Diffeomorphisms and families of Fourier-Mukai transforms in mirror symmetry

Assuming the standard framework of mirror symmetry, a conjecture is formulated describing how the diffeomorphism group of a Calabi-Yau manifold Y should act by families of Fourier-Mukai transforms over the complex moduli space of the mirror X. The conjecture generalizes a proposal of Kontsevich relating monodromy transformations and self-equivalences. Supporting evidence is given in the case of elliptic curves, lattice-polarized K3 surfaces and Calabi-Yau threefolds. A relation to the global Torelli problem is discussed.Comment: Approx. 20 pages LaTeX. One reference adde

Normal subgroups in the Cremona group (long version)

Let k be an algebraically closed field. We show that the Cremona group of all birational transformations of the projective plane P^2 over k is not a simple group. The strategy makes use of hyperbolic geometry, geometric group theory, and algebraic geometry to produce elements in the Cremona group that generate non trivial normal subgroups.Comment: With an appendix by Yves de Cornulier. Numerous but minors corrections were made, regarding proofs, references and terminology. This long version contains detailled proofs of several technical lemmas about hyperbolic space

On the Classification of Quasihomogeneous Functions

We give a criterion for the existence of a non-degenerate quasihomogeneous polynomial in a configuration, i.e. in the space of polynomials with a fixed set of weights, and clarify the relation of this criterion to the necessary condition derived from the formula for the Poincar\'e polynomial. We further prove finiteness of the number of configurations for a given value of the singularity index. For the value 3 of this index, which is of particular interest in string theory, a constructive version of this proof implies an algorithm for the calculation of all non-degenerate configurations.Comment: 12 page

K3-fibered Calabi-Yau threefolds I, the twist map

A construction of Calabi-Yaus as quotients of products of lower-dimensional spaces in the context of weighted hypersurfaces is discussed, including desingularisation. The construction leads to Calabi-Yaus which have a fiber structure, in particular one case has K3 surfaces as fibers. These Calabi-Yaus are of some interest in connection with Type II -heterotic string dualities in dimension 4. A section at the end of the paper summarises this for the non-expert mathematician.Comment: 31 pages LaTeX, 11pt, 2 figures. To appear in International Journal of Mathematics. On the web at http://personal-homepages.mis.mpg.de/bhunt/preprints.html , #

Dark energy domination in the Virgocentric flow

The standard \LambdaCDM cosmological model implies that all celestial bodies are embedded in a perfectly uniform dark energy background, represented by Einstein's cosmological constant, and experience its repulsive antigravity action. Can dark energy have strong dynamical effects on small cosmic scales as well as globally? Continuing our efforts to clarify this question, we focus now on the Virgo Cluster and the flow of expansion around it. We interpret the Hubble diagram, from a new database of velocities and distances of galaxies in the cluster and its environment, using a nonlinear analytical model which incorporates the antigravity force in terms of Newtonian mechanics. The key parameter is the zero-gravity radius, the distance at which gravity and antigravity are in balance. Our conclusions are: 1. The interplay between the gravity of the cluster and the antigravity of the dark energy background determines the kinematical structure of the system and controls its evolution. 2. The gravity dominates the quasi-stationary bound cluster, while the antigravity controls the Virgocentric flow, bringing order and regularity to the flow, which reaches linearity and the global Hubble rate at distances \ga 15 Mpc. 3. The cluster and the flow form a system similar to the Local Group and its outflow. In the velocity-distance diagram, the cluster-flow structure reproduces the group-flow structure with a scaling factor of about 10; the zero-gravity radius for the cluster system is also 10 times larger. The phase and dynamical similarity of the systems on the scales of 1-30 Mpc suggests that a two-component pattern may be universal for groups and clusters: a quasi-stationary bound central component and an expanding outflow around it, due to the nonlinear gravity-antigravity interplay with the dark energy dominating in the flow component.Comment: 7 pages, 2 figures, Astronomy and Astrophysics (accepted

From SICs and MUBs to Eddington

This is a survey of some very old knowledge about Mutually Unbiased Bases (MUB) and Symmetric Informationally Complete POVMs (SIC). In prime dimensions the former are closely tied to an elliptic normal curve symmetric under the Heisenberg group, while the latter are believed to be orbits under the Heisenberg group in all dimensions. In dimensions 3 and 4 the SICs are understandable in terms of elliptic curves, but a general statement escapes us. The geometry of the SICs in 3 and 4 dimensions is discussed in some detail.Comment: 12 pages; from the Festschrift for Tony Sudber

Local dark energy: HST evidence from the expansion flow around Cen A/M83 galaxy group

A structure with a massive group in its center and a cool expansion outflow outside is studied around the Cen A galaxy with the use of the Hubble Space Telescope observations. It is demonstrated that the dynamics of the flow is dominated by the antigravity of the dark energy background. The density of dark energy in the cell is estimated to be near the global cosmological density. This agrees with our previous result from the neighborhood of the Local group. A notion of the ``Hubble cell'' is introduced as a building block of the local structure of the universe

Three embeddings of the Klein simple group into the Cremona group of rank three

We study the action of the Klein simple group G consisting of 168 elements on two rational threefolds: the three-dimensional projective space and a smooth Fano threefold X of anticanonical degree 22 and index 1. We show that the Cremona group of rank three has at least three non-conjugate subgroups isomorphic to G. As a by-product, we prove that X admits a Kahler-Einstein metric, and we construct a smooth polarized K3 surface of degree 22 with an action of the group G.Comment: 43 page

The Volume of some Non-spherical Horizons and the AdS/CFT Correspondence

We calculate the volumes of a large class of Einstein manifolds, namely Sasaki-Einstein manifolds which are the bases of Ricci-flat affine cones described by polynomial embedding relations in C^n. These volumes are important because they allow us to extend and test the AdS/CFT correspondence. We use these volumes to extend the central charge calculation of Gubser (1998) to the generalized conifolds of Gubser, Shatashvili, and Nekrasov (1999). These volumes also allow one to quantize precisely the D-brane flux of the AdS supergravity solution. We end by demonstrating a relationship between the volumes of these Einstein spaces and the number of holomorphic polynomials (which correspond to chiral primary operators in the field theory dual) on the corresponding affine cone.Comment: 25 pp, LaTeX, 1 figure, v2: refs adde