Fedosov differentials and Catalan numbers

Abstract

The aim of the paper is to establish a non-recursive formula for the general solution of Fedosov's 'quadratic' fixed-point equation (Fedosov 1994 J. Diff. Geom. 40 213-38). Fedosov's geometrical fixed-point equation for a differential is rewritten in a form similar to the functional equation for the generating function of Catalan numbers. This allows us to guess the solution. An adapted example for Kaehler manifolds of constant sectional curvature is considered in detail. Also for every connection on a manifold a familiar classical differential will be introduced.

Wick Rotations in Deformation Quantization

“…Note that for complex projective spaces and hyperbolic discs this formula coincides (up to rescaling the formal parameter) with the formula derived in [18,Thm. 3.2.4] for a Fedosov star product with form Ω = 0.…”

“…This formula was already known in the special case of È n and n , [18], where it was derived from the Fedosov construction. Our result therefore allows to compare this approach with phase space reduction without appealing to any abstract classification results, and generalizes it to a larger class of manifolds.…”

“…Proposition 5. 18 The reduced exterior covariant derivative D red on M red is the one for the Levi-Civita connection associated to the (not necessarily definite) reduced metric g red ∈ S 2 (M red ), which is defined by…”

“…Alternatively, one can e.g. use Berezin dequantization [9], a Lie algebraic approach [1] or an explicit solution of the recursive equations coming from the Fedosov construction [18].…”

Moyal-Weyl deformations of $\mathrm{DGA}$ and $\mathrm{DGCA}$

“…Formula 3 defines a deformed associative product [10], for a detailed proof of the associativity of 3 we refer to [15]. This product is an important ingredient in the Fedosov deformation quantization of symplectic manifolds, see [12] for a construction of Fedosov ⋆ products on Kähler manifolds of constant sectional curvature.…”