The Mukai pairing—II: the Hochschild-Kostant-Rosenberg isomorphism

We continue the study of the Hochschild structure of a smooth space that we began in our previous paper, examining implications of the Hochschild-Kostant-Rosenberg theorem. The main contributions of the present paper are:

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we introduce a generalization of the usual notions of Mukai vector and Mukai pairing on differential forms that applies to arbitrary manifolds;

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we give a proof of the fact that the natural Chern character map K₀(X)→HH₀(X) becomes, after the HKR isomorphism, the usual one ; and

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we present a conjecture that relates the Hochschild and harmonic structures of a smooth space, similar in spirit to the Tsygan formality conjecture.