An algebraic index theorem for Poisson manifolds

The formality theorem for Hochschild chains of the algebra of functions on a smooth manifold gives us a version of the trace density map from the zeroth Hochschild homology of a deformation quantization algebra to the zeroth Poisson homology. We propose a version of the algebraic index theorem for a Poisson manifold which is based on this trace density map.

💡 Research Summary

The paper develops a new algebraic index theorem for arbitrary Poisson manifolds by exploiting the formality theorem for Hochschild chains of the algebra of smooth functions. Starting from Kontsevich’s formality, which provides an L∞‑quasi‑isomorphism between multivector fields and multidifferential operators, the authors extend the construction to the Hochschild chain complex C·(Aℏ,Aℏ) of a deformation quantization algebra Aℏ on a manifold M. Using the Shoikhet‑Willwacher chain‑to‑chain formality map, they obtain an L∞‑module morphism 𝒰 that intertwines the Hochschild differential with the Poisson differential. In particular, on degree zero they define a linear trace‑density map

 τ : HH₀(Aℏ) → HP₀(M,π),

which sends a Hochschild 0‑cycle (i.e., a trace on the quantized algebra) to a Poisson 0‑cycle, i.e., a density on the underlying Poisson manifold. The map τ respects the ℏ‑formal expansion, reduces to the classical Poisson trace as ℏ→0, and commutes with the Batalin‑Vilkovisky operator, thereby preserving the full BV‑algebra structure.

With τ in hand, the authors formulate an index theorem that parallels the classical Atiyah‑Singer/Alvarez‑Gaumé‑Witten picture but works without any symplectic or Kähler hypothesis. Given a K‑theory class