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- W1979142664 abstract "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 proof of the Witten holonomy formula (Theorem 1.7) in the special case above. 1. MAIN RESULTS Let M be a closed oriented smooth manifold, G a compact Lie group consisting of diffeomorphisms of M, P -* Z a principal G-bundle over a smooth manifold Z with a connection and D a G-equivariant elliptic operator. Then, for any g e G, the index of D evaluated at g, Index(D, g), is defined by Index(D, g) = tr(glker D) tr(glcokerD) and can be calculated by the well-known fixed point formula (cf. [1], [2] or [5]). On the other hand, the determinant of D evaluated at g, det(D, g), is defined by det(D, g) = det(glker D)/ det(gIcokerD). Then the next proposition is an immediate consequence of the elementary result of Lemma 1 in Appendix. Proposition 1.1. Let g e G be any element of finite order p. Then the next equality holds: det(D, g) = exp p E 1 -e-2?iklp {Index(D) -Index(D, gk)} where Index(D) = Index(D, 1) is the numerical index of D. Received by the editors November 2, 1993. 1991 Mathematics Subject Classification. Primary 58G26; Secondary 58G10." @default.
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- W1979142664 date "1995-07-01" @default.
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- W1979142664 title "On the determinant and the holonomy of equivariant elliptic operators" @default.
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- W1979142664 doi "https://doi.org/10.1090/s0002-9939-1995-1260183-9" @default.
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