Logarithmic Connections, WZNW Action, and Moduli of Parabolic Bundles on the Sphere

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