Toric K\"ahler metrics seen from infinity, quantization and compact tropical amoebas

We consider the metric space of all toric K\"ahler metrics on a compact toric manifold; when "looking at it from infinity" (following Gromov), we obtain the tangent cone at infinity, which is parametrized by equivalence classes of complete geodesics. In the present paper, we study the associated limit for the family of metrics on the toric variety, its quantization, and degeneration of generic divisors. The limits of the corresponding K\"ahler polarizations become degenerate along the Lagrangian fibration defined by the moment map. This allows us to interpolate continuously between geometric quantizations in the holomorphic and real polarizations and show that the monomial holomorphic sections of the prequantum bundle converge to Dirac delta distributions supported on Bohr-Sommerfeld fibers. In the second part, we use these families of toric metric degenerations to study the limit of compact hypersurface amoebas and show that in Legendre transformed variables they are described by tropical amoebas. We believe that our approach gives a different, complementary, perspective on the relation between complex algebraic geometry and tropical geometry.Comment: v1: 32 pages, 5 figures; v2: 1 figure added; v3: 1 reference added; v4: some reorganization, 1 theorem (now 1.1) added; v5: final version, to appear in JD

Laughlin states change under large geometry deformations and imaginary time Hamiltonian dynamics

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

Coherent state transforms and vector bundles on elliptic curves

AbstractWe extend the coherent state transform (CST) of Hall to the context of the moduli spaces of semistable holomorphic vector bundles with fixed determinant over elliptic curves. We show that by applying the CST to appropriate distributions, we obtain the space of level k, rank n and genus one non-abelian theta functions with the unitarity of the CST transform being preserved. Furthermore, the shift in the level k→k+n appears in a natural way in this finite-dimensional framework

On the support of the Ashtekar-Lewandowski measure

We show that the Ashtekar-Isham extension of the classical configuration space of Yang-Mills theories (i.e. the moduli space of connections) is (topologically and measure-theoretically) the projective limit of a family of finite dimensional spaces associated with arbitrary finite lattices. These results are then used to prove that the classical configuration space is contained in a zero measure subset of this extension with respect to the diffeomorphism invariant Ashtekar-Lewandowski measure. Much as in scalar field theory, this implies that states in the quantum theory associated with this measure can be realized as functions on the ``extended" configuration space.Comment: 22 pages, Tex, Preprint CGPG-94/3-

Fishers' knowledge and seahorse conservation in Brazil

From a conservationist perspective, seahorses are threatened fishes. Concomitantly, from a socioeconomic perspective, they represent a source of income to many fishing communities in developing countries. An integration between these two views requires, among other things, the recognition that seahorse fishers have knowledge and abilities that can assist the implementation of conservation strategies and of management plans for seahorses and their habitats. This paper documents the knowledge held by Brazilian fishers on the biology and ecology of the longsnout seahorse Hippocampus reidi. Its aims were to explore collaborative approaches to seahorse conservation and management in Brazil; to assess fishers' perception of seahorse biology and ecology, in the context evaluating potential management options; to increase fishers' involvement with seahorse conservation in Brazil. Data were obtained through questionnaires and interviews made during field surveys conducted in fishing villages located in the States of Piauí, Ceará, Paraíba, Maranhão, Pernambuco and Pará. We consider the following aspects as positive for the conservation of seahorses and their habitats in Brazil: fishers were willing to dialogue with researchers; although captures and/or trade of brooding seahorses occurred, most interviewees recognized the importance of reproduction to the maintenance of seahorses in the wild (and therefore of their source of income), and expressed concern over population declines; fishers associated the presence of a ventral pouch with reproduction in seahorses (regardless of them knowing which sex bears the pouch), and this may facilitate the construction of collaborative management options designed to eliminate captures of brooding specimens; fishers recognized microhabitats of importance to the maintenance of seahorse wild populations; fishers who kept seahorses in captivity tended to recognize the condtions as poor, and as being a cause of seahorse mortality