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Let \(P\) be a Delzant polytope. We show that the quantization of the corresponding toric manifold \(X_{P}\) in toric Kähler polarizations and in the toric real polarization are related by analytic continuation of Hamiltonian flows evaluated at time \(t = {\sqrt{-1}}s\). We relate the quantization of \(X_{P}\) in two different toric Kähler polarizations by taking the time-\({\sqrt{-1}}s\) Hamiltonian “flow” of strongly convex functions on the moment polytope \(P\). By taking \(s\) to infinity, we obtain the quantization of \(X_{P}\) in the (singular) real toric polarization. Recall that \(X_{P}\) has an open dense subset which is biholomorphic to \(({\mathbb {C}}^{*})^{n}\). The quantization of \(X_{P}\) in a toric Kähler polarization can also be described by applying the complexified Hamiltonian flow of the Abreu–Guillemin symplectic potential \(g\), at time \(t={\sqrt{-1}}\), to an appropriate finite-dimensional subspace of quantum states in the quantization of \(T^{*}{\mathbb {T}}^{n}\) in the vertical polarization. By taking other imaginary times, \(t= k {\sqrt{-1}}, k\in {\mathbb {R}}\), we describe toric Kähler metrics with cone singularities along the toric divisors in \(X_{P}\). For convex Hamiltonian functions and sufficiently negative imaginary part of the complex time, we obtain degenerate Kähler structures which are negative definite in some regions of \(X_{P}\). We show that the pointwise and \(L^2\)-norms of quantum states are asymptotically vanishing on negative-definite regions.

Complex symplectomorphisms and pseudo-Kähler islands in the quantization of toric manifolds