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

TL;DR

The article develops a half-form corrected quantization framework for toric manifolds along Mabuchi geodesics, producing a singular mixed polarization \\mathcal{P}_{\\infty} that combines p real directions with n-p holomorphic directions. A T^n-equivariant generalized coherent state transform (gCST) links the holomorphic quantization associated with an initial toric Kähler polarization to the mixed polarization, yielding a basis of distributional sections indexed by integral points of the moment polytope and a unitary isomorphism to the direct sum of Kähler reductions across all Bohr–Sommerfeld levels. The key contribution is that quantization commutes with reduction for the mixed polarization, with a precise unitary (up to a universal constant) correspondence to the reductions M_{\\underline m}, while highlighting that the same unitarity fails generically between the initial Kähler polarization and the mixed polarization. The work provides a natural perspective in which unitarity questions in quantization with reduction are recast as unitary equivalence questions between different polarizations, suggesting broader applicability to Fourier-type (Fourier) polarizations beyond the toric setting.

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 Kähler polarization $\mathcal{P}_0$ and then following Mabuchi rays of toric Kähler 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 Kähler 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 Kähler reductions of the initial Kähler 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.

Theorems & Definitions (39)

• Theorem 1.1: Theorem \ref{['prop:evol-polarization']}
• Definition 1.1: Definition \ref{['correctedQS']}
• Theorem 1.2: Theorem\ref{['thm_polsectionsinside']}
• Theorem 1.3: Theorem \ref{['convsections']}
• Theorem 1.4: Theorem \ref{['lemma-norms']}
• Theorem 1.5: Theorem \ref{['thmqcr']}
• Corollary 1.1: Corollary \ref{['cor_qrunitary']}
• Definition 2.1
• Definition 2.2
• Remark 2.1