Monomial transformations of the projective space

We prove that, over any field, the dimension of the indeterminacy locus of a rational transformation $f$ of $P^n$ which is defined by monomials of the same degree $d$ with no common factors is at least $(n-2)/2$, provided that the degree of $f$ as a map is not divisible by $d$. This implies upper bounds on the multidegree of $f$

Tau anomalous magnetic moment form factor at Super B/Flavor factories

The proposed high-luminosity B/Flavor factories offer new opportunities for the improved determination of the fundamental physical parameters of standard heavy leptons. Compared to the electron or the muon case, the magnetic properties of the $\tau$ lepton are largely unexplored. We show that the electromagnetic properties of the $\tau$, and in particular its magnetic form factor, may be measured competitively in these facilities, using unpolarized or polarized electron beams. Various observables of the $\tau$'s produced on top of the $\Upsilon$ resonances, such as cross-section and normal polarization for unpolarized electrons or longitudinal and transverse asymmetries for polarized beams, can be combined in order to increase the sensitivity on the magnetic moment form factor. In the case of polarized electrons, we identify a special combination of transverse and longitudinal $\tau$ polarizations able to disentangle this anomalous magnetic form factor from both the charge form factor and the interference with the Z-mediating amplitude. For an integrated luminosity of $15 \times 10^{18} b^{-1}$ one could achieve a sensitivity of about $10^{-6}$, which is several orders of magnitude below any other existing high- or low-energy bound on the magnetic moment. Thus one may obtain a QED test of this fundamental quantity to a few % precision.Comment: 20 pages, 4 figure

Chen-Ruan cohomology of ADE singularities

We study Ruan's \textit{cohomological crepant resolution conjecture} for orbifolds with transversal ADE singularities. In the $A_n$-case we compute both the Chen-Ruan cohomology ring $H^*_{\rm CR}([Y])$ and the quantum corrected cohomology ring $H^*(Z)(q_1,...,q_n)$. The former is achieved in general, the later up to some additional, technical assumptions. We construct an explicit isomorphism between $H^*_{\rm CR}([Y])$ and $H^*(Z)(-1)$ in the $A_1$-case, verifying Ruan's conjecture. In the $A_n$-case, the family $H^*(Z)(q_1,...,q_n)$ is not defined for $q_1=...=q_n=-1$. This implies that the conjecture should be slightly modified. We propose a new conjecture in the $A_n$-case which we prove in the $A_2$-case by constructing an explicit isomorphism.Comment: This is a short version of my Ph.D. Thesis math.AG/0510528. Version 2: chapters 2,3,4 and 5 has been rewritten using the language of groupoids; a link with the classical McKay correpondence is given. International Journal of Mathematics (to appear

Quantum affine Cartan matrices, Poincare series of binary polyhedral groups, and reflection representations

We first review some invariant theoretic results about the finite subgroups of SU(2) in a quick algebraic way by using the McKay correspondence and quantum affine Cartan matrices. By the way it turns out that some parameters (a,b,h;p,q,r) that one usually associates with such a group and hence with a simply-laced Coxeter-Dynkin diagram have a meaningful definition for the non-simply-laced diagrams, too, and as a byproduct we extend Saito's formula for the determinant of the Cartan matrix to all cases. Returning to invariant theory we show that for each irreducible representation i of a binary tetrahedral, octahedral, or icosahedral group one can find a homomorphism into a finite complex reflection group whose defining reflection representation restricts to i.Comment: 19 page

Self-gravitating branes of codimension 4 in Lovelock gravity

We construct a familly of exact solutions of Lovelock equations describing codimension four branes with discrete symmetry in the transverse space. Unlike what is known from pure Einstein gravity, where such brane solutions of higher codimension are singular, the solutions we find, for the complete Lovelock theory, only present removable singularities. The latter account for a localised tension-like energy-momentum tensor on the brane, in analogy with the case of a codimension two self-gravitating cosmic string in pure Einstein gravity. However, the solutions we discuss present two main distinctive features : the tension of the brane receives corrections from the induced curvature of the brane's worldsheet and, in a given Lovelock theory, the spectrum of possible values of the tension is discrete. These solutions provide a new framework for the study of higher codimension braneworlds.Comment: 22 page

Quivers from Matrix Factorizations

We discuss how matrix factorizations offer a practical method of computing the quiver and associated superpotential for a hypersurface singularity. This method also yields explicit geometrical interpretations of D-branes (i.e., quiver representations) on a resolution given in terms of Grassmannians. As an example we analyze some non-toric singularities which are resolved by a single CP1 but have "length" greater than one. These examples have a much richer structure than conifolds. A picture is proposed that relates matrix factorizations in Landau-Ginzburg theories to the way that matrix factorizations are used in this paper to perform noncommutative resolutions.Comment: 33 pages, (minor changes

Higgs boson production in photon-photon collision at ILC: a comparative study in different little Higgs models

We study the process \gamma\gamma->h->bb_bar at ILC as a probe of different little Higgs models, including the simplest little Higgs model (SLH), the littlest Higgs model (LH), and two types of littlest Higgs models with T-parity (LHT-I, LHT-II). Compared with the Standard Model (SM) prediction, the production rate is found to be sizably altered in these little Higgs models and, more interestingly, different models give different predictions. We find that the production rate can be possibly enhanced only in the LHT-II for some part of the parameter space, while in all other cases the rate is suppressed. The suppression can be 10% in the LH and as much as 60% in both the SLH and the LHT-I/LHT-II. The severe suppression in the SLH happens for a large \tan\beta and a small m_h, in which the new decay mode h->\eta\eta (\eta is a light pseudo-scalar) is dominant; while for the LHT-I/LHT-II the large suppression occurs when f and m_h are both small so that the new decay mode h->A_H A_H is dominant. Therefore, the precision measurement of such a production process at the ILC will allow for a test of these models and even distinguish between different scenarios.Comment: Version in JHEP (h-g-g & h-gamma-gamma expressions added

Chromomagnetic Dipole Moment of the Top Quark Revisited

We study the complete one-loop contributions to the chromagnetic dipole moment $\Delta\kappa$ of the top quark in the Standard Model, two Higgs doublet models, topcolor assited technicolor models (TC2), 331 models and extended models with a single extra dimension. We find that the SM predicts $\Delta\kappa = - 0.056$ and that the predictions of the other models are also consitent with the constraints imposed on $\Delta\kappa$ by low-energy precision measurements.Comment: 20 pages, 5 figures, Updat