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| $\eta$-INVARIANT AND CHERN-SIMONS CURRENT |
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Citation： |
ZHANG Weiping.$\eta$-INVARIANT AND CHERN-SIMONS CURRENT[J].Chinese Annals of Mathematics B,2005,26(1):45~56 |
| Page view： 1083
Net amount： 863 |
Authors： |
ZHANG Weiping; |
Foundation： |
Project supported by the Cheung-Kong Scholarship, the Key Laboratory of Pure Mathematics and
Combinatorics of the Ministry of Education of China, and the 973 Project of the Ministry of Science
and Technology of China. |
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| Abstract： |
The author presents an alternate proof of the Bismut-Zhang localization formula
of $\eta$ invariants, when the target manifold is a sphere, by using ideas of mod k index
theory instead of the difficult analytic localization techniques of Bismut-Lebeau. As a
consequence, it is shown that the R/Z part of the analytically defined $\eta$ invariant of
Atiyah-Patodi-Singer for a Dirac operator on an odd dimensional closed spin manifold
can be expressed purely geometrically through a stable Chern-Simons current on a
higher dimensional sphere. As a preliminary application, the author discusses the relation
with the Atiyah-Patodi-Singer R/Z index theorem for unitary flat vector bundles,
and proves an R refinement in the case where the Dirac operator is replaced by the
Signature operator. |
Keywords： |
Direct image, $\eta$-Invariant, Chern-Simons current, mod k index theorem |
Classification： |
58J |
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