A canonical connection on bundles on Riemann surfaces and Quillen connection on the theta bundle

We investigate the symplectic geometric and also differential aspects of moduli space connections on a compact connected Riemann surface X . Fix theta characteristic K 1 / 2 ; it defines divisor M stable vector bundles rank r degree zero. Given bundle E ? lying outside divisor, we construct natural holomorphic connection that depends holomorphically Using this connection, canonical isomorphism between following two: (1) C pairs ( , D ) where is (2) Conn ? given by sheaf line associated to divisor. The above structure preserving, moves as runs over family surfaces.