$Local^{3}$ Index Theorem
$Local^{3}$ Index Theorem
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$局部^{3}$索引定理
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发表时间：
2011
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通讯作者：
Nicolae Teleman
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作者：
Nicolae Teleman
$Local^{3}$索引定理意味着$Local(Local(Local \;Index \; Theorem)))$。 $本地\;指数 \; Theorem$ 是 Connes-Moscovici 局部索引定理 \cite{Connes-Moscovici1}, \cite{Connes-Moscovici2}。第二个“局部”是指定位于基础代数的某个可分离子环的循环同调，而最后一个是指Alexander-Spanier型循环同调。 Connes-Moscovici 的工作基于与光滑流形 $M$ 上的椭圆伪微分算子 $A$ 相关的算子 $R(A) = \mathbf{P} - \mathbf{e}$，其中 $\mathbf{P}$、$\mathbf{e}$ 是幂等的，请参见 \cite{Connes-Moscovici1}，第 12 页。 353. 算子$R(A)$有两个主要优点：它是一个平滑算子，其分布核位于$M \times M$对角线的任意小邻域内。运算符 $R(A)$ 也有两个缺点： -i) 它不是幂等的（因此它不具有真正的 Connes-Chern 特征）； -ii) 即使它是幂等的，它的 Connes-Chern 特征也属于平滑算子代数的循环同调（有 \emph{任意} 支持，即 \emph{trivial}。本文针对这两个挫折提出的困难提出了一种新的解决方案。对于这一点 -i)，我们证明虽然 $R(A)$ 不是幂等的，但它满足恒等式 $ (\mathbf{R}(A))^{2} \;=\; \mathbf{R}(A) - [\mathbf{R}(A) 。 e + e .\mathbf{R}(A) ]。 $ 我们证明，如果平滑算子代数的循环同调复形对于可分离子代数 $\Lambda = \mathbb{C} + \mathbb{C} 而言，算子 $R(A)$ 具有真正的陈氏特征。 e$，参见第 1 节。 8.1.对于-ii)，我们引入了\emph{local}循环同调的概念；这是在 Alexander-Spanier 同调的基础上构建的，即通过平滑算子的分布支持来过滤代数的循环同调复形链，请参见第 1 节。 7. 使用这些新工具，我们重新表述了 Connes-Moscovici 局部指数定理，请参阅定理 23，第 2 节。 9. 作为该定理的推论，我们证明平滑算子代数的 \emph{local} 循环同源性至少与基流形的 Alexander-Spanier 同源性一样大。 Connes-Moscovici 局部指数定理目前的重新表述为新的研究开辟了道路，请参见第 2 节。 10.
$Local^{3}$ Index Theorem means $Local(Local(Local \;Index \; Theorem)))$. $Local \; Index \; Theorem$ is the Connes-Moscovici local index theorem \cite{Connes-Moscovici1}, \cite{Connes-Moscovici2}. The second "Local" refers to the cyclic homology localised to a certain separable subring of the ground algebra, while the last one refers to Alexander-Spanier type cyclic homology. The Connes-Moscovici work is based on the operator $R(A) = \mathbf{P} - \mathbf{e}$ associated to the elliptic pseudo-differential operator $A$ on the smooth manifold $M$, where $\mathbf{P}$, $\mathbf{e}$ are idempotents, see \cite{Connes-Moscovici1}, Pg. 353. The operator $R(A)$ has two main merits: it is a smoothing operator and its distributional kernel is situated in an arbitrarily small neighbourhood of the diagonal in $M \times M$. The operator $R(A)$ has also two setbacks: -i) it is not an idempotent (and therefore it does not have a genuine Connes-Chern character); -ii) even if it were an idempotent, its Connes-Chern character would belong to the cyclic homology of the algebra of smoothing operators (with \emph{arbitrary} supports, which is \emph{trivial}. This paper presents a new solution to the difficulties raised by the two setbacks. For which concerns -i), we show that although $R(A)$ is not an idempotent, it satisfies the identity $ (\mathbf{R}(A))^{2} \;=\; \mathbf{R}(A) - [\mathbf{R}(A) . e + e .\mathbf{R}(A) ]. $ We show that the operator $R(A)$ has a genuine Chern character provided the cyclic homology complex of the algebra of smoothing operators is \emph{localised} to the separable sub-algebra $\Lambda = \mathbb{C} + \mathbb{C} . e$, see Sect. 8.1. For which concerns -ii), we introduce the notion of \emph{local} cyclic homology; this is constructed on the foot-steps of the Alexander-Spanier homology, i.e. by filtering the chains of the cyclic homology complex of the algebra of smoothing operators by their distributional support, see Sect. 7. Using these new instruments, we give a reformulation of the Connes-Moscovici local Index Theorem, see Theorem 23, Sect. 9. As a corollary of this theorem, we show that the \emph{local} cyclic homology of the algebra of smoothing operators is at least as big as the Alexander-Spanier homology of the base manifold. The present reformulation of Connes-Moscovici local index theorem opens the way to new investigations, see Sect. 10.