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

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 $$\mathscr {N}_{0}$$ N 0 of the open stratum depending explicitly on a choice of parabolic weights, a real-valued function $$\mathscr {S}$$ S is defined as the regularized critical value of the non-compact Wess–Zumino–Novikov–Witten action functional. The definition of $$\mathscr {S}$$ S depends on a suitable notion of parabolic bundle ‘uniformization map’ following from the Mehta–Seshadri and Birkhoff–Grothendieck theorems. It is shown that $$-\mathscr {S}$$ - S is a primitive for a (1,0)-form $$\vartheta $$ ϑ on $$\mathscr {N}_{0}$$ N 0 associated with the uniformization data of each intrinsic irreducible unitary logarithmic connection. Moreover, it is proved that $$-\mathscr {S}$$ - S is a Kähler potential for $$(\Omega -\Omega _{\mathrm {T}})|_{\mathscr {N}_{0}}$$ ( Ω - Ω T ) | N 0 , where $$\Omega $$ Ω is the Narasimhan–Atiyah–Bott Kähler form in $$\mathscr {N}$$ N and $$\Omega _{\mathrm {T}}$$ Ω 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 $$[\Omega ]$$ [ Ω ] and tautological classes, which holds globally over certain open chambers of parabolic weights where $$\mathscr {N}_{0} = \mathscr {N}$$ N 0 = N .