A New Look at Unitarity in Quantization Commutes with Reduction for Toric Manifolds

Abstract

Abstract

Abstract For a symplectic manifold M with the Hamiltonian action of a compact Lie group G , we give a new perspective on the problem of unitarity in the quantization commutes with reduction correspondence. We argue that this correspondence leads naturally to the introduction of what we call Fourier polarizations, whose real directions include the Hamiltonian vector fields of all invariant functions of the moment map. For a Fourier polarization, quantization commutes with reduction unitarily. For a general G -invariant polarization, $$\mathcal {P}$$ P , unitarity in the quantization commutes with reduction correspondence can then be formulated in terms of the existence of a Fourier polarization, $$\mathcal {P}_F$$ P F , such that the quantizations with respect to $$\mathcal {P}$$ P and to $$\mathcal {P}_F$$ P F are unitarily equivalent. We describe this perspective in detail for symplectic toric manifolds, by considering symplectic reductions associated to the action of a subtorus $$T^p\subset T^n$$ T p ⊂ T n . We consider half-form quantization in Fourier (mixed) polarizations $$\mathcal {P}_\infty $$ P ∞ , whose real directions are generated by (the Hamiltonian vector fields of the) components of the $$T^p$$ T p moment map. We show that Fourier mixed polarizations of this type can be obtained by Mabuchi geodesic rays of toric Kähler structures, starting at a given initial structure and with velocity given by the norm square of the moment map of the torus subgroup. We lift these geodesic rays to the quantum bundle via generalized coherent state transforms, which define equivariant isomorphisms between quantum Hilbert spaces for the Kähler and Fourier polarizations, where the later are obtained at infinite geodesic time. Unitarity in quantization commutes with reduction in these cases then becomes equivalent to the unitarity of these isomorphisms. In general, however, in the toric case, these isomorphisms are not unitary.