Obstructed D-Branes in Landau-Ginzburg Orbifolds

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.Comment: 44 pages; v3: typos correcte

Continuous Curvelet Transform: II. Discretization and Frames

We develop a unifying perspective on several decompositions exhibiting directional parabolic scaling. In each decomposition, the individual atoms are highly anisotropic at fine scales, with effective support obeying the parabolic scaling principle length ≈ width^2. Our comparisons allow to extend Theorems known for one decomposition to others. We start from a Continuous Curvelet Transform f → Γ_f (a, b, θ) of functions f(x_1, x_2) on R^2, with parameter space indexed by scale a > 0, location b ∈ R^2, and orientation θ. The transform projects f onto a curvelet γ_(abθ), yielding coefficient Γ_f (a, b, θ) = f, _(γabθ); the corresponding curvelet γ_(abθ) is defined by parabolic dilation in polar frequency domain coordinates. We establish a reproducing formula and Parseval relation for the transform, showing that these curvelets provide a continuous tight frame. The CCT is closely related to a continuous transform introduced by Hart Smith in his study of Fourier Integral Operators. Smith’s transform is based on true affine parabolic scaling of a single mother wavelet, while the CCT can only be viewed as true affine parabolic scaling in euclidean coordinates by taking a slightly different mother wavelet at each scale. Smith’s transform, unlike the CCT, does not provide a continuous tight frame. We show that, with the right underlying wavelet in Smith’s transform, the analyzing elements of the two transforms become increasingly similar at increasingly fine scales. We derive a discrete tight frame essentially by sampling the CCT at dyadic intervals in scale a_j = 2^−j, at equispaced intervals in direction, θ_(jℓ), = 2π2^(−j/2)ℓ, and equispaced sampling on a rotated anisotropic grid in space. This frame is a complexification of the ‘Curvelets 2002’ frame constructed by Emmanuel Candès et al. [1, 2, 3]. We compare this discrete frame with a composite system which at coarse scales is the same as this frame but at fine scales is based on sampling Smith’s transform rather than the CCT. We are able to show a very close approximation of the two systems at fine scales, in a strong operator norm sense. Smith’s continuous transform was intended for use in forming molecular decompositions of Fourier Integral Operators (FIO’s). Our results showing close approximation of the curvelet frame by a composite frame using true affine paraboblic scaling at fine scales allow us to cross-apply Smith’s results, proving that the discrete curvelet transform gives sparse representations of FIO’s of order zero. This yields an alternate proof of a recent result of Candès and Demanet about the sparsity of FIO representations in discrete curvelet frames

Evolution along the sequence of S0 Hubble types induced by dry minor mergers. II - Bulge-disk coupling in the photometric relations through merger-induced internal secular evolution

Galaxy mergers are considered as questionable mechanisms for the evolution of lenticular galaxies (S0's), on the basis that even minor ones induce structural changes that are difficult to reconcile with the strong bulge-disk coupling observed in the photometric scaling relations of S0's. We check if the evolution induced onto S0's by dry intermediate and minor mergers can reproduce their photometric scaling relations, analysing the bulge-disk decompositions of the merger simulations presented in Eliche-Moral et al. (2012). The mergers induce an evolution in the photometric planes compatible with the data of S0's, even in those ones indicating a strong bulge-disk coupling. The mergers drive the formation of the observed photometric relation in some cases, whereas they induce a slight dispersion compatible with data in others. Therefore, this evolutionary mechanism tends to preserve these scaling relations. In those photometric planes where the morphological types segregate, the mergers always induce evolution towards the region populated by S0's. The structural coupling of the bulge and the disk is preserved or reinforced because the mergers trigger internal secular processes in the primary disk that induce significant bulge growth, even although these models do not induce bars. Intermediate and minor mergers can thus be considered as plausible mechanisms for the evolution of S0's attending to their photometric scaling relations, as they can preserve and even strengthen any pre-existing structural bulge-disk coupling, triggering significant internal secular evolution (even in the absence of bars or dissipational effects). This means that it may be difficult to isolate the effects of pure internal secular evolution from those of the merger-driven one in present-day early-type disks (abridged).Comment: Accepted for publication in Astronomy & Astrophysics, 13 pages, 8 figures. Definitive version after proofs. Added references and corrected typo