Invariant star products on \(S^2\) and the canonical trace (Q703012)

In the article under review the authors consider star products on \((S^2,\omega)\), where the symplectic form \(\omega\) is the usual volume form on \(S^2\). As a first step they show that the star products on \((S^2,\omega)\) obtained from other constructions in [\textit{K. Hayasaka, R. Nakayama} and \textit{Y. Takaya}, Phys. Lett. B 553, No. 1--2, 109--118 (2003; Zbl 1006.81086)] and in [\textit{A. Y. Alekseev} and \textit{A. Lachowska}, ``Invariant \(\ast\)-products on coadjoint orbits and the Shapovalov pairing'', preprint math.QA/0308100], which are both known to be SO(3)-invariant, actually coincide. Using the index theorem of Fedosov-Nest-Tsygan [cf. \textit{B. Fedosov}, Deformation quantization and index theory, Mathematical Topics. 9. Berlin: Akademie Verlag (1996; Zbl 0867.58061) and \textit{R. Nest} and \textit{B. Tsygan}, Commun. Math. Phys. 172, No. 2, 223--262 (1995; Zbl 0887.58050)], which establishes a relation between \(\text{Tr}_{\text{can}}(1)\) and the characteristic class of a star product [cf. \textit{P. Deligne}, Sel. Math., New Ser. 1, No. 4, 667--697 (1995; Zbl 0852.58033)] on compact symplectic manifolds, the characteristic class of the considered star product \(\star\) on \((S^2,\omega)\) is computed. Here \(\text{Tr}_{\text{can}}\) denotes the canonical trace on the star product algebra as constructed in [\textit{A. V. Karabegov}, Lett. Math. Phys. 45, No. 3, 217--228 (1998; Zbl 0943.53052)]. To perform the computation of the characteristic class they use the fact that \(\star\) is stronlgy closed [cf. \textit{A. Connes, M. Flato} and \textit{D. Sternheimer}, Lett. Math. Phys. 24, No. 1, 1--12 (1992; Zbl 0767.55005)] and construct explicitly local equivalence transformations from \(\star\) to the Weyl-Moyal star product in local Darboux coordinates, which enables them to find an explicit formula for the trace density \(\mu_{\text{can}}\) which determines \(\text{Tr}_{\text{can}}(f) = \int_{S^2} f \mu_{\text{can}}\) for \(f \in C^\infty (S^2)[[\hbar]]\). The main result of the paper is that the characteristic class of \(\star\) is given by \(\frac{[\omega]}{2\pi \hbar} + \frac{1}{2}c_1 (S^2)\), where \(c_1(S^2)\) denotes the first Chern class of \(S^2\).