A Gauge-Theoretic Approach to the Chern Form of the Canonical Bundle on the Moduli Space of Stable Parabolic Bundles
Wentworth, Richard A
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In this thesis we apply a gauge-theoretic approach to construct the moduli space of stable parabolic bundles on a closed Riemann surface using weighted Sobolev spaces. We study the metric properties of the moduli space, and in particular, we compute the L2 curvature of its canonical bundle. By identifying the canonical bundle with the index bundle of a suitable family of Dolbeault operators, we define a spectral Quillen metric on the canonical bundle via a relative analytic torsion construction first introduced by Müller. We compute the curvature of the canonical bundle with respect to this Quillen metric and find that it consists of the standard Atiyah-Singer term along with a cuspidal contribution coming from the parabolic structure and depending upon the parabolic weights. This gives a new proof of a result of Zograf-Takhtajan.