Дисертації з теми "String dualities"

In the present thesis, we offer an introduction to type IIB string compactifications on $\mathbb{T}^{d}/\Gamma$ toroidal orbifolds. We first describe the technical method to construct these spaces and reduce the string background on it. We will have (non)-geometrical fluxes arising from these spaces which decorate with discrete deformations our four $\mathcal{N}=1$ dimensional supergravity theory. Solving its equations of motion, we find several families of supersymmetric AdS vacua with fixed moduli, which can be related through a set of  $SL(2,\mathbb{Z})$ symmetries.

This thesis is devoted to various questions connected with duality. It is composed of two parts.The first part discusses some aspects of timelike T-duality. We explore the possibility of compactification of supergravity theories with various signatures(low energy limit of M-theories which are dual under timelike T-dualities) on parallelizable internal seven dimensional (pseudo-)spheres. We show that, beside the standard theory, only one of the dual theories known as M'-theory can admit such a solution. The effective four dimensional theory is non-supersymmetric and due to the presence of torsion the symmetry of seven dimensional (pseudo-)sphere breaks down to Spin(3, 4). In the second part, in an attempt to have a systematic discussion of gaugings in supergravity, we show the isomorphism between the space of local deformations of the appropriate zero coupling limit of the embedding tensor Lagrangian and that of the second-order scalar-vector Lagrangian, describing the bosonic sector of supergravity ignoring gravity, in a chosen duality frame determined by embedding tensors. We analyze the BV-BRST deformation of a class of scalar-vector coupled Lagrangians, which contains supergravity Lagrangians as examples, and find a set of constraints that guarantee the consistency of the deformations of the Lagrangians. We show in principle that for a large class of theories considered in this thesis, the only deformations are those of the Yang-Mills type associated with a subgroup of the rigid symmetries.
Doctorat en Sciences
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Understanding the dynamics of strongly coupled gauge theories is one of the greatest challenges in modern theoretical physics. A new hope in attacking this problem was brought by the surprising discovery of integrability in a special four-dimensional gauge theory { the N = 4 supersymmetric Yang-Mills theory (SYM) in the limit of large number of colors. Quantum integrability manifests itself as a powerful hidden symmetry which allows to explore the theory far beyond the conventional perturbative regime, and may even lead to its exact solution. Integrability should also shed light on the striking gauge/string duality, which holographically relates N = 4 SYM with a string theory in curved geometry. In this thesis we focus on one of the key quantities in the N = 4 SYM theory { its spectrum of conformal dimensions, which correspond to string state energies. The study of integrability has culminated in reformulation of the spectral problem as a compact set of Riemann-Hilbert type equations known as the Quantum Spectral Curve (QSC). We demonstrate the power of this framework by applying it to study the spectrum in a wide variety of settings. The new methods which we present allow to explore previously unreachable regimes. We rst discuss an all-loop solution in a near-BPS limit, leading also to new strong coupling predictions. Next we describe an ecient numerical algorithm which allows to compute the nite-coupling spectrum with nearly unlimited precision (e.g. 60 digits in some important cases). We also present a universal analytic iterative method, which in particular allows to solve a longstanding open problem related to the BFKL limit in which N = 4 SYM develops close links with QCD. Finally we propose the extension of the QSC to the deformed case corresponding to a cusped Wilson line, uncovering new algebraic features of the construction. This allows to systematically study the generalized quark-antiquark potential and generate numerous new results.

In this thesis the recently developed duality covariant approach to string and Mtheory is investigated. In this formalism the U-duality symmetry of M-theory or Tduality symmetry of Type II string theory becomes manifest upon extending coordinates that describe a background. The effective potential of Double Field Theory is formulated only up to a boundary term and thus does not capture possible topological effects that may come from a boundary. By introducing a generalised normal we derive a manifestly duality covariant boundary term that reproduces the known Gibbons-Hawking action of General Relativity, if the section condition is imposed. It is shown that the full potential can be represented as a sum of the scalar potential of gauged supergravity and a topological term that is a full derivative. The latter is written totally in terms of the geometric flux and the non-geometric Q-flux integrated over the doubled torus. Next we show that the Scherk-Schwarz reduction of M-theory extended geometry successfully reproduces known structures of maximal gauged supergravities. Local symmetries of the extended space defined by a generalised Lie derivatives reduce to gauge transformations and lead to the embedding tensor written in terms of twist matrices. The scalar potential of maximal gauged supergravity that follows from the effective potential is shown to be duality invariant with no need of section condition. Instead, this condition, that assures the closure of the algebra of generalised diffeomorphisms, takes the form of the quadratic constraints on the embedding tensor.

In this work, we extend the well-known spectral cover construction first developed by Friedman, Morgan, and Witten to describe more general vector bundles on elliptically fibered Calabi-Yau geometries. In particular, we consider the case in which the Calabi-Yau fibration is not in Weierstrass form but can rather contain fibral divisors or multiple sections (i.e., a higher rank Mordell-Weil group). In these cases, general vector bundles defined over such Calabi-Yau manifolds cannot be described by ordinary spectral data. To accomplish this, we employ well-established tools from the mathematics literature of Fourier-Mukai functors. We also generalize existing tools for explicitly computing Fourier-Mukai transforms of stable bundles on elliptic Calabi-Yau manifolds. As an example of these new tools, we produce novel examples of chirality changing small instanton transitions. Next, we provide a geometric formalism that can substantially increase the understood regimes of heterotic/F-theory duality. We consider heterotic target space dual (0,2) GLSMs on elliptically fibered Calabi-Yau manifolds. In this context, each half of the ``dual" heterotic theories must, in turn, have an F-theory dual. Moreover, the apparent relationship between two heterotic compactifications seen in (0,2) heterotic target space dual pairs should, in principle, induce some putative correspondence between the dual F-theory geometries. It has previously been conjectured in the literature that (0,2) target space duality might manifest in F-theory as multiple $K3$-fibrations of the same elliptically fibered Calabi-Yau manifold. We investigate this conjecture in the context of both 6-dimensional and 4-dimensional effective theories and demonstrate that in general, (0,2) target space duality cannot be explained by such a simple phenomenon alone. In all cases, we provide evidence that non-geometric data in F-theory must play at least some role in the induced F-theory correspondence while leaving the full determination of the putative new F-theory duality to the future work. Finally, we consider F-theory over elliptically fibered manifolds, with a general conic base. Such manifolds are quite standard in F-theory sense, but our goal is to explore the extent of the heterotic/F-theory duality over such manifolds. We consider heterotic target space dual (0,2) GLSMs on elliptically fibered Calabi-Yau manifolds. In this context, each half of the ``dual" heterotic theories must, in turn, have an F-theory dual. Moreover, the apparent relationship between two heterotic compactifications seen in (0,2) heterotic target space dual pairs should, in principle, induce some putative correspondence between the dual F-theory geometries. It has previously been conjectured in the literature that (0,2) target space duality might manifest in F-theory as multiple $K3$-fibrations of the same elliptically fibered Calabi-Yau manifold. We investigate this conjecture in the context of both 6-dimensional and 4-dimensional effective theories and demonstrate that in general, (0,2) target space duality cannot be explained by such a simple phenomenon alone. In all cases, we provide evidence that non-geometric data in F-theory must play at least some role in the induced F-theory correspondence while leaving the full determination of the putative new F-theory duality to the future work. Finally, we consider F-theory over elliptically fibered manifolds, with a general conic base. Such manifolds are quite standard in F-theory sense, but our goal is to explore the extent of the heterotic/F-theory duality over such manifolds.
Doctor of Philosophy
String theory is the only physical theory that can lead to self-consistent, effective quantum gravity theories. However, quantum mechanics restricts the dimension of the effective spacetime to ten (and eleven) dimensions. Hence, to study the consequences of string theory in four dimensions, one needs to assume the extra six dimensions are curled into small compact dimensions. Upon this ``compactification," it has been shown (mainly in the 1990s) that different classes of string theories can have equivalent four-dimensional physics. Such classes are called dual. The advantage of these dualities is that often they can map perturbative and non-perturbative limits of these theories. The goal of this dissertation is to explore and extend the geometric limitations of the duality between heterotic string theory and F-theory. One of the main tools in this particular duality is the Fourier-Mukai transformation. In particular, we consider Fourier-Mukai transformations over non-standard geometries. As an application, we study the F-theory dual of a heterotic/heterotic duality known as target space duality. As another side application, we derive new types of small instanton transitions in heterotic strings. In the end, we consider F-theory compactified over particular manifolds that if we consider them as a geometry dual to a heterotic string, can lead to unexpected consequences.

We consider string vacua formed by compactifying Type II string theories on toroidal orbifolds and generalised Calabi-Yau manifolds and their transformations under a set of non-perturbative dualities. The dualities are the Type IIA-IIB exchanging T duality, the self-symmetry of Type IIB S duality, the non-trivial combination of the two, U duality, and the generalisation of T duality to include Calabi-Yaus, mirror symmetry. The requirement of the eective theory superpotential being invariant under these dualities is used to justify additional fluxes which do not descend via compactication from the ten dimensional action, which form an N = 2 theory. Their non-geometric structures, Bianchi constraints and tadpoles are determined and then classied in terms of modular S duality induced multiplets. The Z2 Z2 orientifold is used as an explicit example of the general methods, with N = 1 Type IIB non-geometric vacua which possess T and S duality invariance also constructed. These are then used to motivate the existence of an exchange between moduli spaces on self mirror dual manifolds with N = 2. Such an exchange is seen to result in flux structures which are schematically the same as the standard formulation but with inequivalent flux constraints.

In the following dissertation, we explore the applicability of Yangian symmetry to various integrable models, in particular, in relation with S-matrices. One of the main themes in this dissertation is that, after a careful study of the mathematics of the symmetry algebras one finds that in an integrable model, one can directly reconstruct S-matrices just from the algebra. It has been known for a long time that S-matrices in integrable models are fixed by symmetry. However, Lie algebra symmetry, the Yang-Baxter equation, crossing and unitarity, which are what constrains the S-matrix in integrable models, are often taken to be separate, independent properties of the S-matrix. Here, we construct scattering matrices purely from the Yangian, showing that the Yangian is the right algebraic object to unify all required symmetries of many integrable models. In particular, we reconstruct the S-matrix of the principal chiral field, and, up to a CDD factor, of other integrable field theories with su(n) symmetry. Furthermore, we study the AdS/CFT correspondence, which is also believed to be integrable in the planar limit. We reconstruct the S-matrices at weak and at strong coupling from the Yangian or its classical limit. This version of the thesis includes minor corrections following the viva on 17 September 2010.

Motivated by the AdS/CFT correspondence, new supersymmetric solutions to Type IIB and Type IIA supergravity are presented. These solutions contain $AdS_5$ or $AdS_4$ factors and are generated using T-duality symmetries of supergravity. The technique used to generate these solutions consists of performing a series of non-Abelian and Abelian T-dualities, sometimes with coordinate shifts in-between, to Freund-Rubin type seed backgrounds. An added bonus of the gauge fixing procedure inherent in non-Abelian T-Duality is the freedom to generate backgrounds with extra free parameters, some examples of which are presented. Aspects of the dual field theories of these new solutions are analyzed using holography techniques. The supersymmetry of these new backgrounds is also discussed. In addition to supergravity backgrounds with AdS, the study of generalized Calabi-Yau manifolds in the context of flux compactifications is briefly reviewed. The particular case of the resolved cone over $Y^{p,q}$ and its admission of generalized SU(3) structure solutions is examined. Contrary to geometries with $AdS$ factors, whose field theory duals are conformal field theories, these types of geometries can be phenomenologically interesting to study, as their gauge theory duals are minimally supersymmetric and confining, thus they could someday help aid our understanding of strongly-coupled QCD (Quantum Chromodynamics).

Bosonic string theory and superstring theory are briefly overviewed. Three dimensional field theories are similarly discussed, with a focus on effective $N=2$ supersymmetric theories. It is shown how to induced contributions to the Chern-Simons level of the low energy theory, by integrating out massive matter. Such effective field theories are then shown to arise from type IIB brane configurations based on the Hanany-Witten brane configuration. Strong-weak dualities are overviewed, leading to a discussion of the three dimensional strong-weak dualities: Aharony duality for theories with zero Chern-Simons level, and Giveon-Kutasov duality for theories with non-zero Chern-Simons level. In the results section, brane configurations corresponding to three-dimensional $N=2$ $U(N_{c})$ field theories with various numbers of flavour of massive matter are investigated. The resulting low energy field theories are explained, and the flows between Aharony and Giveon-Kutasov dualities are catalogued. Three dimensional $N=2$ effective field theories obtained through the inclusion of massive adjoint matter are also examined, with the flows between Aharony and Giveon-Kutasov dualities, again, catalogued. Finally, the significance of the results and the possibilities for future research, are discussed.

In this thesis we study gauge/gravity duals in the 5d/6d AdS/CFT correspondence. We start with field theories defined on squashed five-spheres with SU(3) × U(1) symmetry. These five-sphere backgrounds are continuously connected to the round sphere. We find a one-parameter family of 3/4 BPS deformations and a two-parameter family of (generically) 1/4 BPS deformations. The gravity duals are constructed in Euclidean Romans F(4) gauged supergravity in six dimensions, and uplift to massive type IIA supergravity. We holographically renormalize the Romans theory, and use our general result to compute the renormalized on-shell actions for the solutions. The results agree perfectly with the large N limit of the dual gauge theory partition function, which we compute using large N matrix model techniques. In addition we compute BPS Wilson loops in these backgrounds, both in supergravity and in the large N matrix model, again finding precise agreement. We conjecture a general formula for the partition function on any five-sphere background, which for fixed gauge theory depends only on a certain supersymmetric Killing vector. We then proceed to study Euclidean Romans supergravity in six dimensions with a non-trivial Abelian R-symmetry gauge field. We show that supersymmetric solutions are in one-to-one correspondence with solutions to a set of differential constraints on an SU(2) structure. As an application of our results we (i) show that this structure reduces at a conformal boundary to the five-dimensional rigid supersymmetric geometry previously studied, (ii) find a general expression for the holographic dual of the VEV of a BPS Wilson loop, matching an exact field theory computation, (iii) construct holographic duals to squashed Sasaki-Einstein backgrounds, again matching to a field theory computation, and (iv) find new analytic solutions to the squashed five-sphere background. We also analyse the classification of gravity duals with zero B-field.

L'une des théories les plus prometteuses qui vise à unifier la mécanique quantique et la relativité générale est actuellement la théorie des cordes. La recherche d'une formulation supersymétrique des cordes a conduit à cinq théories de supercordes cohérentes en dix dimensions qui ont ensuite été unifiées dans le cadre de la théorie M. L'aspect fondamental de cette unification est la découverte d'un réseau de dualités entre les cinq théories des supercordes et la supergravité à onze dimensions.Cette thèse aborde les dualités dans le contexte de la théorie F en huit dimensions. La théorie F est douze dimensionnelle et fournit une formulation non perturbative de la supergravité de type IIB avec 7-branes. La théorie des champs exceptionnels, quant à elle, fournit une formulation U-dual de la supergravité de type IIB. Nous nous concentrons donc sur les liens possibles entre ces deux formulations. La théorie F est également supposée être duale à la théorie des cordes hétérotique en 8 dimensions. La structure des groupes de jauge apparaît radicalement différemment dans ces deux formulations. Dans la théorie F, elle est interprétée comme un choix particulier de structure algébrique d'une surface K3 elliptique, tandis que dans le cadre de la corde hétérotique, elle est principalement déterminée par les lignes de Wilson. Bien qu'étudiée dans le contexte de la théorie des cordes de type IIB, l'identification entre les modules de la théorie F et de la théorie des cordes hétérotique n'est que peu connue.Dans la première partie de cette thèse, nous présentons les notions de base de la théorie des cordes, des compactifications, des branes et des dualités. Dans la seconde, nous montrons que la théorie des champs exceptionnels R+ X E₃(₃)en huit dimensions présente des aspects de la théorie F dans un cadre spécifique, et permet en particulier de décrire les monodromies des(p; q) 7-branes. Enfin, dans la troisième partie, nous étudions la dualité entre la compactification de la théorie F sur une surface K3 elliptique et la corde hétérotique sur un deux-tore. Nous présentons comment construire des surfaces K3 elliptiques via des polyèdres réflexifs qui peuvent être interprétés en termes de lignes de Wilson dans la théorie des cordes hétérotique duale
One of the most promising theories to unify quantum mechanics and general relativity is currently string theory. The search for a supersymmetric formulation of strings led to five consistent ten dimensional superstring theories which were later unified under the scope of M-theory. The fundamental aspect of this unification is the discovery of a web of dualities between the five superstring theories and eleven dimensional supergravity.This thesis addresses dualities in the context of F-theory in eight dimensions. F-theory is twelve dimensional and provides a non-perturbative formulation of type IIB supergravity with 7-branes. On the other hand exceptional field theory provides a U-dual formulation of type IIB supergravity and we therefore focus on the possible links between these two formulations. F-theory is also conjectured to be dual to the heterotic string in 8 dimensions. The gauge group appears radically differently in these two formulations. In F-theory it is interpreted as a particular algebraic structure of an elliptically fibered K3 surface, while on the heterotic string it is principally determined the Wilson lines. Although studied in the context of type IIB string theory, the explicit map between the moduli in F-theory and its heterotic dual are still quite unknown.In the first part of this thesis we present basic notions of string theory, compactifications, branes and dualities. In the second one, we show that R+ X E₃(₃) exceptional field theory in eight dimensions can incorporate aspects of F-theory in a specific setting, and in particular describes the monodromies of (p; q) 7-branes. Finally, in the third part we study the duality between F-theory compactified on an elliptic K3 surface and the heterotic string on a two-torus. We present how to construct elliptically fibered K3 surfaces via reflexive polytopes which can be understood in terms of Wilson lines in the dual heterotic string theory

We discuss various aspects of D-branes in string theory and holography in string theory and loop quantum gravity.

One way to study D-branes is from a microscopic perspective, using conformal field theory techniques. For example, we investigate the question of how D-branes can be introduced into orbifolded theories. Another way to study D-branes is from a space-time perspective. An example is provided by unstable D-branes, where we compute an effective action describing the decay of a bosonic D-brane.

The holographic principle is a proposed duality which suggests that a theory in any region has a dual description on the boundary. We explore two examples: (1) The area law for the entropy of a black hole in the framework of loop quantum gravity, related to particular regularizations of the area operator. (2) The AdS/CFT correspondence proposal, where we investigate a string pulsating on AdS using spin chains.

The goal of the original research presented in this thesis is to study the strong coupling regime of Quantum Field Theories (QFTs) with different methods, making concrete predictions about the phase structure and the dynamics of these theories, and on their observables. The focus is on (gauge) field theories in three spacetime dimensions, which are an interesting laboratory to understand the properties of strong coupling in setups that are usually simpler than in the more familiar case of gauge theories in four dimensions. Importantly, topological effects play a relevant role in three dimensions, thanks to the presence of the so-called Chern-Simons term.The thesis contains a short introduction to QFTs in 3d, principles and applications of infrared dualities, large N techniques, and holography. Indeed, the web of infrared dualities, the large N expansion, and the holographic correspondence between QFT and gravity are the main tools which we use to investigate the strongly coupled regimes of 3d QFTs.Then, the original material is presented. In a first line of research, we focus on the study of the phase diagram of a 3d gauge theory making use of conjectured infrared dualities, extending such dualities to the case where more than one mass parameter can be dialed. In a second line of research, we study a class of 3d gauge theories by engineering their gravity dual in a string theory setup. We prove the existence of multiple phase transitions between phases characterized by both massless particles and topological sectors. In a third line of research, we use holography as a tool to explore the interplay between the physics of 4d QCD and 3d gauge theories. In particular, we analyze the properties of 3d domain walls, which appear as soliton-like solutions of 4d QCD in specific parametric regimes. Finally, we propose a boundary construction of 3d large N vector models, which appear as critical points of theories obtained by coupling degrees of freedom localized on a 3d boundary to a 4d bulk theory. This construction allows to prove new dualities and uncovers a new computational tool for 3d vector models.
Doctorat en Sciences
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La théorie des cordes est un de nos meilleurs candidats pour une théorie quantique de la gravité. A ce jour elle n'a pas encore été conclusive à propose de ce sujet. Malgré cela, on a réalisé qu'on peut en tirer des informations sur tout une variété de sujets, dont notamment les théories de jauges supersymétriques, en étudiant la limite de basse énergie dans le volume d'univers des branes. Cette immersion des théories de jauges en théorie des cordes nous fournit un autre point de vue. Ce dernier nous permet souvent de prendre une approche plus géométrique pour obtenir de nouveaux résultats sinon inaccessible par des méthodes plus conventionnelles. Même en absence de vérification expérimentale de la supersymétrie, sa présence dans cette classe de théories de jauge nous fournit un terrain de jeux propice pour tester de nouvelles méthodes d'une manière efficace. En effet, la présence de la supersymétrie donne une structure additionnelle qui rend la théorie plus rigide. Cela simplifie les calculs et rend des résultats plus accessibles. On peut oser de dire que si on n'arrive pas à calculer un certain résultat en présence de supersymétrie, il y a très peu de chance d'y arriver sans. L'approche par la théorie des cordes le rend possible de découvrir des symétries cachées ou de comprendre des symétries connues d'une autre manière.Une classe de théories quantique intéressantes qui sont présentes en théories des cordes, c'est les théories de petites cordes. Ces dernières ont été découverte il y a deux décennies. Ces théories en six dimensions ont été construite une première fois comme théories dans le volume d'universe de branes NS5 dans le cadre de la théorie des cordes IIB en prenant la limite du couplage de la corde qui tend vers zéro. Dans cette limite, la théorie résultant reste non-trivial mais les interactions en dehors de la brane sont supprimées, notamment la gravité. Comme le nom le suggère, ces théories contiennent des cordes qu'on appelle petites cordes. La tension des petites cordes est proportionnelle à l'échelle naturelle de la corde. En plus, ces théories profitent de la T-dualité comme les théories de cordes critiques. Elle sont donc des théories quantiques non-locales. Leur complexité se situe entre celle des théories quantiques locales et celle de la théorie des cordes complète. Elles sont donc des candidates intéressantes pour étudier la dynamique dans le volume d'univers de la brane NS5. Pour des énergies inférieures à l'échelle de la corde, elles ont une description en termes de théories de jauges symétriques de type quiver. On peut donc également obtenir des informations sur ces dernières. Cette description locale n'est plus valable une fois l'échelle de la corde atteinte.Le but principal de cette thèse est d'étudier des dualités entre le théories de petites cordes en utilisant différentes constructions disponible en théorie des cordes. Cela nous permet d'attaquer le problème d'angles différents et de faire un lien avec des structures géométriques. En conséquence on peut analyser différentes relation parmi les théories de petites cordes. On confirme ensuite la validité des dualités qu'on obtient en utilisant la fonction de partition instantonique. Cet object est complètement non-perturbative et établit ces dualités comme résultat exact. Cette structure de dualités s'étend naturellement aux descriptions de basse énergie en terme de théories de jauges supersymétriques. De plus, on étudie les conséquences directes du réseaux de dualités qu'on a découvert
String theory remains one of our best candidates for a theory of quantum gravity. Until now it has not lived up to this goal. However, along the way it was realized that string theory can give us valu-able insights into a variety of subjects among which supersymmetric gauge theories by studying the low-energy worldvolume dynamics of branes. This embedding of gauge theories into string theory provides us with a different viewpoint that often allows us to use powerful geometric considerati-ons in order to obtain new results that are inaccessible from conventional methods. Even in the ab-sence of experimental confirmation of supersymmetry, its presence in this class of gauge theories provides us with a playground where different methods can be tested in an efficient way. Indeed, supersymmetry provides additional structure, rendering the underlying theory more rigid and thus simplifying computations and making results more accessible. One could dare to say that when a certain result can not be calculated in the presence of supersymmetry, there is probably not much hope of achieving it without supersymmetry. This stringy approach to gauge theories makes it pos-sible to unravel hidden dualities or to understand already known ones from a different perspective. An interesting class of quantum theories that are embedded into string theory are the so called little string theories. They have been discovered two decades ago. These six-dimensional theories were first obtained as the worldvolume theory of a stack of NS5 branes in the context of Type II string theory trough a particular decoupling limit that sends the string coupling constant to zero while kee-ping at the same time the string scale finite. In this limit, the resulting theory remains interacting but the bulk dynamics is decoupled, in particular gravity. As their name suggests, they contain strings. The tension of the little strings is proportional to the string scale, which is the only intrinsic scale in the theory. Furthermore, the little string theories enjoy T-duality similar to the critical string theory. They are thus non-local quantum theories. So the complexity of little string theory lies between that of local quantum field theories and full fledged critical string theory. This makes them interesting candidates for studying stringy phenomena in an easier setup where gravity is absent and to learn more about the worldvolume dynamics of the NS5 brane. At energies far below the string scale, they have a low-energy description in terms of quiver gauge theories, so their study can also give us insights into these kinds of theories. This local description breaks down as we reach the string scale and we must rely on the full little string theories. The main goal of this thesis is to study dualities between little string theories by using different dual constructions available in string theory. These allow us to attack the problem from different angles and they establish also a connection to geometric structures. This makes it possible to systematically analyse relations among different little string theories. We then confirm the validity of the newly found duality relations by using the so called instanton partition function. The latter is a completely non-perturbative object allowing us to establish the dualities as an exact result. This duality structure naturally extends to the low-energy description in terms of supersymmetric quiver gauge theories. Furthermore, we study the direct consequences of this duality web. We find interesting cases where the dimensional reduction from six to five dimensions simultaneously reduces the rank of the group and changes the matter content. Another result that we find is the presence of a hidden dihedral symmetry which acts in a highly non-trivial fashion on the spectrum of the underlying gauge theories

We investigate the phenomenon of large N duality in topological string theory from three different perspectives: worldsheets, matrix models, and melting crystals.

In the first part, we utilize the technique of mirror symmetry to generalize the worldsheet derivation of the duality, originally given by Ooguri and Vafa for the A-model on the conifold, to the A-model on more general geometries. We also explain how the Landau-Ginzburg models can be used to perform the worldsheet derivation of the B-model large N dualities.

In the second part, we consider a class of A-model large N dualities where the open string theory reduces through the Chern-Simons theory on a lens space to a matrix model. We compute and compare the matrix model spectral curve and the Calabi-Yau geometry mirror to the closed string geometry, confirming the predictions of the duality.

Finally in the third part, we propose a crystal model that describes the A-model on the resolved conifold. This is a generalization of the crystal for C³. We also consider a novel unitary matrix model for the Chern-Simons theory on the three-sphere and show how the crystal model for the resolved conifold is derived from the matrix model. Certain non-compact D-branes are naturally incorporated into the crystal and the matrix model.

The primary focus of this thesis is to investigate the mathematical and physical properties of spaces that are related by T-duality and its generalisations. In string theory, T-duality is a relationship between two a priori different string backgrounds which nevertheless behave identically from a physical point of view. These backgrounds can have different geometries, different fluxes, and even be topologically distinct manifolds. T-duality is a uniquely `stringy' phenomenon, since it does not occur in a theory of point particles, and together with other dualities has been incredibly useful in elucidating the nature of string theory and M-theory. There exist various generalisations of the usual T-duality, some of which are still putative, and none of which are fully understood. Some of these dualities are inspired by mathematics and some are inspired by physics. These generalisations include non-abelian T-duality, Poisson-Lie T-duality, non-isometric T-duality, and spherical T-duality. In this thesis we review T-duality and its various generalisations, studying the geometric, topological, and physical properties of spaces related by these dualities.

We probe string dualities by using the orientifold and F-theory, and by investigating world volume actions of super D-branes and super M-branes.

We first study orientifolds in various dimensions. We construct orientifolds dual to M-theory compactified on the Klein bottle and on the Möbius band, respectively. Six-dimensional orientifolds with N=1 supersymmetry are constructed. They have multiple tensor multiplets, which cannot be obtained by the conventional Calabi-Yau compactifications. We find F-theory duals for some of these models, thereby making manifest the phase transitions involving the tensionless strings these models can have.

We construct orientifold and F-theory duals of the heterotic string models constructed by Chaudhuri, Hockney and Lykken (CHL) and study N=2 supersymmetric F-theory vacua in six dimensions.

Next, we construct the supersymmetric world volume action of the M-theory 5-brane in a flat eleven-dimensional background. Finally, dual D-brane actions are obtained by carrying out a duality transformation of the world volume gauge field of the D-brane and their properties are studied.