Reduction of prequantized ${\rm Mp}\sp c$ structures

Abstract

The Marsden-Weinstein reduction procedure fashions new symplectic manifolds from Hamiltonian actions on old symplectic manifolds. It is of considerable interest to determine the way in which quantization of the reduced phase spaces relates to that of the original phase spaces. In particular, it is natural to ask for conditions under which prequantization data may be passed through the operation of reduction. Note that reduced phase spaces can fail to be metaplectic, so in general it cannot be expected that traditional prequantization data (Kostant-Souriau line bundle plus metaplectic structure: see [2, 3] for example) will pass to reduced phase spaces; moreover, even when reduced phase spaces admit metaplectic structures, these do not generally arise from metaplectic structures on the original phase spaces in a natural manner. Following Hess, we take the view that prequantization data consists of a prequantized Mpc structure: see [6] for full details. In this note, we describe when and how such prequantization data may be passed to reduced phase spaces . En route, we encounter a generalized version of the Bohr-Wilson-Sommerfeld rule as an obstruction to this passage. We illustrate our account by reference to coadjoint orbits as reduced phase spaces.