Effective SO Superpotential for N=1 Theory with N_f Fundamental Matter

Motivated by the duality conjecture of Dijkgraaf and Vafa between supersymmetric gauge theories and matrix models, we derive the effective superpotential of N=1 supersymmetric gauge theory with gauge group SO(N_c) and arbitrary tree level polynomial superpotential of one chiral superfield in the adjoint representation and N_f fundamental matter multiplets. For a special point in the classical vacuum where the gauge group is unbroken, we show that the effective superpotential matches with that obtained from the geometric engineering approach.Comment: LaTeX, 1+19 pages, To appear in Nucl.Phys.

Composite Representation Invariants and Unoriented Topological String Amplitudes

Sinha and Vafa \cite {sinha} had conjectured that the $SO$ Chern-Simons gauge theory on $S^3$ must be dual to the closed $A$-model topological string on the orientifold of a resolved conifold. Though the Chern-Simons free energy could be rewritten in terms of the topological string amplitudes providing evidence for the conjecture, we needed a novel idea in the context of Wilson loop observables to extract cross-cap $c=0,1,2$ topological amplitudes. Recent paper of Marino \cite{mar9} based on the work of Morton and Ryder\cite{mor} has clearly shown that the composite representation placed on the knots and links plays a crucial role to rewrite the topological string cross-cap $c=0$ amplitude. This enables extracting the unoriented cross-cap $c=2$ topological amplitude. In this paper, we have explicitly worked out the composite invariants for some framed knots and links carrying composite representations in $U(N)$ Chern-Simons theory. We have verified generalised Rudolph's theorem, which relates composite invariants to the invariants in $SO(N)$ Chern-Simons theory, and also verified Marino's conjectures on the integrality properties of the topological string amplitudes. For some framed knots and links, we have tabulated the BPS integer invariants for cross-cap $c=0$ and $c=2$ giving the open-string topological amplitude on the orientifold of the resolved conifold.Comment: 1+17 pages, condensed version of arXiv/1003.5282 to appear in Nucl. Phys.

U(N) Framed Links, Three-Manifold Invariants, and Topological Strings

Three-manifolds can be obtained through surgery of framed links in $S^3$. We study the meaning of surgery procedures in the context of topological strings. We obtain U(N) three-manifold invariants from U(N) framed link invariants in Chern-Simons theory on $S^3$. These three-manifold invariants are proportional to the trivial connection contribution to the Chern-Simons partition function on the respective three-manifolds. Using the topological string duality conjecture, we show that the large $N$ expansion of U(N) Chern-Simons free energies on three-manifolds, obtained from some class of framed links, have a closed string expansion. These expansions resemble the closed string $A$-model partition functions on Calabi-Yau manifolds with one Kahler parameter. We also determine Gopakumar-Vafa integer coefficients and Gromov-Witten rational coefficients corresponding to Chern-Simons free energies on some three-manifolds.Comment: Some clarifications added. Final version to appear in NP

MODELLING AND VIBRATION ANALYSIS OF A ROAD PROFILE MEASURING SYSTEM

During a vehicle development program, load data representing severe customer usage is required. The dilemma faced by a design engineer during the design process is that during the initial stage, only predicted loads estimated from historical targets are available, whereas the actual loads are available only at the fag end of the process. At the same time, changes required, if any, are easier and inexpensive during the initial stages of the design process whereas they are extremely costly in the latter stages of the process. The use of road profiles and vehicle models to predict the load acting on the whole vehicle is currently being researched. This work hinges on the ability to accurately measure road profiles. The objective of the work is to develop an algorithm, using MATLAB Simulink software, to convert the input signals into measured road profile. The algorithm is checked by the MATLAB Simulink 4 degrees of freedom half car model. To make the whole Simulink model more realistic, accelerometer and laser sensor properties are introduced. The present work contains the simulation of the mentioned algorithm with a half car model and studies the results in distance, time, and the frequency domain

Chern-Simons Invariants of Torus Links

We compute the vacuum expectation values of torus knot operators in Chern-Simons theory, and we obtain explicit formulae for all classical gauge groups and for arbitrary representations. We reproduce a known formula for the HOMFLY invariants of torus links and we obtain an analogous formula for Kauffman invariants. We also derive a formula for cable knots. We use our results to test a recently proposed conjecture that relates HOMFLY and Kauffman invariants.Comment: 20 pages, 5 figures; v2: minor changes, version submitted to AHP. The final publication is available at http://www.springerlink.com/content/a2614232873l76h6