Dyon Spectrum in N=4 Supersymmetric Type II String Theories

We compute the spectrum of quarter BPS dyons in freely acting Z_2 and Z_3 orbifolds of type II string theory compactified on a six dimensional torus. For large charges the result for statistical entropy computed from the degeneracy formula agrees with the corresponding black hole entropy to first non-leading order after taking into account corrections due to the curvature squared terms in the effective action. The result is significant since in these theories the entropy of a small black hole, computed using the curvature squared corrections to the effective action, fails to reproduce the statistical entropy associated with elementary string states.Comment: LaTeX file, 32 pages; v2:minor change

Black Holes, Elementary Strings and Holomorphic Anomaly

In a previous paper we had proposed a specific route to relating the entropy of two charge black holes to the degeneracy of elementary string states in N=4 supersymmetric heterotic string theory in four dimensions. For toroidal compactification this proposal works correctly to all orders in a power series expansion in inverse charges provided we take into account the corrections to the black hole entropy formula due to holomorphic anomaly. In this paper we demonstrate that similar agreement holds also for other N=4 supersymmetric heterotic string compactifications.Comment: LaTeX file, 28 pages, reference added, minor changes in appendix

Seven-branes and Supersymmetry

We re-investigate the construction of half-supersymmetric 7-brane solutions of IIB supergravity. Our method is based on the requirement of having globally well-defined Killing spinors and the inclusion of SL(2,Z)-invariant source terms. In addition to the well-known solutions going back to Greene, Shapere, Vafa and Yau we find new supersymmetric configurations, containing objects whose monodromies are not related to the monodromy of a D7-brane by an SL(2,Z) transformation.Comment: 31 pages, 3 figure

Representations of an integer by some quaternary and octonary quadratic forms

In this paper we consider certain quaternary quadratic forms and octonary quadratic forms and by using the theory of modular forms, we find formulae for the number of representations of a positive integer by these quadratic forms.Comment: 20 pages, 4 tables. arXiv admin note: text overlap with arXiv:1607.0380

An Effective Superstring Spectral Action

A supersymmetric theory in two-dimensions has enough data to define a noncommutative space thus making it possible to use all the tools of noncommutative geometry. In particular, we apply this to the N=1 supersymmetric non-linear sigma model and derive an expression for the generalized loop space Dirac operator, in presence of a general background, using canonical quantization. The spectral action principle is used to show that the superstring partition function is also a spectral action valid for the fluctuations of the string modes.Comment: 31 pages, Latex fil

On Type IIB Vacua With Varying Coupling Constant

We describe type IIB compactifications with varying coupling constant in d=6,7,8,9 dimensions, where part of the ten-dimensional SL(2,Z) symmetry is broken by a background with Gamma_0(n) or Gamma(n) monodromy for n=2,3,4. This extends the known class of F-theory vacua to theories which are dual to heterotic compactifications with reduced rank. On compactifying on a further torus, we obtain a description of the heterotic moduli space of G bundles over elliptically fibered manifolds without vector structure in terms of complex geometries.Comment: 32 pages, 5 eps figure

Modular Groups, Visibility Diagram and Quantum Hall Effect

We consider the action of the modular group $\Gamma (2)$ on the set of positive rational fractions. From this, we derive a model for a classification of fractional (as well as integer) Hall states which can be visualized on two ``visibility" diagrams, the first one being associated with even denominator fractions whereas the second one is linked to odd denominator fractions. We use this model to predict, among some interesting physical quantities, the relative ratios of the width of the different transversal resistivity plateaus. A numerical simulation of the tranversal resistivity plot based on this last prediction fits well with the present experimental data.Comment: 17 pages, plain TeX, 4 eps figures included (macro epsf.tex), 1 figure available from reques

On multiplicities in length spectra of arithmetic hyperbolic three-orbifolds

Asymptotic laws for mean multiplicities of lengths of closed geodesics in arithmetic hyperbolic three-orbifolds are derived. The sharpest results are obtained for non-compact orbifolds associated with the Bianchi groups SL(2,o) and some congruence subgroups. Similar results hold for cocompact arithmetic quaternion groups, if a conjecture on the number of gaps in their length spectra is true. The results related to the groups above give asymptotic lower bounds for the mean multiplicities in length spectra of arbitrary arithmetic hyperbolic three-orbifolds. The investigation of these multiplicities is motivated by their sensitive effect on the eigenvalue spectrum of the Laplace-Beltrami operator on a hyperbolic orbifold, which may be interpreted as the Hamiltonian of a three-dimensional quantum system being strongly chaotic in the classical limit.Comment: 29 pages, uuencoded ps. Revised version, to appear in NONLINEARIT

A pp-adic Approach to the Weil Representation of Discriminant Forms Arising from Even Lattices

Suppose that $M$ is an even lattice with dual $M^{*}$ and level $N$. Then the group $Mp_{2}(\mathbb{Z})$, which is the unique non-trivial double cover of $SL_{2}(\mathbb{Z})$, admits a representation $\rho_{M}$, called the Weil representation, on the space $\mathbb{C}[M^{*}/M]$. The main aim of this paper is to show how the formulae for the $\rho_{M}$-action of a general element of $Mp_{2}(\mathbb{Z})$ can be obtained by a direct evaluation which does not depend on ``external objects'' such as theta functions. We decompose the Weil representation $\rho_{M}$ into $p$-parts, in which each $p$-part can be seen as subspace of the Schwartz functions on the $p$-adic vector space $M_{\mathbb{Q}_{p}}$. Then we consider the Weil representation of $Mp_{2}(\mathbb{Q}_{p})$ on the space of Schwartz functions on $M_{\mathbb{Q}_{p}}$, and see that restricting to $Mp_{2}(\mathbb{Z})$ just gives the $p$-part of $\rho_{M}$ again. The operators attained by the Weil representation are not always those appearing in the formulae from 1964, but are rather their multiples by certain roots of unity. For this, one has to find which pair of elements, lying over a matrix in $SL_{2}(\mathbb{Q}_{p})$, belong to the metaplectic double cover. Some other properties are also investigated.Comment: 29 pages, shortened a lo

Nonlinear Differential Equations Satisfied by Certain Classical Modular Forms

A unified treatment is given of low-weight modular forms on \Gamma_0(N), N=2,3,4, that have Eisenstein series representations. For each N, certain weight-1 forms are shown to satisfy a coupled system of nonlinear differential equations, which yields a single nonlinear third-order equation, called a generalized Chazy equation. As byproducts, a table of divisor function and theta identities is generated by means of q-expansions, and a transformation law under \Gamma_0(4) for the second complete elliptic integral is derived. More generally, it is shown how Picard-Fuchs equations of triangle subgroups of PSL(2,R) which are hypergeometric equations, yield systems of nonlinear equations for weight-1 forms, and generalized Chazy equations. Each triangle group commensurable with \Gamma(1) is treated.Comment: 40 pages, final version, accepted by Manuscripta Mathematic