On the determinant and the holonomy of equivariant elliptic operators

Author:
Kenji Tsuboi

Journal:
Proc. Amer. Math. Soc. **123** (1995), 2275-2281

MSC:
Primary 58G26; Secondary 58G10

DOI:
https://doi.org/10.1090/S0002-9939-1995-1260183-9

MathSciNet review:
1260183

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Abstract: Let *M* be a closed oriented smooth manifold, *G* a compact Lie group consisting of diffeomorphisms of 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.

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1995-1260183-9

Keywords:
The determinant and the holonomy of elliptic operators

Article copyright:
© Copyright 1995
American Mathematical Society