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

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 $t=\sqrt{-1}s$. 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 $({\mathbb{C}}^{\ast }{)}^n$. 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 $T^{\ast }{\mathbb{T}}^n$ 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.

25 pages