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We study the change of the Laughlin states under large deformations of the geometry of the sphere and the plane, associated with Mabuchi geodesics on the space of metrics with Hamiltonian $S^1$-symmetry. For geodesics associated with the square of the symmetry generator, as the geodesic time goes to infinity, the geometry of the sphere becomes that of a thin cigar collapsing to a line and the Laughlin states become concentrated on a discrete set of $S^1$--orbits, corresponding to Bohr-Sommerfeld orbits of geometric quantization. The lifting of the Mabuchi geodesics to the bundle of quantum states, to which the Laughlin states belong, is achieved via generalized coherent state transforms, which correspond to the KZ parallel transport of Chern-Simons theory.

The Cauchy problem for (real analytic) geodesics in the space of Kahler metrics with a fixed cohomology class on a compact complex manifold M can be effectively reduced to the problem of finding the flow of a related Hamiltonian vector field $$X_H$$, followed by analytic continuation of the time to complex time. This opens the possibility of expressing the geodesic $$\omega _t$$ in terms of Grobner Lie series of the form $$\exp (\sqrt{-1} \, tX_H)(f)$$, for local holomorphic functions f. The main goal of this paper is to use truncated Lie series as a new way of constructing approximate solutions to the geodesic equation. For the case of an elliptic curve and H a certain Morse function squared, we approximate the relevant Lie series by the first twelve terms, calculated with the help of Mathematica. This leads to approximate geodesics which hit the boundary of the space of Kahler metrics in finite geodesic time. For quantum mechanical applications, one is interested also on the non-Kahler polarizations that one obtains by crossing the boundary of the space of Kahler structures.

Let $$P$$ be a Delzant polytope. We show that the quantization of the corresponding toric manifold $$X_{P}$$ in toric Kahler 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 Kahler 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 Kahler 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 Kahler 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 Kahler 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.

We address quantization of the natural symplectic structure on a moduli space of parabolic vector bundles of parabolic degree zero over CP 1 with four parabolic points and parabolic weights in {0,1/2}. Identifying such parabolic bundles as vector bundles on an elliptic curve, we obtain explicit expressions for the corresponding non-abelian theta func- tions. These non-abelian theta functions are described in terms of certain naturally defined distributions on the compact group SU(2).