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

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 Kähler potential for (Omega -Omega _{mathrm {T}})|_{mathscr {N}_{0}}, where Omega is the Narasimhan–Atiyah–Bott Kähler 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}.