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

2014/11/11 by William D. Kirwin, Kirwin, William D., José M. Mourão +3
Mathematics · #14M25 #32G05 #53C55 #53D50 #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #Differential Geometry (math.DG) #FOS: Mathematics #Geometry and complex manifolds #math.DG #msc:14M25 #msc:32G05 #msc:53C55 #msc:53D50

paper · pdf · doi:10.48550/arxiv.1411.2793

25 pages

arxiv created 2014/11/11 · openalex publication_date 2014/11/11 · arxiv updated 2014/11/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

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 = √(-1) s. We relate the quantization of X_P in two different toric Kähler polarizations by taking the time-√(-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 (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=√(-1), to an appropriate finite-dimensional subspace of quantum states in the quantization of T^*\mathbbTⁿ in the vertical polarization. By taking other imaginary times, t= k √(-1), k∈ 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 L²-norms of quantum states are asymptotically vanishing on negative-definite regions.

Citations

• The exponential map of the complexification of \em Ham in the real-analytic case
• Connecting toric manifolds by conical Kahler-Einstein metrics
• Complex structures on tangent bundles of Riemannian manifolds
• On complexified analytic Hamiltonian flows and geodesics on the space of Kahler metrics

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