We use a generalization of the Gibbons-Hawking ansatz to study the behavior of certain non-compact Calabi-Yau manifolds in the large complex structure limit. This analysis provides an intermediate step toward proving the metric collapse conjecture for toric hypersurfaces and complete intersections.

We start by pointing out that certain Riemann surfaces appear rather naturally in the context of wave equations in the black hole background. For a given black hole there are two closely related surfaces. One is the Riemann surface of complexified "tortoise" coordinate. The other Riemann surface appears when the radial wave equation is interpreted as the Fuchsian differential equation. We study these surfaces in detail for the BTZ and Schwarzschild black holes in four and higher dimensions. Topologically, in all cases both surfaces are a sphere with a set of marked points; for BTZ and 4D Schwarzschild black holes there is 3 marked points. In certain limits the surfaces can be characterized very explicitly. We then show how properties of the wave equation (quasi-normal modes) in such limits are encoded in the geometry of the corresponding surfaces. In particular, for the Schwarzschild black hole in the high damping limit we describe the Riemann surface in question and use this to derive the quasi-normal mode frequencies with the log 3 as the real part. We then argue that the surfaces one finds this way signal an appearance of an effective string. We propose that a description of this effective string propagating in the black hole background can be given in terms of the Liouville theory living on the corresponding Riemann surface. We give such a stringy description for the Schwarzschild black hole in the limit of high damping and show that the quasi-normal modes emerge naturally as the poles in 3-point correlation function in the effective conformal theory.

We construct fractional branes in Landau-Ginzburg orbifold categories and study their behavior under marginal closed string perturbations. This approach is shown to be more general than the rational boundary state construction. In particular we find new D-branes on the quintic -- such as a single D0-brane -- which are not restrictions of bundles on the ambient projective space. We also exhibit a family of deformations of the D0-brane in the Landau-Ginzburg category parameterized by points on the Fermat quintic.

We study deformations of Landau-Ginzburg D-branes corresponding to obstructed rational curves on Calabi-Yau threefolds. We determine D-brane moduli spaces and D-brane superpotentials by evaluating higher products up to homotopy in the Landau-Ginzburg orbifold category. For concreteness we work out the details for lines on a perturbed Fermat quintic. In this case we show that our results reproduce the local analytic structure of the Hilbert scheme of curves on the threefold.

The main purpose in the present paper is to build a Hamiltonian theory for fields which is consistent with the principles of relativity. For this we consider detailed geometric pictures of Lepage theories in the spirit of Dedecker and try to stress out the interplay between the Lepage-Dedecker (LP) description and the (more usual) De Donder- Weyl (DDW) one. One of the main points is the fact that the Legendre transform in the DDW approach is replaced by a Legendre correspondence in the LP theory (this correspondence behaves differently: ignoring the singularities whenever the Lagrangian is degenerate).