The noncommutative family Atiyah-Patodi-Singer index theorem

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• 作者：Yong Wang ^{wangy581@nenu.edu.cn" class="auth_mail" title="E-mail the corresponding author}
• 关键词：58J20 ; 19K56
• 刊名：Journal of Geometry and Physics
• 出版年：2016
• 出版时间：December 2016
• 年：2016
• 卷：110
• 期：Complete
• 页码：44-59
• 全文大小：509 K

文摘

In this paper, we define the eta cochain form and prove its regularity when the kernel of a family of Dirac operators is a vector bundle. We decompose the eta form as a pairing of the eta cochain form with the Chern character of an idempotent matrix and we also decompose the Chern character of the index bundle for a fibration with boundary as a pairing of the family Chern–Connes character for a manifold with boundary with the Chern character of an idempotent matrix. We define the family formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0393044016301759&_mathId=si1.gif&_user=111111111&_pii=S0393044016301759&_rdoc=1&_issn=03930440&md5=8b53532ef67f596030043501689f40e8" title="Click to view the MathML source">bflow="scroll">b-Chern–Connes character and then we prove that it is entire and give its variation formula. By this variation formula, we prove another noncommutative family Atiyah–Patodi–Singer index theorem. Thus, we extend the results of Getzler and Wu to the family case.