ON THE FUNCTORIALITY OF THE CHERN–SIMONS LINE BUNDLE AND THE DETERMINANT LINE BUNDLE

Abstract

We investigate functorial properties of two hermitian line bundles over the moduli space of flat SU(n)-connections on a closed oriented surface; that is, of the Chern–Simons line bundle and the determinant line bundle. We investigate actions of cyclic subgroups of the mapping class group on them. As a consequence, we show that if we modify the determinant line bundle by the Hodge bundle over the moduli space of Riemann surfaces, then these line bundles are functorially isomorphic. This implies two quantum Hilbert spaces defined by the Chern–Simons line bundle and the modified determinant line bundle are functorially isomorphic.