A new look at unitarity in quantization commutes with reduction for toric manifolds

TL;DR

This paper explores how a new class of polarizations in toric manifolds offers a fresh perspective on the unitarity of quantization commutes with reduction, revealing conditions under which this property holds or fails.

Contribution

It introduces mixed polarizations associated with subtorus actions and demonstrates their role in establishing unitarity of quantization commutes with reduction in toric manifolds.

Findings

Polarizations $\u211d$ decompose Hilbert space into reductions.

Quantization commutes unitarily with reduction for $ty$ polarization.

Unitarity for initial K"ahler polarization depends on equivalence with $ty$ polarization.

Abstract

For a symplectic toric manifold we consider half-form quantization in mixed polarizations $\mathcal{P}_\infty$, associated to the action of a subtorus $T^p\subset T^n$. The real directions in these polarizations are generated by components of the $T^p$ moment map. Polarizations of this type can be obtained by starting at a toric K\"ahler polarization $\mathcal{P}_0$ and then following Mabuchi rays of toric K\"ahler polarizations generated by the norm square of the moment map of the torus subgroup. These geodesic rays are lifted to the quantum bundle via a generalized coherent state transform (gCST) and define equivariant isomorphisms between Hilbert spaces for the K\"ahler polarizations and the Hilbert space for the mixed polarization. The polarizations $\mathcal{P}_\infty$ give a new way of looking at the problem of unitarity in the quantization commutes with reduction with respect to the $T^p$-action, as follows. The prequantum operators for the components of the moment map of the $T^p$-action act diagonally with discrete spectrum corresponding to the integral points of the moment polytope. The Hilbert space for the quantization with respect to $\mathcal{P}_\infty$ then naturally decomposes as a direct sum of the Hilbert spaces for all its quantizable coisotropic reductions which, in fact, are the K\"ahler reductions of the initial K\"ahler polarization $\mathcal{P}_0$. This will be shown to imply that, for the polarization $\mathcal{P}_\infty$, quantization commutes unitarily with reduction. The problem of unitarity in quantization commutes with reduction for $\mathcal{P}_0$ is then equivalent to the question of whether quantization in the polarization $\mathcal{P}_0$ is unitarily equivalent with quantization in the polarization $\mathcal{P}_\infty$. In fact, this does not hold in general in the toric case.