arXiv Analytics

Sign in

arXiv:1407.6752 [math.CV]AbstractReferencesReviewsResources

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

Claudio Meneses, Leon A. Takhtajan

Published 2014-07-24, updated 2019-03-25Version 2

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 $\mathscr{N}_{0} = \mathscr{N}$.

Comments: 29 pages. Simplified version with substantial modifications. The proof of theorem 2 has been corrected
Categories: math.CV, hep-th, math.AG
Subjects: 32G13, 81T40
Related articles:
arXiv:1410.6207 [math.CV] (Published 2014-10-22)
Parabolicity of the regular locus of complex varieties
arXiv:1612.01374 [math.CV] (Published 2016-12-05)
Applications of singular connections in symplectic and almost complex geometry
arXiv:1011.5257 [math.CV] (Published 2010-11-23)
Lelong numbers on projective varieties