The author presents an alternate proof of the Bismut-Zhang localization formula of η invariants, when the target manifold is a sphere, by using ideas of mod k index theory instead of the difficult analytic localizati...The author presents an alternate proof of the Bismut-Zhang localization formula of η 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 η 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.展开更多
Around the central theme of 'square root' of the Laplace operator it is shown that the classical Riesz potentials of the first and of the second kind allow for an explicit expression of so-called Hilbert-Dirac...Around the central theme of 'square root' of the Laplace operator it is shown that the classical Riesz potentials of the first and of the second kind allow for an explicit expression of so-called Hilbert-Dirac convolution operators involving natural and complex powers of the Dirac operator.展开更多
基金Project supported by the Cheung-Kong Scholarshipthe Key Laboratory of Pure MathematicsCombinatorics of the Ministry of Education of Chinathe 973 Project of the Ministry of Science and Technology of China.
文摘The author presents an alternate proof of the Bismut-Zhang localization formula of η 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 η 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.
文摘Around the central theme of 'square root' of the Laplace operator it is shown that the classical Riesz potentials of the first and of the second kind allow for an explicit expression of so-called Hilbert-Dirac convolution operators involving natural and complex powers of the Dirac operator.
