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Let M be a closed oriented smooth manifold, G a compact Lie group consisting of diffeomorphisms of M, P -> Z a principal G-bundle with a connection and D a G-equivariant elliptic operator. Then a locally constant family of elliptic operators and its determinant line bundle over Z are naturally defined by D . Moreover the holonomy of the determinant line bundle is defined by the connection in P . In this note, we give an explicit formula to calculate the holonomy (Theorem 1.4) and give a proofdoi:10.2307/2160967 fatcat:tq6g7v2gm5hfpkkuej4er5piju