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 ${\mathcal{N}}_0$ of the open stratum depending explicitly on a choice of parabolic weights, a real-valued function $\mathcal{S}$ is defined as the regularized critical value of the non-compact Wess--Zumino--Novikov--Witten action functional. The definition of $\mathcal{S}$ depends on a suitable notion of parabolic bundle `uniformization map' following from the Mehta--Seshadri and Birkhoff--Grothendieck theorems. It is shown that $-\mathcal{S}$ is a primitive for a (1,0)-form $\vartheta$ on ${\mathcal{N}}_0$ associated with the uniformization data of each intrinsic irreducible unitary logarithmic connection. Moreover, it is proved that $-\mathcal{S}$ is a Kähler potential for $(\Omega -{\Omega }_{\mathrm{T}}){|}_{{\mathcal{N}}_0}$, where $\Omega$ is the Narasimhan--Atiyah--Bott Kähler form in $\mathcal{N}$ 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 ${\mathcal{N}}_0=\mathcal{N}$.

30 pages. Final published version

References in corpus (2)

• On vector-valued Poincaré series of weight 2
• Remarks on groups of bundle automorphisms over the Riemann sphere