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I googled the title on the internet, and arrived at the following result

$$HH_k(D)\cong H_{DR}^{2n-k}(M).$$

Here $M$ is a smooth manifold of dimension $n$, and $D$ is the ring of differential operators on $M$. My first question is

(A): is there a known chain map realizing the above isomorphism?

I think the above isomorphism remains true if we replace $D$ by the ring of differential operators on $M$ with coefficients in a vector bundle $E$. But I am not so sure. If this is indeed the case, my second question is

(B): question (A) with coefficients in $E$?

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    $\begingroup$ If I remember right this was computed in Wodzicki's paper "Cyclic homology of differential operators", but I don't have access right now to know if there is an explicit chain map. The way I think about this for Hochschild cohomology is via deformation quantization for the cotangent bundle, namely the Hochschild cohomology should be (T_poly(T^*(X), [\omega*,) which is Poisson cohomology with a symplectic form. It is then fairly easy to see gives the same thing as cohomology by using the Poisson structure to turn differential forms into polyvector fields. Maybe that can help construct your map? $\endgroup$
    – Daniel Pomerleano
    Commented Mar 11, 2012 at 6:25
  • $\begingroup$ Hi Daniel, I have the paper with me. The argument there was indeed along your comments. But this is exactly where the problem is: to reduce to the cotangent bundle, one uses the spectral sequence associated to the order filtration on D, which does not hint on a chain level map. $\endgroup$
    – tu_junwu
    Commented Mar 11, 2012 at 8:17
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    $\begingroup$ Do you care whether the chain map goes (Hochschild -> de Rham) or (de Rham -> Hochschild)? Or would it be useful to have a third complex with quasi-isomorphisms to (or from) both? The homology isomorphism is natural with respect to open embeddings (in particular diffeomorphisms), but a nontrivial chain map cannot be natural, at least in the direction (de Rham -> Hochschild). You can see this by letting $n=1$, $M=\mathbb R$, and considering the subcomplexes of chains invariant under affine motions of the line. $\endgroup$
    – Tom Goodwillie
    Commented Mar 11, 2012 at 13:08
  • $\begingroup$ Actually I was even expecting maps in both directions, but this is a good example to know. $\endgroup$
    – tu_junwu
    Commented Mar 11, 2012 at 18:32
  • $\begingroup$ Nice question. Also, since $HH_k\cong HH^{2n-k}$ for the algebra of differential operators, I once thought that the most natural map should occur between Hochschild COhomology of differential operators and de rham cohomology of the cotangent space. One can guess a natural formula in degree one, but I have no clue for proving it is a quasi-isomorphism. $\endgroup$
    – Marco Farinati
    Commented Jun 17, 2019 at 1:09

1 Answer 1

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It is not literally what you want, but very close: check Proposition 2.3 and related results in "A Riemann-Roch-Hirzebruch formula for traces of differential operators" by Markus Engeli and Giovanni Felder (http://arxiv.org/abs/math/0702461).

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  • $\begingroup$ This is indeed very close. I will check it out. The only problem from first sight is there the authors consider holomorphic bundles over complex manifolds. $\endgroup$
    – tu_junwu
    Commented Mar 11, 2012 at 18:35
  • $\begingroup$ That is true. However, I have a feeling that might be instructional enough. Sorry for being brief for the moment! $\endgroup$
    – Vladimir Dotsenko
    Commented Mar 11, 2012 at 21:19

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