Morita equivalence bimodules for Wick type star products. (English) Zbl 1091.53063

Neumaier, Nikolai; Waldmann, Stefan

J. Geom. Phys. 47, No. 2-3, 177-196 (2003).

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.

Reviewer: Weiping Li (Stillwater)

MSC:

53D55	Deformation quantization, star products
81S10	Geometry and quantization, symplectic methods

Keywords:

deformation quantization; Fedosov construction; Weyl and (anti)-Wick products; Kähler manifold; Morita equivalence

Citations:

Zbl 0812.53034; Zbl 1009.53063

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