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The characteristic classes of Morita equivalent star products on symplectic manifolds. (English) Zbl 1036.53068
The paper provides a characterization of Morita equivalent star products on symplectic manifolds in terms of Deligne’s relative class [S. Gutt and J. Rawnsley, J. Geom. Phys. 29, 347–392 (1999; Zbl 1024.53057)]. The main result is that two star products \(\star\) and \(\star'\) on a symplectic manifold \((M,\omega)\) are Morita equivalent if and only if there exists a symplectomorphism \(\psi: M \to M\) such that the relative class is \(2\pi i\)-integral. The proof is based on the identification of the deformation space \(\text{Def} (M,\omega)\) with \(\frac{1}{\i \lambda}[\omega] + H^2_{dR}(M)[[\lambda]]\) [R. Nest and B. Tsygan, Commun. Math. Phys. 172, 223–262 (1995; Zbl 0887.58050)] and on the author’s explicit description of the action of \(\text{Pic} (M) \cong H^2(M,\mathbb Z)\) on \(\text{Def}(M,\omega)\): for the line bundle \(L\) with the first Chern class \(c_1(L)\), the action of \(L\) on \(\text{Def} (M,\omega)\) is given by \(\Phi_L([\omega_\lambda]) = [\omega_\lambda] + 2\pi ic_1(L)\). The authors derive this using a local description of deformed line bundles over \(M\) and the Čech-cohomological approach to Deligne’s relative class developed in (S. Gutt and J. Rawnsley, loc. cit.).

53D55 Deformation quantization, star products
81R60 Noncommutative geometry in quantum theory
81S10 Geometry and quantization, symplectic methods
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