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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.).

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