## 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

### Citations:

Zbl 1024.53057; Zbl 0887.58050
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