The main objective of the thesis was to extend the definition of an M-Group to what is called an M-Ring. From this extension a system called an expanded ring follows naturally. To facilitate the development of the expanded ring, chapter I is devoted to developing properties on systems that are not quite rings where many interesting examples are constructed. In chapter II the definition of an M-Ring is given and some of its properties are derived. In chapter III some of the properties of expanded rings are proved, and examples of expanded rings are given to show their existence.

The goal of this thesis is to investigate a conjecture about Mirror Symmetry for Landau Ginzburg (LG) models with non-abelian gauge groups. The conjecture predicts that the LG A-model for a polynomial-group pair $(W,G)$ is equivalent to the LG B-model for the dual pair $(W^*, G^*)$. In particular, the A-model and B-model include the construction of a Frobenius algebra. The LG mirror symmetry conjecture predicts that the A-model Frobenius algebra for $(W,G)$ will be isomorphic to the B-model Frobenius algebra for the dual pair $(W^*,G^*)$. Part of the conjecture includes a rule describing how to construct the dual pair. Until now, no examples of this phenomenon have been verified. In this thesis we will verify the conjecture for the polynomial $W(x_1,x_2,x_3,x_4) = x_1^4+x_2^4+x_3^4+x_4^4$ with a maximal admissible non-abelian group. I present a supplementary guide along with a worked example to compute the state spaces of each of the A and B models with non-abelian groups. This includes formalizing G-actions to take invariants, computing each state space, formalizing the product on each state space, and as the main result, showing there indeed exists an isomorphism of Graded Frobenius Algebras between the LG A-model and dual LG B-model.

Quantum holonomies are non-Abelian Geometric Phases predominantly observed in adiabatic, non-adiabatic, or measurement-based quantum evolutions. Their significance lies in their potential utility within quantum computing due to their robustness against noise throughout the parameter path. In this report, we detail the foundational methods necessary for constructing holonomic non-Abelian gates specifically designed for tripartite states <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%7CW%3E" data-classname="equation" data-title="" />and <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%7C%5Cbar%7BW%7D%3E" data-classname="equation" data-title="" />, which serve as the logical qubits in our project. Given that the existence of a universal set of gates has already been demonstrated for each of these evolution types, our project delves into the advantages of applying these basis states across the three evolution categories. We have reformulated the Nuclear Quadrupole Resonance (NQR) Hamiltonian to be exclusively composed of two-body terms, thus rendering it more experimentally feasible. Furthermore, we have connected the W states with the remaining tripartite states to construct a four-level model system and generalized gates within this framework. Lastly, we introduce a measurement-based method that maintains its non-Abelian attributes even in the Zeno limit, where the process of projective measurement gradually approaches the adiabatic model. / Icke-Abelska geometriska faser, så kallade kvantholonomier, observeras huvudsakligen i adiabatiska, icke-adiabatiska eller mätningsbaserade manipulationer av kvanttillstånd. De har stor potential till användning inom kvantdatorberäkningar på grund av deras robusthet mot olika typer av brus. I den här rapporten beskriver vi de grundläggande metoderna som är nödvändiga för att konstruera holonoma kvantgrindar som är speciellt utformade för trekroppstillstånden <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%7CW%3E" data-classname="equation" /> och <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%7C%5Cbar%7BW%7D%3E" data-classname="equation" data-title="" />, som fungerar som de logiska kvantbitarna i projektet, givet att det är redan bevisat att alla dessa modeller kan klara kraven för universalitet. Den här rapporten fokuserar på fördelarna med att tillämpa dessa logiska kvantbitar för tre olika evolutionskategorier. Vi har omformulerat kärnkvadrupolresonans Hamiltonianen så att den uteslutande består av tvåkroppstermer, vilket gör den mer experimentellt genomförbar för att realisera adiabatiska holonoma kvantgrindar. Vidare har vi kopplat <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?W" data-classname="equation" data-title="" />-tillstånden med andra trekroppstillstånden för att konstruera ett så kallat sammanflätat <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%5CLambda" data-classname="equation" data-title="" />-system och icke-adiabatiska holonoma kvantgrindar inom detta ramverk. Slutligen introducerar vi en mätningsbaserad metod som, till skillnad från tidigare resultat, bibehåller sina icke-Abelska attribut även i Zeno-gränsen, där processen med projektiv mätning gradvis närmar sig den adiabatiska kärnkvadrupolresonans-modellen.

In this thesis, we study the degree of the canonical map of surfaces of general type. In particular, we give the first examples known in the literature of surfaces having degree d=10,11, 13, 14, 15, and 18 of the canonical map. They are presented in a self-contained and independent way from the rest of the thesis. We show also how we have discovered them. These surfaces are product-quotient surfaces. In this thesis, we study the theory of product-quotient surfaces giving also some new results and improvements. As a consequence of this, we have written and run a MAGMA script to produce a list of families of product-quotient surfaces having geometric genus three and a self-intersection of the canonical divisor large. After that, we study the canonical map of product-quotient surfaces and we apply the obtained results to the list of product-quotient surfaces just mentioned. In this way, we have discovered the examples of surfaces having degree d=10,11,14, and 18 of the canonical map. The remaining ones with degrees 13 and 15 do not satisfy the assumptions to compute the degree of the canonical map directly. Hence we have had to compute the canonical degree of these two families of product-quotient surfaces in a very explicit way through the equations of the pair of curves defining them. Another work of this thesis is the classification of all smooth surfaces of general type with geometric genus three which admits an action of a group G isomorphic to \mathbb Z_2^k and such that the quotient is a projective plane. This classification is attained through the theory of abelian covers. We obtained in total eleven families of surfaces. We compute the canonical map of all of them, finding in particular a family of surfaces with a canonical map of degree 16 not in the literature. We discuss the quotients by all subgroups of G finding several K3 surfaces with symplectic involutions. In particular, we show that six families are families of triple K3 burgers in the sense of Laterveer. Finally, in another work we study also the possible accumulation points for the slopes K^2/ \chi of unbounded sequences of minimal surfaces of general type having a degree d of the canonical map. As a new result, we construct unbounded families of minimal (product-quotient) surfaces of general type whose degree of the canonical map is 4 and such that the limits of the slopes K^2/ \chi assume countably many different values in the closed interval [6+2/3, 8].

No description available.

In the first part of this thesis, we investigate a way to find the complex structure moduli, for a given background of type IIB string theory in the presence of flux in special cases. We introduce a way to compute the complex structure and axion dilaton moduli explicitly. In the second part, we discuss $(0,2)$ supersymmetric versions of some recent exotic $mathcal{N}=(2,2)$ supersymmetric gauged linear sigma models, describing intersections of Grassmannians. In the next part, we consider mirror symmetry for certain gauge theories with gauge groups $F_4$, $E_6$, and $E_7$. In the last part of this thesis, we study whether certain branched-double-cover constructions in Landau-Ginzburg models can be extended to higher covers. / Doctor of Philosophy / This thesis concerns string theory, a proposal for unification of general relativity and quantum field theory. In string theory, the building block of all the particles are strings, such that different vibrations of them generate particles. String theory predicts that spacetime is 10-dimensional. In string theorist's intuition, the extra six-dimensional internal space is so small that we haven't detected it yet. The physics that string theory predicts we should observe, is governed by the shape of this six-dimensional space called a `compactification manifold.' In particular, the possible ways in which this geometry can be deformed give rise to light degrees of freedom in the associated observable physical theory. In the first part of this thesis, we determine these degrees of freedom, called moduli, for a large class of solutions of the so-called type IIB string theory. In the second part, we focus on constructing such spaces explicitly. We also show that there can be different equivalent ways of constructing the same internal space. The third part of the thesis concerns mirror symmetry. Two compactification manifolds are called mirror to each other, when they both give the same four-dimensional effective theory. In this part, we describe the mirror of two-dimensional gauge theories with $F_4$, $E_6$, and $E_7$ gauge group, using the Gu-Sharpe proposal.

This thesis introduces the emerging field of quantum computing, emphasizing its capability to surpass traditional computing by solving complex problems that are beyond the reach of classical computers. Unlike classical systems that operate with bits and logic gates, quantum computing utilizes qubits and quantum gates, exploiting the vast computational space offered by quantum mechanics. A focal point of this study is topological quantum computing, a novel approach designed to overcome the inherent vulnerability of quantum systems to errors, such as decoherence and operational inaccuracies. At the heart of this method lies the use of non-Abelian anyons, with a particular focus on Fibonacci anyons, whose unique topological characteristics and braiding operations present a viable path to fault-tolerant quantum computation. This thesis aims to elucidate how the braiding of Fibonacci anyons can be employed to construct the necessary quantum gates for topological quantum computing. By offering a foundational exploration of quantum computing principles, especially topological quantum computing, and detailing the process for creating quantum gates through braiding of Fibonacci anyons, the work sets the stage for further research and development in this transformative computing paradigm.

We investigate various relationships between categories of functors. The major examples are given by extending some duality to a larger structure, such as an adjunction or a recollement of abelian categories. We prove a theorem which provides a method of constructing recollements which uses 0-th derived functors. We will show that the hypotheses of this theorem are very commonly satisfied by giving many examples. In our most important example we show that the well-known Auslander-Gruson-Jensen equivalence extends to a recollement. We show that two recollements, both arising from different characterisations of purity, are strongly related to each other via a commutative diagram. This provides a structural explanation for the equivalence between two functorial characterisations of purity for modules. We show that the Auslander-Reiten formulas are a consequence of this commutative diagram. We define and characterise the contravariant functors which arise from a pp-pair. When working over an artin algebra, this provides a contravariant analogue of the well-known relationship between pp-pairs and covariant functors. We show that some of these results can be generalised to studying contravariant functors on locally finitely presented categories whose category of finitely presented objects is a dualising variety.

Στην παρούσα διδακτορική διατριβή μελετάμε εφαρμογές οι οποίες σχετίζονται με την μη-Αβελιανή Τ-δυϊκότητα και την αντιστοιχία AdS/CFT. Στο πρώτο μέρος, το οποίο αντιστοιχεί στο πρώτο κεφάλαιο της διατριβής, παρουσιάζουμε συνοπτικά τα απαραίτητα μαθηματικά εργαλεία που απαιτούνται για την καλύτερη κατανόηση των κεφαλαίων που ακολουθούν. Στο δεύτερο μέρος, το οποίο αποτελείται από τα κεφάλαια 2,3 και 4, ασχολούμαστε με την έννοια της μη-Αβελιανής Τ-δυϊκότητας. Ποιο συγκεκριμένα, στο δεύτερο κεφάλαιο παρουσιάζουμε τους κανόνες Buscher της Αβελιανής Τ-δυϊκότητα καθώς και την γενίκευση τους στην μη-Αβελιανή περίπτωση. Επίσης στο κεφάλαιο αυτό δείχνουμε τον τρόπο με τον οποίο μπορούμε να εφαρμόσουμε τους κανόνες της μη-Αβελιανής Τ-δυϊκότητας σε υπόβαθρα υπερβαρύτητας τύπου II τα οποία περιλαμβάνουν πεδία Ramond-Ramond. Η διαδικασία αυτή μπορεί να θεωρηθεί σαν μια τεχνική κατασκευής νέων λύσεων υπερβαρύτητας. Στο τρίτο κεφάλαιο θεωρούμε μια γενική κατηγορία υποβάθρων υπερβαρύτητας με ισομετρία SO(4) στα οποία εφαρμόζουμε τον μετασχηματισμό της μη-Αβελιανής Τ-δυϊκότητας ως προς την υποομάδα SU(2) της ομάδας ισομετρίας. Πραγματοποιώντας διαστατική ελάττωση στην αρχική και την δυϊκή θεωρία καταλήγουμε στην ίδια επταδιάστατη θεωρία. Ως αποτέλεσμα, οποιαδήποτε λύση αυτής της επταδιάστατης θεωρίας μπορεί να ανυψωθεί ταυτόχρονα στο αρχικό και στο δυϊκό υπόβαθρο. Η παρατήρηση αυτή μας παρέχει μια αντιστρεπτή απεικόνιση μεταξύ δυο λύσεων υπερβαρύτητας τύπου II οι οποίες συνδέονται με έναν μετασχηματισμό μη-Αβελιανής Τ-δυϊκότητας. Επίσης, για την συγκεκριμένη περίπτωση υποβάθρων αποδεικνύουμε ότι το δυϊκό υπόβαθρο διατηρεί τη μισή υπερσυμμετρία σε σχέση με το αρχικό. Στο τέταρτο κεφάλαιο μελετάμε τη δράση της μη-Αβελιανής Τ-δυϊκότητας σε μια σειρά από υπόβαθρα με υπερσυμμετρία N=1 των οποίων οι δυϊκές θεωρίες πεδίου είναι γνωστές. Σκοπός του κεφαλαίου αυτού είναι η μελέτη της μη-Αβελιανής Τ-δυϊκότητας στα πλαίσια της αντιστοιχίας AdS/CFT. Αυτό το επιτυγχάνουμε μελετώντας διάφορες ποσότητες των θεωριών πεδίου που αντιστοιχούν στο αρχικό και στο δυϊκό υπόβαθρο. Τέλος, στο πέμπτο κεφάλαιο κάνουμε χρήση τεχνικών ολογραφίας προκειμένου να μελετήσουμε το φαινόμενο της εισαγωγής φερμιονικών προσμίξεων σε τρισδιάστατες θεωρίες ύλης τύπου Chern-Simons, οι οποίες περιλαμβάνουν μεγάλο αριθμό γεύσεων. / In this thesis, we study applications which are related to the non-Abelian T-duality and the AdS/CFT correspondence. In the first part, which corresponds to the first chapter of the present thesis, we briefly present the basic mathematical tools required for the better understanding of the material included in the next chapters. In the second part, which consists of the chapters 2,3 and 4, we deal with the concept of non-Abelian T-duality. More concretely, in the second chapter we present the Buscher rules of the Abelian T-duality and we generalize them to the non-Abelian case. We also show how to implement the rules of non-Abelian T-duality in backgrounds of type II supergravity with non-vanishing Ramond-Ramond fields. This proccess can be seen as a generating technique of new supergravity backgrounds. In the third chapter we consider a general class of supergravity backgrounds with SO(4) isometry and we perform the non-Abelian T-duality transforation with respect to the SU(2) subgroup of SO(4). After a consistent truncation to seven dimensions of both the original and the dual background we reproduce the same seven dimensional theory. As a result, any solution of this seven dimensional theory can be uplifted simultaneously to both the original and the dual background. This provides an invertible map between two solutions of type II supergravity which are related with a non-Abelian T-duality transformation. Also, in the case of supergravity backgrounds considered here, we show that the dual background preserves the half supersymmetry with respect to the original one. In chapter 4, we study the action of non-Abelian T-duality on a series of N=1 supersymmetric backgrounds whose field theory duals are well understood. The aim of this chapter is to study the transformation of non-Abelian T-duality within the framework of the AdS/CFT correspondence. This is done by considering several observables of the field theories that correspond to both the original and the dual backgrounds. Finally, in chapter 5 we use holographic techniques in order to study the effect of the addition of fermionic impurities to the three dimensional Chern-Simons matter theories with a large number of flavors.

Submitted by Vasti Diniz (vastijpa@hotmail.com) on 2017-09-12T13:00:44Z No. of bitstreams: 1 arquivototal.pdf: 2334284 bytes, checksum: c3d087bb8f68c3b8f619b05f161a3e77 (MD5) / Made available in DSpace on 2017-09-12T13:00:44Z (GMT). No. of bitstreams: 1 arquivototal.pdf: 2334284 bytes, checksum: c3d087bb8f68c3b8f619b05f161a3e77 (MD5) Previous issue date: 2016-02-25 / Conselho Nacional de Pesquisa e Desenvolvimento Científico e Tecnológico - CNPq / In this thesis, we study the influence of gravitating non-Abelian cosmic strings on the spacetime geomerty. In order to develop this analysis, we constructed a set of coupled non-linear differential equations. Because there is no closed solution for this set of equations, we solve it numerically to determine the behaviour for the Higgs, gauge and metric fields. This model under consideration present two bosonic sectors, besides the non-Abelian gauge field. The two bosonic sectors may present a direct coupling. So, we investigate the relevance of this coupling on the system, specifically in the linear energy density of the string and on the planar angle deficit. We also analyze the behaviors of these quantities as function of the energy scale where the gauge symmetry is spontaneously broken. We have extented this analysis to de Sitter and anti-de Sitter spacetimes. In order to do that we construct the complete set of equations of motion considering the presence of a cosmological constant. By using numerical analysis we provide the behavior of the Higgs and gauge fields and also for the metric tensor for specific values of the physical parameters of the theory. For de Sitter case, we find the appearance of horizons that although being consequence of the presence of the cosmological constant it strongly depends on the value of the gravitational coupling. In the anti-de Sitter case, we find that the system does not present horizons. In fact the new feature of this system is related with the behavior of the (tt) and (zz) components of the metric tensor. They present a strongly increasing for large distance from the string. / Nesta tese estudamos a influência das cordas cósmicas não-Abelianas na geometria do espaço-tempo. Para este fim, utilizamos um modelo de Higgs não-Abeliano acoplado com a gravidade e obtemos um sistema de equações diferenciais não-lineares. Como este sistema de equações diferenciais não possui solução analítica, realizamos análise numérica para obter o comportamento dos campos de Higgs, de gauge e métricos em função da distância à corda cósmica. O modelo considerado apresenta dois campos bosônicos e um campo de gauge não-Abeliano. Como os dois setores bosônicos podem apresentar um acoplamento direto, investigamos a relevância deste acoplamento no sistema, especificamente na densidade linear de energia e no déficit de ângulo planar. Também analisamos o comportamento destas quantidades como função da escala de energia onde a simetria de gauge é espontaneamente quebrada. Ampliamos este estudo para as cordas cósmicas não-Abelianas no espaço-tempo de de Sitter e anti-de Sitter. Para isto, construímos um sistema de equações de campo considerando a presença da constante cosmológica. Utilizando a análise numérica, fornecemos o comportamento dos campos de Higgs, de gauge e dos campos métricos para valores específicos dos parâmetros físicos do modelo. Para o caso do espaço-tempo de de Sitter, salientamos o surgimento do horizonte cosmológico que, embora seja consequência da constante cosmológica, está fortemente relacionado ao acoplamento gravitacional. Para o espaço-tempo de anti-de Sitter, encontramos que o sistema não apresenta horizonte. Esta característica do sistema está relacionada às componentes (tt) e (zz) do tensor métrico, que divergem para grandes distâncias da corda cósmica.