The determinant bundle on the moduli space of stable triples over a curve

Abstract

We construct a holomorphic Hermitian line bundle over the moduli space of stable triples of the form (E₁, E₂, Φ), where E₁ and E₂ are holomorphic vector bundles over a fixed compact Riemann surface X, and Φ: E₂ → E₁ is a holomorphic vector bundle homomorphism. The curvature of the Chern connection of this holomorphic Hermitian line bundle is computed. The curvature is shown to coincide with a constant scalar multiple of the natural Kahler form on the moduli space. The construction is based on a result of Quillen on the determinant line bundle over the space of Dolbeault operators on a fixed C^∞ Hermitian vector bundle over a compact Riemann surface.