Mathematical Research Letters

Volume 12 (2005)

Number 5

A Simple Algebraic Proof of the Algebraic Index Theorem

Pages: 655 – 672



PoNing Chen

Vasiliy Dolgushev


In math.QA/0311303 B. Feigin, G. Felder, and B. Shoikhet proposed an explicit formula for the trace density map from the quantum algebra of functions on an arbitrary symplectic manifold $\M$ to the top degree cohomology of $\M$. They also evaluated this map on the trivial element of $K$-theory of the algebra of quantum functions. In our paper we evaluate the map on an arbitrary element of $K$-theory, and show that the result is expressed in terms of the $\hA$-genus of $\M$, the Deligne-Fedosov class of the quantum algebra, and the Chern character of the principal symbol of the element. For a smooth (real) symplectic manifold (without a boundary), this result implies the Fedosov-Nest-Tsygan algebraic index theorem.

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