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**Quantization and the index.**
*(English.
Russian original)*
Zbl 0635.58019

Sov. Phys., Dokl. 31, 877-878 (1986); translation from Dokl. Akad. Nauk SSSR 291, 82-86 (1986).

Some results about a quantization construction, previously developed by the author, are presented. He considers a \(2n\)-dimensional smooth symplectic manifold \(M\) and constructs algebra \(W\) whose elements are sections of the fibrations of (formal) Weyl algebras. The subalgebra \(W_ D\subset W\) of flat sections is defined and the quantizations are constructed as mappings from the set \(C_ h^{\infty}(M)\) of formal power series in \(h\) (Planck’s constant) to the \(W_ D\) algebra. It is shown that the proposed construction has the property that the deformational quantization is possible on any symplectic manifold without any additional conditions.

The integrability conditions which are necessary for the algebra \(W_ D\) to admit asymptotic operational representation are obtained. The trace and the index in the algebra \(W_ D\) are defined and it is shown that the necessary condition mentioned above requires integrability of the index. A formula for the index, which is similar to the Atiyah-Singer formula, is also derived.

The integrability conditions which are necessary for the algebra \(W_ D\) to admit asymptotic operational representation are obtained. The trace and the index in the algebra \(W_ D\) are defined and it is shown that the necessary condition mentioned above requires integrability of the index. A formula for the index, which is similar to the Atiyah-Singer formula, is also derived.

Reviewer: Gh.Zet