[Paper Review] Almost Kaehler deformation quantization

Alexander Karabegov, Martin Schlichenmaier|ArXiv.org|Feb 21, 2001

Homotopy and Cohomology in Algebraic Topology19 references3 citations

TL;DR

This paper constructs a Fedosov-type deformation quantization on arbitrary almost-Kähler manifolds using a natural affine connection with torsion that respects the almost-Kähler structure. The key result is that the characteristic class of the resulting star-product is $[\varkappa] = -(1/2i)\varepsilon$, where $\varepsilon$ is the canonical class of the almost complex structure, generalizing earlier results from Kähler to non-integrable almost-Kähler settings.

ABSTRACT

We use a natural affine connection with nontrivial torsion on an arbitrary almost-Kaehler manifold which respects the almost-Kaehler structure to construct a Fedosov-type deformation quantization on this manifold.

Motivation & Objective

• To extend Fedosov's deformation quantization framework to almost-Kähler manifolds, which are not necessarily Kähler due to non-integrable almost complex structures.
• To construct a differential star-product on arbitrary almost-Kähler manifolds using a natural affine connection with nontrivial torsion that preserves the almost-Kähler structure.
• To compute the characteristic class of the resulting star-product and show it depends on the canonical class $\varepsilon$ of the almost complex manifold.
• To generalize the known result for Kähler manifolds, where the star-product has characteristic class $(1/i\nu)[\omega] - (1/2i)\varepsilon$, to the almost-Kähler case.

Proposed method

• Use of the Yano connection—a natural affine connection on almost-Kähler manifolds that preserves the almost complex structure and metric but has nontrivial torsion when the complex structure is non-integrable.
• Adaptation of Fedosov's formalism to connections with torsion by modifying the differential complex and curvature terms in the star-product construction.
• Construction of a twisted differential operator $D'$ on the Weyl algebra bundle, incorporating the curvature and torsion of the Yano connection.
• Definition of a flat connection $\nabla'$ on the Weyl algebra bundle via a modified curvature term $r'$, ensuring the existence of a unique flat section $\tau'(f)$ for each smooth function $f$.
• Use of the symbol map $\tau'$ to define the star-product $f \ast g = \tau'(f) \cdot \tau'(g) \big|_{y=0}$, with the product computed via the Weyl multiplication.
• Computation of the characteristic class via the curvature term $\varkappa = \delta(r'^{(3)}_1)$, leading to the expression $[\varkappa] = -(1/2i)\varepsilon$.

Experimental results

Research questions

• RQ1 Can Fedosov's deformation quantization be generalized to almost-Kähler manifolds, which lack integrable complex structures?
• RQ2 What is the characteristic class of the star-product constructed on an almost-Kähler manifold using a torsionful connection?
• RQ3 How does the canonical class $\varepsilon$ of the almost complex structure appear in the characteristic class of the star-product?
• RQ4 Does the star-product constructed via the Yano connection reduce to the known star-product with separation of variables in the Kähler case?
• RQ5 Is the characteristic class $[\varkappa]$ expressible purely in terms of geometric invariants of the almost-Kähler structure?

Key findings

• The star-product constructed on an almost-Kähler manifold using the Yano connection is differential and normalized, with $C_1(f,g) = (i/2)\{f,g\}$.
• The characteristic class of the star-product is $[\varkappa] = -(1/2i)\varepsilon$, where $\varepsilon$ is the canonical class of the almost complex structure.
• The curvature term $\varkappa$ is computed as $\varkappa = -i\Delta(R + \nabla r^{(2)})$, with $\Delta$ denoting the Hochschild degree projection.
• The form $\lambda = \Delta(\nabla r^{(2)})$ is exact, equal to $d\mu$ for a globally defined one-form $\mu = (1/6)J^t_s T^s_{tl} dx^l$, ensuring consistency in the characteristic class computation.
• In the Kähler case, the star-product coincides with the known star-product with separation of variables, confirming consistency with prior results.
• The canonical class $\varepsilon$ is defined as the first Chern class of the $(1,0)$-subbundle of the complexified tangent bundle, and its appearance in the characteristic class generalizes the Kähler case formula.