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Morita equivalence bimodules for Wick type star products. (English) Zbl 1091.53063
Deformation quantization is a deformation algebraic structures from classical physics to quantum physics. On Kähler manifolds $$M$$, one has a compatible complex structure which carries holomorphic and anti-holomorphic bi-differential operators. There are globally well-defined operators $P = g^{k\overline{l}}i_s(z_k) \otimes i_s(\overline{z}_l), \;\;\;\overline{P} = g^{k\overline{l}}i_s(\overline{z}_k)\otimes i_s(z_l),$ for the Kähler metric $$g=(g_{kl})$$ and the fiber-wise products $a \circ_k b = \mu \circ e^{(k+1)\lambda P + (k-1)\lambda \overline{P}} a\otimes b$ on a complex bundle $$E \to M$$ through the Fedosov construction. Note that $$k=0$$ the product gives the Weyl product $$\star_{\text{Weyl}}$$, $$k=1$$ the Wick product $$\star_{\text{Wick}}$$ and $$k=-1$$ the anti-Wick product $$\star_{\text{Wick}^{-1}}$$ on the $$C[[\lambda]]$$-module $$\prod_{s=0}^{\infty} \Gamma^{\infty}(V^sT^*M \otimes \wedge^{\bullet}T*M \otimes F)[[\lambda]]$$ for $$F = E$$ or $$\text{End}(E)$$.
On Kähler manifolds, the three canonical star products are related by their characteristic classes $c(\star_{\text{Wick}^{\pm}}) = c(\star_{\text{Weyl}} \pm \pi i c_1(\wedge^{(n, 0)}T^*M).$ A deformation quantization of $$E$$ is a deformation right module structure $$\circ_k$$ for $$\Gamma^{\infty}(E)[[\lambda]]$$ with respect to $$\star_k$$, where $$\circ_k$$ always exists uniquely up to equivalence, and a deformation left module structure $$\circ_k^{'}$$ for $$\Gamma^{\infty}(\text{End}(E))[[\lambda]]$$ with respect to $$\star_k^{'}$$. The bi-module structure gives a Morita equivalence bi-module $$(\Gamma^{\infty}(\text{End}(E))[[\lambda]], \circ_k^{'}, \circ_k)$$ for the deformed algebras $$(\Gamma^{\infty}(\text{End}(E))[[\lambda]] , \circ_k^{'})$$ and $$(\Gamma^{\infty}(E)[[\lambda]], \circ_k)$$.
In a symplectic case with almost complex structure, two stars $$\star_k$$ and $$\star_k^{'}$$ are Morita equivalent if and only if there exists a symplectic diffeomorphism $$\psi$$ of $$M$$ such that $$\psi^* c(\star_k^{'}) - c(\star_k)$$ is an integral de Rham class. In B. V. Fedosov [J. Differ. Geom. 40, 213–238 (1994; Zbl 0812.53034)] and S. Waldmann [Lett. Math. Phys. 60, 157–170 (2002; Zbl 1009.53063)], the concrete bi-module structure necessarily exists and depends on non-canonical choices.
The main result of the paper under review shows that using the Kähler geometry there is a canonical construction of a Morita equivalence structure for the canonical line bundle on the Kähler manifold in Theorem 4 and Corollary 3. The local expressions for the bi-module multiplications $$\star_{\text{Wick}^{\pm}} = \star_{\pm 1}$$ are computed through the Fedosov construction (section 2) and the star products (section 3), hence explicit existence of bi-module structure. The 1-parameter family of star products links the Weyl product and (anti)-Wick products. The canonical Morita equivalence is contained in section 7.

##### MSC:
 53D55 Deformation quantization, star products 81S10 Geometry and quantization, symplectic methods
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##### References:
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