In this dissertation, we investigatethe supersymmetric completion of curvature squared invariants in five and six dimensionsas well as the construction of off-shell Poincar´e supergravities and their matter couplings. We use superconformal calculus in fiveand six dimensions, which are an off- shell formalisms. In fivedimensions,there are twoinequivalentWeyl multiplets: the standard Weyl multiplet and the dilaton Weyl multiplet.The main difference betweenthese twoWeyl multiplets is thatthe dilaton Weyl multipletcontains a graviphoton in its field content whereas the standard Weyl multiplet does not.A supergravity theory based on the standard Weyl multiplet requires coupling to an external vector multiplet. In five dimensions,we construct two new formulations for 2-derivative off-shell Poincar´e supergravity theories and present the internally gauged models. We also construct supersymmetric completions of all curvature squared terms in five dimensional supergravity with eight supercharges.Adopting the dilaton Weyl multiplet, we construct a Weyl squared invariant, the supersymmetric combination of Gauss-Bonnet combination and the Ricci scalar squared invariant as well as all vector multiplets coupled curvature squared invariants. Since the minimal off-shell supersymmetric Riemann tensor squared invariant has been obtained before, both the minimal off-shell and the vector multiplets coupled curvature squared invariants in the dilation Weyl multiplet are complete. We also constructedan off-shell Ricci scalar squared invariant utilizing the standard Weyl multiplet.The supersymmetric Ricci scalar squared in the standard Weyl multiplet is coupled to n number of vector multiplets by construction, and it deforms the very special geometry. We found that in the supersymmetric AdS5 vacuum, the very special geometry defined on the moduli space is modified in a simple way. We study the vacuum solutions with AdS2 × S3 and AdS3 × S2 structures. We also analyze the spectrum around a maximally supersymmetric Minkowski5, and study the magnetic string and electric black hole. Finally, we generalize our procedure for the construction of an off-shell Ricci scalar squared invariant in five dimensions to N = (1, 0), D = 6 supergravity.

In this thesis, higher derivative theories and constrained dynamics are investigated in detail. In the first part of the thesis, we discuss how the Ostrogradski instability emerges in non-degenerate higher derivative theories in the context of a one-dimensional point particle where the position of the particle is a function only dependent on time. We show that the instabilities can only be removed by the addition of constraints if the original theory’s phase space is reduced. We then generalize this formalism to the most general higher derivative gravity theory where the action is not only linearly dependent on the Ricci scalar but also the quadratic curvature invariants in four-dimensional spacetime. We find that the instabilities can be removed by the judicious addition of constraints at the quadratic level of metric fluctuations around Minkowski and de Sitter backgrounds while the dimensionality of the original phase space is reduced. The constrained higher derivative gravity theory is ghost free as well as preserves the renormalization properties of higher derivative gravity, at the price of giving up the Lorentz invariance. In the second part of the thesis, we study the spherically symmetric static solution of a class of two scalar-field theory, where one of them is a Lagrange multiplier enforcing a constraint relating the value of the other scalar field to the norm of its derivative. We find the spherically symmetric static solution of the theory with an exponential potential. However, when we investigate the stability issue of the solution, the perturbation with the odd type symmetry is stable, while the even modes always contain one ghostlike degree of freedom.

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In this thesis massive higher derivative gravity theories are analyzed in some detail. One-particle scattering amplitude between two covariantly conserved sources mediated by a graviton exchange is found at tree-level in D dimensional (Anti)-de Sitter and flat spacetimes for the most general quadratic curvature theory augmented with the Pauli-Fierz mass term. From the amplitude expression, the Newtonian potential energies are calculated for various cases. Also, from this amplitude and the propagator structure, a three dimensional unitary theory is identified. In the second part of the thesis, the found three dimensional unitary theory is studied in more detail from a canonical point of view. The general higher order action is written in terms of gauge-invariant functions both in flat and de Sitter backgrounds. The analysis is extended by adding static sources, spinning masses and the gravitational Chern-Simons term separately to the theory in the case of flat spacetime. For all cases the microscopic spectrum and the masses are found. In the discussion of curved spacetime, the masses are found in the relativistic and non-relativistic limits. In the Appendix, some useful calculations that are frequently used in the bulk of the thesis are given.

In this thesis, higher derivative theories are investigated. Ostrogradskian instability of higher derivative theories is examined both at the classical and quantum levels. It is shown that avoiding the instability in nondegenerate higher derivative theories is impossible. Moreover, the degenerate model of relativistic particle with a curvature term is studied as a viable higher derivative theory. Most of the work we present here is not original. We give a review of the literature and compile various detached works that already exist.

In this master s thesis, we have studied the possibility of Lorentz invariance violation by considering some terms of the standard-model extesion (SME), specifically, those terms which refer to the quantum electrodynamics extension. We have performed quantum corrections in the fermionic sector of the usual quantum electrodynamics with added terms which violate the Lorentz and CPT symmetries in two different configurations. Firstly, adding operators governed by the coefficient gκλμ and, in a later time, adding operators governed by the coefficient bμ. In the theory with the coefficient gκλμ, we have performed quantum corrections in order to induce, in the bosonic sector of the theory, the conventional Chern-Simons and the higher-derivative terms, and analyze the behavior of both terms in the finite temperature regime, by using the method of derivative expansion and the Matsubara formalism. On the other hand, in the model with the coefficient bμ, we have performed quantum corrections in order to induce, in the bosonic sector, the higher-derivative Chern-Simons term, by using the method of derivative expansion and, subsequently, analyze the behavior of this term in the finite temperatura regime, where we have used the Matsubara formalism again. Thus, for the first case, we have observed that the conventional Chern-Simons term is nonzero only at finite temperature, whereas the higher-derivative term is finite at zero temperature, however, this term goes to zero when the temperature increases infinitely. In the second part of our study, we have shown that the higher-derivative Chern-Simons term is induced at zero temperature, nevertheless, this term goes to zero when the temperature increases too much. / Conselho Nacional de Desenvolvimento Científico e Tecnológico / Nesta dissertação, estudamos a possibilidade de violação da invariância de Lorentz levando em conta alguns termos do modelo padrão estendido, mais especificamente, da parte deste modelo que trata da eletrodinâmica quântica estendida. Realizamos correções quânticas no setor fermiônico da eletrodinâmica quântica usual adicionada de termos que violam as simetrias de Lorentz e CPT em duas configurações diferentes. Primeiramente, adicionando operadores governados pelo coeficiente gκλμ e, em um momento posterior, adicionando operadores governados pelo coeficiente bμ. Para a teoria com o coeficiente gκλμ, realizamos correções quânticas com a intenção de induzir, no setor bosônico da teoria, os termos de Chern-Simons convencional e o de derivada superior, e analisar o comportamento de ambos os termos no regime de temperatura finita, fazendo o uso do método da expansão derivativa e do formalismo de Matsubara. Já para o modelo com o coeficiente bμ, realizamos correções quânticas a fim de induzir, no setor bosônico, o termo de Chern-Simons de derivada superior, usando o método da expansão derivativa e, posteriormente, analisar o comportamento deste termo no regime de temperatura finita, ao utilizarmos novamente o formalismo de Matsubara. Assim, para o primeiro caso, notamos que o termo de Chern-Simons convencional é diferente de zero apenas à temperatura finita, enquanto que, o termo de derivada superior é finito em temperatura zero, porém, tal termo vai a zero quando a temperatura cresce infinitamente. Na segunda parte do nosso estudo, mostramos que o termo de Chern-Simons de derivada superior é induzido no regime de temperatura zero, contudo, tal termo vai a zero quando a temperatura cresce muito.

In dieser Arbeit untersuchen wir dualitätsinvariante höher-abgeleitete (alpha') Korrekturen an effektiven Stringtheorien mit niedriger Energie. Wir beschränken uns auf den universellen masselosen Sektor, bestehend aus Graviton, B-Feld und Dilaton, und spezialisieren uns auf Hintergründe mit d abelschen Isometrien, die O(d, d)-Invarianz genießen, und schließen Szenarien wie Kosmologie und Schwarze Löcher als Spezialfälle ein. Es wird erwartet, dass die O(d, d)-Symmetrie für beliebige Ableitungsordnungen erhalten bleibt, eine Tatsache, die Hohm und Zwiebach dazu motivierte, die kosmologische Klassifikation zu finden. Eine solche Konstruktion parametrisiert eine große Klasse rein zeitabhängiger dualitätsinvarianter Theorien für alle Ableitungsordnungen in Form einiger Koeffizienten. Stringtheorien stellen einzelne Punkte in diesem Theorieraum dar, die durch bestimmte Konfigurationen dieser Koeffizienten bestimmt werden. Wir berechnen die ersten Koeffizienten mit zwei Methoden: Verdichtungen und Beta-Funktionsberechnungen aus einer kosmologischen Weltenblattaktion. Anschließend untersuchen wir die Hohm-, Siegel- und Zwiebach-Theorie (HSZ) im kosmologischen Hintergrund, die die masselosen Stringmoden und neuartige massive Felder kodiert. Wir integrieren letztere heraus, um eine effektive Theorie für die masselosen Felder zu erhalten und lesen die entsprechenden Koeffizienten ab. Anschließend behalten wir die massiven Felder bei, finden eine zweifach abgeleitete Neuformulierung der Theorie, bestimmen die alpha'-genauen Friedmann-Gleichungen und erforschen die spannungsfreie Grenze. Wir überprüfen die Klassifikation für FRW-Hintergründe und stellen fest, dass alle Korrekturen trivial sind. Wir erweitern die Klassifikation auf zweidimensionale Hintergründe mit zeitähnlicher Isometrie, einem Raum, der Lösungen für schwarze Löcher zulässt. Wir finden alpha'-deformierte Schwarze Löcher mit und ohne Singularitäten. Letztere entsprechen regulären Kosmologien. / In this thesis we study duality-invariant higher-derivative (alpha') corrections to string low energy effective theories. We restrict to the universal massless sector, consisting of the graviton, B-field and dilaton, and specialize to backgrounds with d abelian isometries, which enjoy O(d, d)-invariance and contain scenarios such as cosmology and black holes as particular cases. The O(d, d)-symmetry is expected to be preserved to arbitrary orders in derivatives, a fact that motivated Hohm and Zwiebach to arrive at the cosmological classification. Such construction parameterizes a large class of purely-time dependent duality-invariant theories to all orders in derivatives in terms of a countable infinite number of coefficients. String theories represent single points in this theory space, determined by specific configurations of these coefficients. We compute the first coefficients by following two methods: compactifications and beta-function computations from a worldsheet action already in cosmological backgrounds. We then study the Hohm, Siegel, and Zwiebach (HSZ) theory in cosmological backgrounds, which encodes the massless string modes plus novel massive fields. We integrate out the latter to arrive at an effective theory for the massless fields and read the corresponding coefficients. We then keep the massive fields, find a two-derivative reformulation of the theory, determine alpha'-exact Friedmann equations and explore the tensionless limit. Coming back to generic duality-invariant theories, we revisit the classification for FRW backgrounds and find that all corrections are trivial. We extend the classification to two-dimensional backgrounds with time-like isometry, a space that admits black-hole solutions. We prove that the dual of a solution with a regular horizon must have a curvature singularity, and find alpha'-deformed black holes with and without singularities. The latter correspond to regular cosmologies.