regular locus within the moduli space of stable parabolic bundles of parabolic degree zero.
• To construct a Kähler potential on this regular locus using the regularized WZNW functional on singular Hermitian metrics.
• To establish that this potential is an antiderivative of a (1,0)-form associated with solutions of the Riemann-Hilbert problem.
• To connect geometric structures on the moduli space with conformal field theory and representation theory via singular connections.

Proposed method

• The regular locus is defined as the subset of parabolic bundles with minimal Birkhoff-Grothendieck decomposition and generic Bruhat type at infinity.
• The WZNW functional is regularized and evaluated on singular Hermitian metrics in the corresponding vector bundles to produce a Kähler potential.
• The potential is shown to be a well-defined function on the regular locus, derived from the asymptotic behavior of the WZNW action.
• A (1,0)-form is constructed from solutions of the Riemann-Hilbert problem associated with the monodromy data of the bundle.
• The potential is proven to be an antiderivative of this (1,0)-form, linking geometric and analytic structures.
• The construction relies on the interplay between parabolic structures, holomorphic connections, and the geometry of the moduli space.

Experimental results

Research questions

• RQ1 How can a Kähler potential be constructed on the moduli space of parabolic bundles using conformal field theory tools like the WZNW functional?
• RQ2 What is the geometric significance of the regular locus defined by minimal Birkhoff-Grothendieck decomposition and generic Bruhat type at infinity?
• RQ3 How does the regularized WZNW functional on singular Hermitian metrics relate to differential forms on the moduli space?
• RQ4 In what way does the solution of the Riemann-Hilbert problem encode the differential structure of the Kähler potential?
• RQ5 What is the role of the vector bundle analog of Klein's Hauptmodul in organizing the geometry of the moduli space?

Key findings

• A Kähler potential is explicitly constructed on the regular locus of the moduli space of stable parabolic bundles of parabolic degree zero on the Riemann sphere.
• The potential arises as the value of the regularized WZNW functional evaluated on singular Hermitian metrics in the corresponding vector bundles.
• The potential is proven to be an antiderivative of a (1,0)-form defined via solutions of the Riemann-Hilbert problem.
• The regular locus is characterized by minimal Birkhoff-Grothendieck decomposition and generic Bruhat type of the constant term at infinity.
• The construction establishes a direct link between singular connections, conformal field theory, and the differential geometry of moduli spaces.
• The vector bundle analog of Klein's Hauptmodul is introduced as a tool to organize the structure of the moduli space.