The first Chern form on moduli of parabolic bundles

Abstract

For moduli space of stable parabolic bundles on a compact Riemann surface, we derive an explicit formula for the curvature of its canonical line bundle with respect to Quillen's metric and interpret it as a local index theorem for the family of dbar-operators in associated parabolic endomorphism bundles. The formula consists of two terms: one standard (proportional to the canonical Kaehler form on the moduli space), and one nonstandard, called a cuspidal defect, that is defined by means of special values of the Eisenstein-Maass series. The cuspidal defect is explicitly expressed through curvature forms of certain natural line bundles on the moduli space related to the parabolic structure. We also compare our result with Witten's volume computation.