Branes and supersymmetric quantum field theories

Since the discovery of D-branes as non-perturbative, dynamic objects in string theory, various configurations of branes in type IIA/B string theory and M-theory have been considered to study their low-energy dynamics described by supersymmetric quantum field theories.

One example of such a construction is based on the description of Seiberg-Witten curves of four-dimensional N = 2 supersymmetric gauge theories as branes in type IIA string theory and M-theory. This enables us to study the gauge theories in strongly-coupled regimes. Spectral networks are another tool for utilizing branes to study non-perturbative regimes of two- and four-dimensional supersymmetric theories. Using spectral networks of a Seiberg-Witten theory we can find its BPS spectrum, which is protected from quantum corrections by supersymmetry, and also the BPS spectrum of a related two-dimensional N = (2,2) theory whose (twisted) superpotential is determined by the Seiberg-Witten curve. When we don’t know the perturbative description of such a theory, its spectrum obtained via spectral networks is a useful piece of information. In this thesis we illustrate these ideas with examples of the use of Seiberg-Witten curves and spectral networks to understand various two- and four-dimensional supersymmetric theories.

First, we examine how the geometry of a Seiberg-Witten curve serves as a useful tool for identifying various limits of the parameters of the Seiberg-Witten theory, including Argyres-Seiberg duality and Argyres-Douglas fixed points. Next, we consider the low-energy limit of a two-dimensional N = (2, 2) supersymmetric theory from an M-theory brane configuration whose (twisted) superpotential is determined by the geometry of the branes. We show that, when the two-dimensional theory flows to its infra-red fixed point, particular cases realize Kazama-Suzuki coset models. We also study the BPS spectrum of an Argyres-Douglas type superconformal field theory on the Coulomb branch by using its spectral networks. We provide strong evidence of the equivalence of superconformal field theories from different string-theoretic constructions by comparing their BPS spectra.

I. Application of multi-Regge theory to production processes. II. High energy model for proton-proton scattering

This dissertation consists of two parts. The first part presents an explicit procedure for applying multi-Regge theory to production processes. As an illustrative example, the case of three body final states is developed in detail, both with respect to kinematics and multi-Regge dynamics. Next, the experimental consistency of the multi-Regge hypothesis is tested in a specific high energy reaction; the hypothesis is shown to provide a good qualitative fit to the data. In addition, the results demonstrate a severe suppression of double Pomeranchon exchange, and show the coupling of two "Reggeons" to an external particle to be strongly damped as the particle's mass increases. Finally, with the use of two body Regge parameters, order of magnitude estimates of the multi-Regge cross section for various reactions are given.

The second part presents a diffraction model for high energy proton-proton scattering. This model developed by Chou and Yang assumes high energy elastic scattering results from absorption of the incident wave into the many available inelastic channels, with the absorption proportional to the amount of interpenetrating hadronic matter. The assumption that the hadronic matter distribution is proportional to the charge distribution relates the scattering amplitude for pp scattering to the proton form factor. The Chou-Yang model with the empirical proton form factor as input is then applied to calculate a high energy, fixed momentum transfer limit for the scattering cross section, This limiting cross section exhibits the same "dip" or "break" structure indicated in present experiments, but falls significantly below them in magnitude. Finally, possible spin dependence is introduced through a weak spin-orbit type term which gives rather good agreement with pp polarization data.

Randomness-efficient curve sampling

Curve samplers are sampling algorithms that proceed by viewing the domain as a vector space over a finite field, and randomly picking a low-degree curve in it as the sample. Curve samplers exhibit a nice property besides the sampling property: the restriction of low-degree polynomials over the domain to the sampled curve is still low-degree. This property is often used in combination with the sampling property and has found many applications, including PCP constructions, local decoding of codes, and algebraic PRG constructions.

The randomness complexity of curve samplers is a crucial parameter for its applications. It is known that (non-explicit) curve samplers using O(log N + log(1/δ)) random bits exist, where N is the domain size and δ is the confidence error. The question of explicitly constructing randomness-efficient curve samplers was first raised in [TU06] where they obtained curve samplers with near-optimal randomness complexity.

In this thesis, we present an explicit construction of low-degree curve samplers with optimal randomness complexity (up to a constant factor) that sample curves of degree (m log_q(1/δ))^O(1) in F_q^m. Our construction is a delicate combination of several components, including extractor machinery, limited independence, iterated sampling, and list-recoverable codes.

Probabilistic robust control : theory and applications

In this work, the development of a probabilistic approach to robust control is motivated by structural control applications in civil engineering. Often in civil structural applications, a system's performance is specified in terms of its reliability. In addition, the model and input uncertainty for the system may be described most appropriately using probabilistic or "soft" bounds on the model and input sets. The probabilistic robust control methodology contrasts with existing H∞/μ robust control methodologies that do not use probability information for the model and input uncertainty sets, yielding only the guaranteed (i.e., "worst-case") system performance, and no information about the system's probable performance which would be of interest to civil engineers.

The design objective for the probabilistic robust controller is to maximize the reliability of the uncertain structure/controller system for a probabilistically-described uncertain excitation. The robust performance is computed for a set of possible models by weighting the conditional performance probability for a particular model by the probability of that model, then integrating over the set of possible models. This integration is accomplished efficiently using an asymptotic approximation. The probable performance can be optimized numerically over the class of allowable controllers to find the optimal controller. Also, if structural response data becomes available from a controlled structure, its probable performance can easily be updated using Bayes's Theorem to update the probability distribution over the set of possible models. An updated optimal controller can then be produced, if desired, by following the original procedure. Thus, the probabilistic framework integrates system identification and robust control in a natural manner.

The probabilistic robust control methodology is applied to two systems in this thesis. The first is a high-fidelity computer model of a benchmark structural control laboratory experiment. For this application, uncertainty in the input model only is considered. The probabilistic control design minimizes the failure probability of the benchmark system while remaining robust with respect to the input model uncertainty. The performance of an optimal low-order controller compares favorably with higher-order controllers for the same benchmark system which are based on other approaches. The second application is to the Caltech Flexible Structure, which is a light-weight aluminum truss structure actuated by three voice coil actuators. A controller is designed to minimize the failure probability for a nominal model of this system. Furthermore, the method for updating the model-based performance calculation given new response data from the system is illustrated.

The steady-state response of multidegree-of-freedom systems with a spatially localized nonlinearity

This thesis is concerned with the dynamic response of a General multidegree-of-freedom linear system with a one dimensional nonlinear constraint attached between two points. The nonlinear constraint is assumed to consist of rate-independent conservative and hysteretic nonlinearities and may contain a viscous dissipation element. The dynamic equations for general spatial and temporal load distributions are derived for both continuous and discrete systems. The method of equivalent linearization is used to develop equations which govern the approximate steady-state response to generally distributed loads with harmonic time dependence.

The qualitative response behavior of a class of undamped chainlike structures with a nonlinear terminal constraint is investigated. It is shown that the hardening or softening behavior of every resonance curve is similar and is determined by the properties of the constraint. Also examined are the number and location of resonance curves, the boundedness of the forced response, the loci of response extrema, and other characteristics of the response. Particular consideration is given to the dependence of the response characteristics on the properties of the linear system, the nonlinear constraint, and the load distribution.

Numerical examples of the approximate steady-state response of three structural systems are presented. These examples illustrate the application of the formulation and qualitative theory. It is shown that disconnected response curves and response curves which cross are obtained for base excitation of a uniform shear beam with a cubic spring foundation. Disconnected response curves are also obtained for the steady-state response to a concentrated load of a chainlike structure with a hardening hysteretic constraint. The accuracy of the approximate response curves is investigated.

Theory of ozone isotopic effects and various electron transfer reactions

Three separate topics, each stimulated by experiments, are treated theoretically in this dessertation: isotopic effects of ozone, electron transfer at interfaces, and intramolecular directional electron transfer in a supramolecular system.

The strange mass-independent isotope effect for the enrichment of ozone, which has been a puzzle in the literature for some 20 years, and the equally puzzling unconventional strong mass-dependent effect of individual reaction rate constants are studied as different aspects of a symmetry-driven behavior. A statistical (RRKM-based) theory with a hindered-rotor transition state is used. The individual rate constant ratios of recombination reactions at low pressures are calculated using the theory involving (1) small deviation from the statistical density of states for symmetric isotopomers, and (2) weak collisions for deactivation of the vibrationally excited ozone molecules. The weak collision and partitioning among exit channels play major roles in producing the large unconventional isotope effect in "unscrambled" systems. The enrichment studies reflect instead the non-statistical effect in "scrambled" systems. The theoretical results of low-pressure ozone enrichments and individual rate constant ratios obtained from these calculations are consistent with the corresponding experimental results. The isotopic exchange rate constant for the reaction ^(16)O + ^(18)O ^(18)O→+ ^(16)O ^(18)O + ^(18)O provides information on the nature of a variationally determined hindered-rotor transition state using experimental data at 130 K and 300 K. Pressure effects on the recombination rate constant, on the individual rate constant ratios and on the enrichments are also investigated. The theoretical results are consistent with the experimental data. The temperature dependence of the enrichment and rate constant ratios is also discussed, and experimental tests are suggested. The desirability of a more accurate potential energy surface for ozone in the transition state region is also noted.

Electron transfer reactions at semiconductor /liquid interfaces are studied using a tight-binding model for the semiconductors. The slab method and a z-transform method are employed in obtaining the tight-binding electronic structures of semiconductors having surfaces. The maximum electron transfer rate constants at Si/viologen^(2-/+) and InP /Me_(2)Fc^(+/O) interfaces are computed using the tight-binding type calculations for the solid and the extended-Huckel for the coupling to the redox agent at the interface. These electron transfer reactions are also studied using a free electron model for the semiconductor and the redox molecule, where Bardeen's method is adapted to calculate the coupling matrix element between the molecular and semiconductor electronic states. The calculated results for maximum rate constant of the electron transfer from the semiconductor bulk states are compared with the experimentally measured values of Lewis and coworkers, and are in reasonable agreement, without adjusting parameters. In the case of InP /liquid interface, the unusual current vs applied potential behavior is additionally interpreted, in part, by the presence of surface states.

Photoinduced electron transfer reactions in small supramolecular systems, such as 4-aminonaphthalimide compounds, are interesting in that there are, in principle, two alternative pathways (directions) for the electron transfer. The electron transfer, however, is unidirectional, as deduced from pH-dependent fluorescence quenching studies on different compounds. The role of electronic coupling matrix element and the charges in protonation are considered to explain the directionality of the electron transfer and other various results. A related mechanism is proposed to interpret the fluorescence behavior of similar molecules as fluorescent sensors of metal ions.

The theory of long distance electron transfer reactions

The rate of electron transport between distant sites was studied. The rate depends crucially on the chemical details of the donor, acceptor, and surrounding medium. These reactions involve electron tunneling through the intervening medium and are, therefore, profoundly influenced by the geometry and energetics of the intervening molecules. The dependence of rate on distance was considered for several rigid donor-acceptor "linkers" of experimental importance. Interpretation of existing experiments and predictions for new experiments were made.

The electronic and nuclear motion in molecules is correlated. A Born-Oppenheimer separation is usually employed in quantum chemistry to separate this motion. Long distance electron transfer rate calculations require the total donor wave function when the electron is very far from its binding nuclei. The Born-Oppenheimer wave functions at large electronic distance are shown to be qualitatively wrong. A model which correctly treats the coupling was proposed. The distance and energy dependence of the electron transfer rate was determined for such a model.

The Madden-Julian Oscillation : observation, modeling, and theory

The Madden-Julian Oscillation (MJO) is a pattern of intense rainfall and associated planetary-scale circulations in the tropical atmosphere, with a recurrence interval of 30-90 days. Although the MJO was first discovered 40 years ago, it is still a challenge to simulate the MJO in general circulation models (GCMs), and even with simple models it is difficult to agree on the basic mechanisms. This deficiency is mainly due to our poor understanding of moist convection—deep cumulus clouds and thunderstorms, which occur at scales that are smaller than the resolution elements of the GCMs. Moist convection is the most important mechanism for transporting energy from the ocean to the atmosphere. Success in simulating the MJO will improve our understanding of moist convection and thereby improve weather and climate forecasting.

We address this fundamental subject by analyzing observational datasets, constructing a hierarchy of numerical models, and developing theories. Parameters of the models are taken from observation, and the simulated MJO fits the data without further adjustments. The major findings include: 1) the MJO may be an ensemble of convection events linked together by small-scale high-frequency inertia-gravity waves; 2) the eastward propagation of the MJO is determined by the difference between the eastward and westward phase speeds of the waves; 3) the planetary scale of the MJO is the length over which temperature anomalies can be effectively smoothed by gravity waves; 4) the strength of the MJO increases with the typical strength of convection, which increases in a warming climate; 5) the horizontal scale of the MJO increases with the spatial frequency of convection; and 6) triggered convection, where potential energy accumulates until a threshold is reached, is important in simulating the MJO. Our findings challenge previous paradigms, which consider the MJO as a large-scale mode, and point to ways for improving the climate models.

Density functional theory embedding for correlated wavefunctions

Methods that exploit the intrinsic locality of molecular interactions show significant promise in making tractable the electronic structure calculation of large-scale systems. In particular, embedded density functional theory (e-DFT) offers a formally exact approach to electronic structure calculations in which the interactions between subsystems are evaluated in terms of their electronic density. In the following dissertation, methodological advances of embedded density functional theory are described, numerically tested, and applied to real chemical systems.

First, we describe an e-DFT protocol in which the non-additive kinetic energy component of the embedding potential is treated exactly. Then, we present a general implementation of the exact calculation of the non-additive kinetic potential (NAKP) and apply it to molecular systems. We demonstrate that the implementation using the exact NAKP is in excellent agreement with reference Kohn-Sham calculations, whereas the approximate functionals lead to qualitative failures in the calculated energies and equilibrium structures.

Next, we introduce density-embedding techniques to enable the accurate and stable calculation of correlated wavefunction (CW) in complex environments. Embedding potentials calculated using e-DFT introduce the effect of the environment on a subsystem for CW calculations (WFT-in-DFT). We demonstrate that WFT-in-DFT calculations are in good agreement with CW calculations performed on the full complex.

We significantly improve the numerics of the algorithm by enforcing orthogonality between subsystems by introduction of a projection operator. Utilizing the projection-based embedding scheme, we rigorously analyze the sources of error in quantum embedding calculations in which an active subsystem is treated using CWs, and the remainder using density functional theory. We show that the embedding potential felt by the electrons in the active subsystem makes only a small contribution to the error of the method, whereas the error in the nonadditive exchange-correlation energy dominates. We develop an algorithm which corrects this term and demonstrate the accuracy of this corrected embedding scheme.

Advanced density functional theory methods for materials science

In this work we chiefly deal with two broad classes of problems in computational materials science, determining the doping mechanism in a semiconductor and developing an extreme condition equation of state. While solving certain aspects of these questions is well-trodden ground, both require extending the reach of existing methods to fully answer them. Here we choose to build upon the framework of density functional theory (DFT) which provides an efficient means to investigate a system from a quantum mechanics description.

Zinc Phosphide (Zn₃P₂) could be the basis for cheap and highly efficient solar cells. Its use in this regard is limited by the difficulty in n-type doping the material. In an effort to understand the mechanism behind this, the energetics and electronic structure of intrinsic point defects in zinc phosphide are studied using generalized Kohn-Sham theory and utilizing the Heyd, Scuseria, and Ernzerhof (HSE) hybrid functional for exchange and correlation. Novel 'perturbation extrapolation' is utilized to extend the use of the computationally expensive HSE functional to this large-scale defect system. According to calculations, the formation energy of charged phosphorus interstitial defects are very low in n-type Zn₃P₂ and act as 'electron sinks', nullifying the desired doping and lowering the fermi-level back towards the p-type regime. Going forward, this insight provides clues to fabricating useful zinc phosphide based devices. In addition, the methodology developed for this work can be applied to further doping studies in other systems.

Accurate determination of high pressure and temperature equations of state is fundamental in a variety of fields. However, it is often very difficult to cover a wide range of temperatures and pressures in an laboratory setting. Here we develop methods to determine a multi-phase equation of state for Ta through computation. The typical means of investigating thermodynamic properties is via ’classical’ molecular dynamics where the atomic motion is calculated from Newtonian mechanics with the electronic effects abstracted away into an interatomic potential function. For our purposes, a ’first principles’ approach such as DFT is useful as a classical potential is typically valid for only a portion of the phase diagram (i.e. whatever part it has been fit to). Furthermore, for extremes of temperature and pressure quantum effects become critical to accurately capture an equation of state and are very hard to capture in even complex model potentials. This requires extending the inherently zero temperature DFT to predict the finite temperature response of the system. Statistical modelling and thermodynamic integration is used to extend our results over all phases, as well as phase-coexistence regions which are at the limits of typical DFT validity. We deliver the most comprehensive and accurate equation of state that has been done for Ta. This work also lends insights that can be applied to further equation of state work in many other materials.

Yang-Mills theory in six-dimensional superspace

The superspace approach provides a manifestly supersymmetric formulation of supersymmetric theories. For N= 1 supersymmetry one can use either constrained or unconstrained superfields for such a formulation. Only the unconstrained formulation is suitable for quantum calculations. Until now, all interacting N>1 theories have been written using constrained superfields. No solutions of the nonlinear constraint equations were known.

In this work, we first review the superspace approach and its relation to conventional component methods. The difference between constrained and unconstrained formulations is explained, and the origin of the nonlinear constraints in supersymmetric gauge theories is discussed. It is then shown that these nonlinear constraint equations can be solved by transforming them into linear equations. The method is shown to work for N=1 Yang-Mills theory in four dimensions.

N=2 Yang-Mills theory is formulated in constrained form in six-dimensional superspace, which can be dimensionally reduced to four-dimensional N=2 extended superspace. We construct a superfield calculus for six-dimensional superspace, and show that known matter multiplets can be described very simply. Our method for solving constraints is then applied to the constrained N=2 Yang-Mills theory, and we obtain an explicit solution in terms of an unconstrained superfield. The solution of the constraints can easily be expanded in powers of the unconstrained superfield, and a similar expansion of the action is also given. A background-field expansion is provided for any gauge theory in which the constraints can be solved by our methods. Some implications of this for superspace gauge theories are briefly discussed.

Three-dimensional superconformal field theory, Chern-Simons theory, and their correspondence

In this thesis, we discuss 3d-3d correspondence between Chern-Simons theory and three-dimensional N = 2 superconformal field theory. In the 3d-3d correspondence proposed by Dimofte-Gaiotto-Gukov information of abelian flat connection in Chern-Simons theory was not captured. However, considering M-theory configuration giving the 3d-3d correspondence and also other several developments, the abelian flat connection should be taken into account in 3d-3d correspondence. With help of the homological knot invariants, we construct 3d N = 2 theories on knot complement in 3-sphere for several simple knots. Previous theories obtained by Dimofte-Gaiotto-Gukov can be obtained by Higgsing of the full theories. We also discuss the importance of all flat connections in the 3d-3d correspondence by considering boundary conditions in 3d N = 2 theories and 3-manifold.