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.

Mirror symmetry is the phenomenon, originally discovered by physicists, that Calabi-Yau manifolds come in dual pairs, with each member of the pair producing the same physics. Mathematicians studying enumerative geometry became interested in mirror symmetry around 1990, and since then, mirror symmetry has become a major research topic in pure mathematics. One important problem in mirror symmetry is that there may be several ways to construct a mirror dual for a Calabi-Yau manifold. Hence it is a natural question to ask: when two different mirror symmetry constructions apply, do they agree?We specifically consider two mirror symmetry constructions for K3 surfaces known as BHK and LPK3 mirror symmetry. BHK mirror symmetry was inspired by the LandauGinzburg/Calabi-Yau correspondence, while LPK3 mirror symmetry is more classical. In particular, for algebraic K3 surfaces with a purely non-symplectic automorphism of order n, we ask if these two constructions agree. Results of Artebani Boissi`ere-Sarti originally showed that they agree when n = 2, and more recently Comparin-Lyon-Priddis-Suggs showed that they agree when n is prime. However, the n being composite case required more sophisticated methods. Whenever n is not divisible by four (or n = 16), this problem was solved by Comparin and Priddis by studying the associated lattice theory more carefully. In this thesis, we complete the remaining case of the problem when n is divisible by four by finding new isomorphisms and deformations of the K3 surfaces in question, develop new computational methods, and use these results to complete the investigation, thereby showing that the BHK and LPK3 mirror symmetry constructions also agree when n is composite.

京都大学 / 0048 / 新制・課程博士 / 博士(理学) / 甲第19495号 / 理博第4155号 / 新制||理||1597(附属図書館) / 32531 / 京都大学大学院理学研究科物理学・宇宙物理学専攻 / (主査)准教授 福間 將文, 教授 川合 光, 教授 田中 貴浩 / 学位規則第4条第1項該当 / Doctor of Science / Kyoto University / DGAM

Gromov-Witten invariants play a crucial role in symplectic- and enumerative Geometry as well as topological String Theory. Essentially, theseinvariants are a count of (pseudo)holomorphic curves of a given genus,going through n-marked points on a symplectic manifold. In the last fewdecades, this has been a huge research topic for both physicists as well asmathematicians, and breakthroughs in calculation techniques have beenmade using Mirror Symmetry. We investigate and explicitly calculateclosed genus zero Gromov-Witten invariants of toric Calabi-Yau threefolds, namely O(−3) → P2 and the resolved conifold. This will be doneby using localization techniques, mirror symmetry and the so called diskpartition function.

京都大学 / 新制・課程博士 / 博士(理学) / 甲第24412号 / 理博第4911号 / 新制||理||1702(附属図書館) / 京都大学大学院理学研究科物理学・宇宙物理学専攻 / (主査)教授 橋本 幸士, 准教授 福間 將文, 教授 杉本 茂樹 / 学位規則第4条第1項該当 / Doctor of Science / Kyoto University / DFAM

The Swampland program is way of sorting effective field theories based on conjectures of how an effective field theory consistent with quantum gravity should look like. In this thesis we take a closer look at the No Global Symmetries Conjecture, the Weak Gravity Conjecture, the de Sitter Conjecture and the Trans-Planckian Censorship Conjecture. The Weak Gravity Conjecture generalized to arbitrary dimensions and p-forms is tested under dimensional reduction and its sharpened version is used to discuss how under this conjecture non-supersymmetric AdS geometries are unstable. The de Sitter Conjecture and the Trans-Planckian Censorship Conjecture are compared to show that they give similar predictions based on different assumptions. Lastly we provide an example of a theory emerging from the swampland that utilizes the instability of AdS geometries to model our universe as the boundary of a higher dimensional expanding bubble. / Swampland-programmet är ett sätt att sortera effektiva fältteorier baserat på antaganden om hur en effektiv fältteori förenlig med kvantgravitation bör se ut. I den här avhandlingen tittar vi närmare på No Global Symmetries-antagandet, Weak Gravity-antagandet, de Sitter-antagandet och Trans-Planckian Censorship-antagandet. Weak gravity-antagandet generaliserat till godtyckliga dimensioner och p-former testas under dimensionsreduktion och dess skärpta version används för att diskutera hur man via detta antagande finner en instabilitet i icke-supersymmetriska AdS-geometrier. De sitter-antagandet och Trans-Planckian Censorship-antagandet jämförs för att visa att de ger liknande förutsägelser baserat på olika frågeställningar. Slutligen ger vi ett exempel på en teori som växer fram ur programmet som använder instabiliteten hos AdS-geometrier genom att modellera vårt universum som randen på en expanderande bubbla i högre dimensioner.

String theory imposes slight modifications to Einstein's equations of general relativity (GR). In (4), the authors claim that the gravitational field equations in empty space, which in GR are just R [subscript greek letters mu nu ] = 0, should hold one extra term which is first order in the string constant [alpha'] and proportional to the Riemann curvature tensor squared. They do admit, however, that this simple modification is just schematic. In (1) the authors use modified equations which are coupled to the dilation field. We show that equations given in (4) do not admit an isotropic solution; justification of these equations would require sacrificing isotropy. We thus investigate the consequences of the coupled equations from (1) and the black-hole solution they give there. We calculate the additional perihelion precession of Mercury, the added deflection of photons by the sun, and the extra gravitational redshift which should be present if these equations hold. We determine that additional effects due to string theory in each of these cases are quite minuscule.

No description available.

We present a method of quantum renormalization group, which makes it possible to construct a bulk theory for a general conformal field theory in the context of anti-de Sitter/conformal field theory correspondence. We demonstrate that within this method it is possible to construct scalar field theory in anti-de Sitter space. We also demonstrate that from a conformal field theory possessing global symmetry, it is possible to construct non-abelian gauge theory in anti-de Sitter space. / Thesis / Master of Science (MSc)

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.