Charged Macroscopic type II Strings and their Networks

We write down charged macroscopic string solutions in type II string theories, compactified on torii, and present an explicit solution of the spinor Killing equations to show that they preserve 1/2 of the type II supersymmetries. The S-duality symmetry of the type IIB string theory in ten-dimensions is used to write down the SL(2,Z) multiplets of such strings and the corresponding 1/2 supersymmetry conditions. Finally we present examples of planar string networks, using charged macroscopic (p,q)-strings. An interesting feature of some of these networks, which preserve 1/4 supersymmetry, is a required alignment among three parameters, namely the orientation of strings, a U(1) phase associated with the maximal compact subgroup of SL(2, Z), and an (angular) parameter associated with a solution generating transformation, which is responsible for creating charges and currents on the strings.Comment: 26 pages, Latex, typos corrected in few equations, minor change

Non-Planar String Networks on Tori

Type II strings in D=5 contain particle-like 1/8 supersymmetric BPS states. In this note we give a string-network representation of such states by considering (periodic) non-planar $(p,q,r)$-string networks of eight dimensional type II string theory on $T^3$. We obtain the BPS mass formula of such states, in terms of charges and generating-vectors of the torus, and show its invariance under an $SL(3, Z)\times SL(3, Z)$ group of transformations. Results are then generalized to string-networks associated with the $SL(5, Z)$ $U$-duality in seven dimensions. We also discuss reinterpretation of the above $(D=5)$ mass formula in terms of BPS states in world-volume theories of $U2$-branes in D=8.Comment: 11 pages, latex, one figure, very minor English correction

O(d~,d~)O(\tilde d,\tilde d) Transformations and 3D Black Hole

We generalize the results of a previous paper by one of the authors to show a relationship among a class of string solutions through $O(\tilde d, \tilde d)$ transformations. The results are applied to a rotating black hole solution of three dimensional general relativity discussed recently. We extend the black hole solution to string theory and show its connection with the three dimensional black string with nonzero momentum through an $O(\tilde d, \tilde d)$ transformation of the above type.Comment: 12 pages, LaTeX, (December 1992

Export and Economic Growth in India: Causal Interpretation

The present study focuses on the cointegration between Export and Gross Domestic Product and its components at current and constant prices. Time series data for Export and Gross Domestic Product and its components has been taken for the period 1950-51 to 2001-02. In the long run export and GDP reveal that export and GDP at constant prices are not cointegrable while export and GDP at current prices are cointegrable and also the direction of causality is positive. In the short run, through error correction mechanism it has been observed that GDP as dependent variable and export as an independent variable show that short run changes in export have affected positively to GDP and its components.Gross Domestic Product, Export, Unit root test, Cointegration, Error Correction Model, Time series