This thesis is a study of some aspects of string theory solutions that are exact in the inverse string tension ɑ', and thus are valid beyond the low-energy limit. I investigate D-brane interactions in the maximally supersymmetric plane wave solution of type IIb string theory, and study the fate of the stringy halo surrounding D-branes. I find that the halo is like in flat space for Lorentzian D-branes, while it receives a non-trivial modification for Euclidean D-branes. I also comment on the connection between Hagedorn temperature and T-duality, which motivates a more general study of T-duality in null directions. I consider such transformations in a spinning D-brane solution of supergravity, and find that divergences in the field components associated with null T-dualities are invisible to string and brane probes. I also observe that there are closed timelike curves in all the T-dual solutions, but that none of them are geodesies. The second half of the theses is an investigation of the fate of closed tirnelike curves and of cosmological singularities in an exact stringy Taub-NUT solution of heterotic string theory, and in a rotating generalisation of it. I compute the exact spacetime fields, using a description in terms of a gauged Wess-Zumino-Novikov-Witten model and find that the ɑ' corrections are mild. The key features of the Taub-NUT geometry persist, together with the emergence of a new region of space with Euclidean signature. Closed timelike curves are still present, which is inter-preted as a sign that they might be a natural ingredient in string theory, for instance in pre-Big-Bang cosmological scenarios.

We examine one loop amplitudes for open and closed strings in certain D-brane configurations, and investigate the consequences for phenomenology. Initially we consider open strings at D6-brane intersections. We develop techniques for one-loop diagrams. The one-loop propagator of chiral intersection states is calculated exactly and its finiteness is shown to be guaranteed by RR tadpole cancellation. The result is used to demonstrate the expected softening of power law running of Yukawa couplings at the string scale. We also develop methods to calculate arbitrary TV-point functions at one-loop, including those without gauge bosons in the loop. These techniques are also applicable to heterotic orbifold models. One issue of the intersecting D6-brane models is that the Yukawa couplings of the simpler models suffer from the so-called "rank one" problem - there is only a single non-zero mass and no mixing. We consider the one-loop contribution of E2-instantons to Yukawa couplings on intersecting D6-branes, and show that they can provide a solution. In addition they have the potential to provide a geometric explanation for the hierarchies observed in the Yukawa couplings. In order to do this we provide the necessary quantities for instanton calculus in this class of background. We then explore how the IR pathologies of noncommutative field theory are resolved when the theory is realized as open strings in background B-fields: essentially, since the IR singularities are induced by UV/IR mixing, string theory brings them under control in much the same way as it does the uv singularities. We show that at intermediate scales (where the Selberg-Witten limit is a good approximation) the theory reproduces the noncommutative field theory with all the (un)usual features such as UV/IR mixing, but that outside this regime, in the deep infra-red, the theory flows continuously to the commutative theory and normal Wilsoman behaviour is restored. The resulting low energy physics resembles normal commutative physics, but with additional suppressed Lorentz violating operators. We also show that the phenomenon of UV/IR mixing occurs for the graviton as well, with the result that, in configurations where Planck's constant receives a significant one-loop correction (for example brane-induced gravity), the distance scale below which gravity becomes non-Newtonian can be much greater than any compact dimensions.

Bright solitary-waves may be created in dilute Bose-Einstein condensates of at tractively interacting atoms in one dimensional regimes. In integrable systems, such solitary waves are particle-like objects called solitons. We investigate the consequences of non-integrability on the solitary waves in trapped Bose-Einstein condensates caused by an axial harmonic trap, and non-integrability caused by three dimensional effects. To analyse the soliton-like nature of the solitary-waves in an axial harmonic trap, a particle analogy for the solitary-waves is formulated. Exact Soliton solutions exist in the absence of an external trapping potential, which behave in a particle-like manner, and we find the particle analogy we employ to be a good model also when a harmonic trapping potential is present up to a gradual shift in the trajectories when the harmonic trap period is short compared with the, collision time of the solitons. We find that the collision time of the solions is dependent on the relative phase of the solitons as they collide. In the case of two solitons, the particle model is integrable, and the dynamics are completely regular. In the case of a system of two solitary waves of equal norm, the solitons are shown to retain their phase difference for repeated collisions. The extension to three particles supports both regular and chaotic regimes. The trajectory shift observed for two solitons carrier over to the case of three solitons. This shift aside, the agreement between the particle model and the wave dynamics remains good, even in chaotic regimes. We predict that these chaotic regimes will be an indicator of rapid depletion of the condensate due to quantum transitions of the condensate particles into non-condensate modes. To analyse the residual effects of the three dimensional nature of the solitary waves, we use a nonlinear Schrödinger equation with an additional quintic term. We perform variational calculations, and confirm the collapse of a soliton when the number of particles contained therein is increased past a critical number. We investigate the effects of varying the axial trap frequency and scattering length on the critical number. We propose a method to model particle exchange between solitons by extending the variational treatment to two solitons.

In this work we present a variational algorithm to determine the parameters iir(x) and er(x) in the Maxwell system VxE + k xTH = 0, V x H - kerE = 0 in a body Q from boundary measurements of electromagnetic pairs (n x En dci,n x Hn dn), n= 1,2,…, where n is the outer unit normal. We show that this inverse problem can be solved by minimizing a positive functional C7(m,c) and using a conjugate gradient scheme. Apart from implementations with global boundary, we also consider the case of partial boundary, where we have only data available on a subset T C dQ. Further do we develop uniqueness results, to show that the given data (n x En dn, n x Hn dn), n = 1,2,…, is a sufficient basis to solve the inverse problem. We investigate the uniqueness properties of the inverse problem in the case of global boundary data as well as in the case of partial boundary data. To show the effectivness and the stability of our approach we present various numerical results with noisy data. Finally we outline an alternative method, where one is only interested in recovering the support of the functions fi l 1 and er 1.

The present thesis deals with microfluidic systems under the influence of electric fields. The purpose of this research is to identify key behaviours which are highly relevant for applications in lab-on-chip devices such as pH control, patterning, mixing and cell trapping. In the first part of the thesis we consider the electrokinetics of charged, porous membranes and present a mathematical model for the ionic transport under the effects of a horizontal electric field. First, we investigate the behaviour of a system that consists of one anion membrane with two reservoirs and produce numerical solutions with the aim to gain a better understanding of the mechanisms that lead to overlimiting current. We then analyse the features of a system where a bipolar membrane is held in an electrolyte bath with water and a salt. We use our model to confirm findings from experiments such as the hysteretic behaviour of the IV curve and the water splitting phenomenon. In the second part of the thesis we examine the behaviour of interfaces between two fluids that are sandwiched between two electrodes. We find that introducing a constant flow rate into the system leads to time modulated travelling waves. In the case of flat channel walls these look like moving strips which are reminiscent of the patterns found in the no flow case. Adding corrugations on one or both electrodes leads to a rich variety of dynamics. We develop a Floquet stability analysis which takes into account the fact that the base state of the system is nonuniform. This is a very useful tool for identifying the different types of behaviours which arise as we change the applied voltage and the overall flow rate. We examine the streamlines of the fluid to explore the advantages of the different regimes: just by changing the applied voltage it is possible to transition from an environment which is favourable to efficient mixing to one which could enhance cell trapping.

In this thesis, the emerging field of higher gauge theory will be discussed, particularly in relation to problems arising in M-theory, such as selfdual strings and the so-called (2,0) theory. This thesis will begin with a Nahm-like construction for selfdual strings using loop space, the space of loops on spacetime. This construction maps solutions of the Basu-Harvey equation, the BPS equation arising in the description of multiple M2-branes, to solutions of a selfdual string equation on loop space. Furthermore, all ingredients of the construction reduce to those of the ordinary Nahm construction when compactified on a circle with all loops restricted to those wrapping the circle. The rest of this thesis, however, will not involve loop space. We will see a Nahm-like construction for the case of infinitely many selfdual strings, suspended between two M5-branes. This is possible since the limit taken renders the fields describing the M5-branes abelian. This avoids the problem which the rest of this thesis focuses on: What fields describe multiple M5-branes? The answer is likely to involve higher gauge theory, a categorification of gauge theory which describes the parallel transport of extended objects. Any theories which involves 3-algebras, including current M2-brane models and the Lambert-Papageorgakis M5-brane model, are examples of higher gauge theories. Recently, a class of models with N = (1, 0) supersymmetry have been found, with significant overlap with algebraic structures in higher gauge theory. This overlap suggests that the full N = (2, 0) theory could involve semistrict L∞-algebras. Finally, we will see some explicit selfdual string solutions, which may fit into these frameworks.

We study in this thesis the superconformal index of 4-dimensional N 2 SUpNq gauge eld theories on S1 S3. We focus in particular on a reduced version of the index, known as the Schur index, with only one fugacity q for the space-time symmetries. We start by studying circular quiver gauge theories which can be diagrammatically represented in a way reminiscent of A^ Dynkin diagrams. The Schur index of these theories is usually given in terms of a complex matrix model involving various elliptic functions. We use an elliptic determinant identity to simplify the integrand and express it in terms of determinants, allowing us to rewrite the whole index as a weighted sum over partition functions of free Fermi gases living on a circle. Each partition function is then studied in the grand canonical ensemble and we nd exact compact expressions for the analogue of the grand partition function which we dene as the grand index. For short quivers with only one or two nodes we are able to analytically deduce the Schur index exaclty as a q-series, and for small values of N we are able to express it in terms of the complete elliptic integrals of the rst and second kind. For longer quivers, we are able to extract the Schur index in the large N limit, up to non-perturbative corrections. We also investigate another class of theories, namely ^D-type quivers, for which we are able to apply the same techniques. We obtain the grand index as well as a simple and compact expression for its leading term in the large chemical potential limit. This thesis also contains a detailed review of the superconformal index, a comparison of our results with some other previously known expressions for the Schur index obtained through other formalisms, as well as various technical appendices.

No description available.

In this thesis we considered non-Abelian monodromy groups in F-theory, based on SU(5) GUTs, with models presented using the V4, A4, and D4 monodromy actions. The role of these symmetries in generating the observed mixing patterns of neutrinos has been considered in the cases of A4 and D4 monodromy. An older model based on E6 is also considered as a candidate model to explain the recent LHC diphoton excess. R-parity violating processes in F-theory have also been considered, and it has been shown that such effects should be considered generic in F-theory without an ad hoc R-parity. The strengths of such coupling are discussed in the context of local F-theory Yukawa computations.

This thesis concerns research undertaken in two related topics concerning high-energy gravitational physics. The first is the construction of a manifestly diffeomorphisminvariant Exact Renormalization Group (ERG). This is a procedure that constructs effective theories of gravity by integrating out high-energy modes down to an ultraviolet cutoff scale without gauge-fixing. The manifest diffeomorphism invariance enables us to construct a fully background-independent formulation. This thesis will explore both the fixed-background and background-independent forms of the manifestly diffeomorphism-invariant ERG. The second topic is cosmological backreaction, which concerns the effect of averaging over high-frequency metric perturbations to the gravitational field equations describing the universe at large scales. This has been much studied the context of the unmodified form of General Relativity, but has been much less studied in the context of higher-derivative effective theories obtained by integrating out the high-energy modes of some more fundamental (quantum) theory of gravity. The effective stress-energy tensor for backreaction can be used directly as a diffeomorphism-invariant effective stress-energy tensor for gravitational waves without specifying the background metric. This thesis will construct the manifestly diffeomorphism-invariant ERG and compute the effective action at the classical level in two different schemes. We will then turn to cosmological backreaction in higher-derivative gravity, deriving the general form of the effective stress-energy tensor due to inhomogeneity for local diffeomorphism-invariant effective theories gravity. This an exciting research direction, as it begins the construction of a quantum theory of gravity as well as investigating possible implications for cosmology.