Logarithmic connections, WZNW action, and moduli of parabolic bundles on the sphere

TL;DR

This paper establishes a deep connection between the WZNW action functional, Kähler geometry, and the moduli space of parabolic bundles on the sphere, providing explicit formulas linking cohomology and tautological classes.

Contribution

It introduces a new real-valued function on the moduli space derived from the WZNW action, serving as a Kähler potential and relating cohomological classes explicitly.

Findings

$- ext{S}$ is a primitive for a specific (1,0)-form on the moduli space.

$- ext{S}$ acts as a Kähler potential for a modified Kähler form.

Explicit relation between the cohomology class $[ ext{Ω}]$ and tautological classes.

Abstract

Moduli spaces of stable parabolic bundles of parabolic degree $0$ over the Riemann sphere are stratified according to the Harder--Narasimhan filtration of underlying vector bundles. Over a Zariski open subset $\mathscr{N}_{0}$ of the open stratum depending explicitly on a choice of parabolic weights, a real-valued function $\mathscr{S}$ is defined as the regularized critical value of the non-compact Wess--Zumino--Novikov--Witten action functional. The definition of $\mathscr{S}$ depends on a suitable notion of parabolic bundle `uniformization map' following from the Mehta--Seshadri and Birkhoff--Grothendieck theorems. It is shown that $-\mathscr{S}$ is a primitive for a (1,0)-form $\vartheta$ on $\mathscr{N}_{0}$ associated with the uniformization data of each intrinsic irreducible unitary logarithmic connection. Moreover, it is proved that $-\mathscr{S}$ is a K\"ahler potential for $(\Omega-\Omega_{\mathrm{T}})|_{\mathscr{N}_{0}}$, where $\Omega$ is the Narasimhan--Atiyah--Bott K\"ahler form in $\mathscr{N}$ and $\Omega_{\mathrm{T}}$ is a certain linear combination of tautological $(1,1)$-forms associated with the marked points. These results provide an explicit relation between the cohomology class $[\Omega]$ and tautological classes, which holds globally over certain open chambers of parabolic weights where $\mathscr{N}_{0} = \mathscr{N}$.