WZNW actions, holomorphic gauges, and the Kähler structure of moduli spaces

Moduli spaces of stable vector bundles carry a natural Kähler structure, described originally in the Riemann surface case by Narasimhan and in the pioneering work of Atiyah-Bott. Such a Kähler structure is in many ways analogous to the Weil-Petersson metric on moduli spaces of Riemann surfaces, for which a deep relationship with the Liouville functional in Conformal Field Theory was established by Takhtajan and Zograf. In this talk I will describe work in progress on how the ideas of Takhtajan-Zograf can be adapted to vector bundles in three different settings: moduli of stable parabolic bundles in genus 0 and 1, moduli of semistable bundles in genus 1, and Jacobians. In all cases the main tool is an adaptation of the WZNW action of Conformal Field Theory---defined by twisting the so-called chiral models with a topological term---to a functional on singular hermitian metrics on a suitable holomorphic gauge. I will also describe briefly how the previous results can be generalized to moduli spaces of parabolic Higgs bundles.Non UBCUnreviewedAuthor affiliation: University of KielPostdoctora