Wick rotations in deformation quantization

Abstract

We study formal and non-formal deformation quantizations of a family of manifolds that can be obtained by phase space reduction from [Formula: see text] with the Wick star product in arbitrary signature. Two special cases of such manifolds are the complex projective space [Formula: see text] and the complex hyperbolic disc [Formula: see text]. We generalize several older results to this setting: The construction of formal star products and their explicit description by bidifferential operators, the existence of a convergent subalgebra of “polynomial” functions, and its completion to an algebra of certain analytic functions that allow an easy characterization via their holomorphic extensions. Moreover, we find an isomorphism between the non-formal deformation quantizations for different signatures, linking, e.g., the star products on [Formula: see text] and [Formula: see text]. More precisely, we describe an isomorphism between the (polynomial or analytic) function algebras that is compatible with Poisson brackets and the convergent star products. This isomorphism is essentially given by Wick rotation, i.e. holomorphic extension of analytic functions and restriction to a new domain. It is not compatible with the [Formula: see text]-involution of pointwise complex conjugation.