Let X be a compact complex manifold and let E be a holomorphic vector bundle on X. Any global holomorphic differential operator D on E induces an endomorphism of $\\text{H}^{\\bullet}(X,E)$. The super-trace of this endomorphism is the super-trace of D. This is a linear functional on the 0-th Hochschild homology of Diff(E), the algebra of global holomorphic differential operators on E. While the Hochschild homology in the usual sense of Diff(E) is too big for explicit computation, there is a notion of completed Hochschild homology of Diff(E) with a very nice property: If HH_i(Diff(E)) denotes the i-th completed Hochschild homology, then HH_i(Diff(E)) is isomorphic to \\text{H}^{2n-i}(X), the 2n-i th cohomology of X with complex coefficients. We shall attempt to outline how the supertrace mentioned above extends to a linear functional on the 0-th completed Hochschild homology of Diff(E), and thus, on H^{2n}(X). A priori, this linear functional depends on E. It however, can be shown that it is precisely the integral over X. This fact also helps one connect the local Riemann-Roch theorems of Nest-Tsygan to the Hirzebruch Riemann-Roch theorem. Analogous results about similar constructions using cyclic homology instead of Hochschild homology are also available.

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Last updated: 24 Mar 2023