On the determinant and the holonomy of equivariant elliptic operators

Abstract:

Let M be a closed oriented smooth manifold, G a compact Lie group consisting of diffeomorphisms of $M,P \to Z$ 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 proof of the Witten holonomy formula (Theorem 1.7) in the special case above.