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non-commutative global analytic index theory

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The motivation for the development of non-commutative index theory

There has been various non-commutative index theorems, such as for example, the Connes-Moscovisci index theorem?, and the Kashiwara-Schapira index theorem?, the Arthur-Selberg trace formula and the Lefschetz trace formula. All these index type results should be treated in a uniform setting. This ideas was present in the litterature since a long time (Dold maybe; see the references at the end of this section).

It is not yet clear how one may give an optimal setting for a non-commutative global analytic index theory, but here is what one may try to do: using ∞\infty-stacks (e.g., group actions) and formal stacks (foliations), one may try to treat Connes-Moscovisci formulae in a more geometric way. However, if one looks at the π’Ÿ\mathcal{D}-module setting, for example, one sees a deeply non-commutative situation. This means that the following generalized geometry may be use to define a very general notion of trace and Chern character:

  1. First, one may try to work with an ∞\infty-topos XX together with a sheaf of associative algebras π’œ\mathcal{A}. The ideas here extend the work of Kashiwara-Schapira and someone else that will be cited later on: one will use the category BiMod f(π’œ)BiMod_f(\mathcal{A}) of modules on π’œ\mathcal{A} with good finiteness properties (e.g., coherent good in the case of π’Ÿ\mathcal{D}-modules; perfect for π’ͺ\mathcal{O}-modules on a stack; coherent for π’ͺ\mathcal{O}-modules on a usual space, etc…). This category is equipped with two monoidal structure. Let’s use the first (left) one, that is given by

    (β„³,𝒩)β†¦β„³βˆ˜π’©:=β„³βŠ— π’œπ’©. (\mathcal{M},\mathcal{N})\mapsto \mathcal{M}\circ\mathcal{N}:=\mathcal{M}\otimes_\mathcal{A}\mathcal{N}.

    Remark that one may define the notion of direct sum of π’œ\mathcal{A}-bimodule, denoted βŠ•\oplus. We thus actually have a bimonoidal category

    (BiMod f(π’œ),βŠ•,∘)(BiMod_f(\mathcal{A}),\oplus,\circ)

    that is not symmetric in ∘\circ but symmetric in βŠ•\oplus. This is thus a categorification of an associative ring. One may easily define the notion of a seminorm |β‹…||\cdot| on such an associative categorical ring (Ac,βŠ•,∘)(\Ac,\oplus,\circ), following the approach explained in generalized global analytic geometry, and also define a Berkovich spectrum β„³(π’œ,βŠ•,∘)\mathcal{M}(\mathcal{A},\oplus,\circ). This will give a topological space or a GG-topological space (or an ∞\infty-topos, in the Γ©tale topology situation) together with a sheaf Uβ†¦π’œ(U)U\mapsto \mathcal{A}(U) of seminormed associative categorical rings. Remark that to every multiplicative seminorm on π’œ\mathcal{A}, one may associate a prime ideal in π’œ\mathcal{A}. We are thus in a conceptually abstract situation that is very close to the Toen-Vezzosi approach to the Chern character (the monoidal structure is not symmetric).

  2. The definition of a categorical trace for objects of π’œ\mathcal{A} acting on π’œ\mathcal{A} by the left tensor product ∘\circ may be given if π’œ\mathcal{A} is equipped with a kind of rigidity structure. In the π’Ÿ\mathcal{D}-module setting, if we work with bimodules such as β„³βŠ Dβ„³\mathcal{M}\boxtimes D\mathcal{M}, Kashiwara and Schapira use the notion of trace kernel to define the corresponding class in Hochschild homology. One may try to give a similar construction using rigidity, e.g., the fact that a natural dual for the above bimodule is simply Dβ„³Ο‰βŠ Ο‰β„³D\mathcal{M}\omega\boxtimes \omega\mathcal{M}. The diagonal Ξ”\Delta seems to play an important role here.

Now a natural constraint on this situation if we look at the Kashiwara and Schapira results is to use the associated Hochschild or negative cyclic homology to define a trace: one must not suppose that (π’œ,∘)(\mathcal{A},\circ) is a rigid monoidal category, because this is not true in the example of elliptic pairs (Atiyah-Singer) (the corresponding trace kernel is given by tensor product of a π’Ÿ\mathcal{D}-module kernel with a constructible kernel). One should only suppose something weaker, related to the fact that one wants a trace to be defined on cyclic and/or Hochschild cohomology of the situation (a global invariant on XX, that has a meaning in the Atiyah-Singer situation).

To be continued.

Possible references

Connes-Moscovisci

Kashiwara-Schapira

Selberg

Arthur

Ramados-Tang-Tsen Hochschild-Lefschetz class for π’Ÿ\mathcal{D}-modules

PoNing Chen, Vasiliy Dolgushev: A Simple Algebraic Proof of the Algebraic Index Theorem

Last revised on January 1, 2015 at 18:09:25. See the history of this page for a list of all contributions to it.