Singular Poisson-Kähler geometry of stratified Kähler spaces and quantization

Johannes Huebschmann (1) , U Lille

1 LPP - Laboratoire Paul Painlevé - UMR 8524

In the presence of classical phase space singularities the standard methods are insufficient to attack the problem of quantization.In certain situations the difficulties can be overcome by means of Kähler quantization on stratified Kähler spaces. Such a space is a stratified symplectic space together with a complex analytic structure which is compatible with the stratified symplectic structure; in particular each stratum is a Kähler manifold in an obvious fashion. Holomorphic quantization on a stratified Kähler space then yields a costratified Hilbert space, a quantum object having the classical singularities as its shadow. Given a Kähler manifold with a hamiltonian action of a compact Lie group that also preserves the complex structure, reduction after quantization coincides with quantization after reduction in the sense that not only the reduced and unreduced quantum phase spaces correspond but the invariant unreduced and reduced quantum observables as well

Mots clés: Stratified symplectic space; complex analytic space; stratified Kähler space; reduction and quantization; holomorphic quantization; quantization on a stratified Kähler space; constrained system; invariant theory; hermitian Lie algebra; correspondence principle; Lie-Rinehart algebra; adjoint quotient

Domaines: Géométrie symplectique [math.SG]; Physique mathématique [math-ph]; Physique mathématique [math-ph]

Johannes Huebschmann, U Lille. Singular Poisson-Kähler geometry of stratified Kähler spaces and quantization. Travaux mathématiques, 2011, 19, pp.27-63. ⟨hal-00799330⟩