In this thesis, we investigate sigma models and algebraic structures emerging from a Hamiltonian description of their dynamics, both in a classical and in a quantum setup. More specifically, we derive the phase space structures together with the Hamiltonians for the bosonic two-dimensional non-linear sigma model, and also for the N=1 and N=2 supersymmetric models. A convenient framework for describing these structures are Lie conformal algebras and Poisson vertex algebras. We review these concepts, and show that a Lie conformal algebra gives a weak Courant–Dorfman algebra. We further show that a Poisson vertex algebra generated by fields of conformal weight one and zero are in a one-to-one relationship with Courant–Dorfman algebras. Vertex algebras are shown to be appropriate for describing the quantum dynamics of supersymmetric sigma models. We give two definitions of a vertex algebra, and we show that these definitions are equivalent. The second definition is given in terms of a λ-bracket and a normal ordered product, which makes computations straightforward. We also review the manifestly supersymmetric N=1 SUSY vertex algebra. We also construct sheaves of N=1 and N=2 vertex algebras. We are specifically interested in the sheaf of N=1 vertex algebras referred to as the chiral de Rham complex. We argue that this sheaf can be interpreted as a formal quantization of the N=1 supersymmetric non-linear sigma model. We review different algebras of the chiral de Rham complex that one can associate to different manifolds. In particular, we investigate the case when the manifold is a six-dimensional Calabi–Yau manifold. The chiral de Rham complex then carries two commuting copies of the N=2 superconformal algebra with central charge c=9, as well as the Odake algebra, associated to the holomorphic volume form.

京都大学 / 0048 / 新制・課程博士 / 博士(理学) / 甲第21561号 / 理博第4468号 / 新制||理||1641(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)准教授 福間 將文, 教授 田中 貴浩, 准教授 國友 浩 / 学位規則第4条第1項該当 / Doctor of Science / Kyoto University / DGAM

Nonlinear sigma-models in physics have been a source of interesting and important ideas in geometry, topology, and algebra. One such model is the curved beta gamma system. This purely bosonic model studies maps from a Riemann surface to a target complex manifold X. The solutions to the classical equations of motion are holomorphic maps. An extension of this model - the so-called heterotic model, incorporates fermionic fields valued in a holomorphic vector bundle E on the complex manifold. In this thesis, I study this extended model within the framework of effective field theory and BV quantization developed by Kevin Costello. Building on earlier work of Gorbounov-Gwilliam-Williams in the purely bosonic case, my approach uses tools of Gelfand-Kazhdan formal geometry and derived deformation theory to extract obstructions to quantization (anomalies) and identify these with characteristic classes of the target manifold. Specifically, I show that the obstruction to solving the Quantum Master Equation can be identified with the class ch_2 (TX)-ch_2(E), and the obstruction to the quantizing equivariantly with respect to holomorphic vector fields on the source Riemann surface can be identified with c_1 (TX) - c_1(E). By analyzing the theory where the source is an elliptic curve, an explicit geometric construction of the partition function is given.

In this dissertation we considered a nonlinear sigma model with gravitino field. This is a supersymmetric extension of the nonlinear sigma model in the string theory, and we set up the geometric model using commuting variables, such that we could analyze it using the tools from calculus of variations. We introduced an action functional which corresponds to the super harmonic map functional, which has four arguments: a map between Riemannian manifolds, a vector spinor, a Riemannian metric and a gravitino. After getting the total variation formula, we considered the symmetries that the action functional possesses. By Noether's principle these families of symmetries induces conservation laws, which help to interpret the energy-momentum tensor and the supercurrent as holomorphic sections of some complex bundle. We also discussed the supersymmetry of our model. It turns out that the supersymmetry only remains in some particular cases, which is still useful in the analysis. Then we defined the weak solution in the distributional sense, and using Riesz potential estimates and Riviere regularity theory, we could improve the regularity of the weak solutions. More precisely, when the Riemannian metric and the gravitino are smooth, then any weak solution is actually smooth; and when the gravitino are coarse but subcritical, we can still show that the weak solutions are Holder continuous. Next we considered the compactness of solutions with bounded energies. We showed the small energy regularity on local domains and gap properties on the global surface. We also established the Pohozaev identities and thus showed the removable singularity theorem. Finally, for a sequence of solutions of uniformly bounded energies with respect to a converging sequence of gravitino fields, we showed that they converges weakly. Actually away from finite points, the convergence is strong and at those points, the energies concentrate. After a rescaling, each of these points corresponds to finitely some Dirac-harmonic maps with curvature terms defined on the Riemann sphere. Moreover, we established the energy identities along the weakly convergent sequences modulo these bubbles.

京都大学 / 0048 / 新制・課程博士 / 博士(理学) / 甲第18066号 / 理博第3944号 / 新制||理||1568(附属図書館) / 30924 / 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 川合 光, 教授 國廣 悌二, 教授 畑 浩之 / 学位規則第4条第1項該当 / Doctor of Science / Kyoto University / DGAM

This thesis is devoted to a study of applications of gauged linear sigma models. First, by constructing (0,2) analogues of Hori-Vafa mirrors, we have given and checked proposals for (0,2) mirrors to projective spaces, toric del Pezzo and Hirzebruch surfaces with tangent bundle deformations, checking not only correlation functions but also e.g. that mirrors to del Pezzos are related by blowdowns in the fashion one would expect. Also, we applied the recent proposal for mirrors of non-Abelian (2,2) supersymmetric two-dimensional gauge theories to examples of two-dimensional A-twisted gauge theories with exceptional gauge groups G_2 and E_8. We explicitly computed the proposed mirror Landau-Ginzburg orbifold and derived the Coulomb ring relations (the analogue of quantum cohomology ring relations). We also studied pure gauge theories, and provided evidence (at the level of these topologicalfield-theory-type computations) that each pure gauge theory (with simply-connected gauge group) flows in the IR to a free theory of as many twisted chiral multiplets as the rank of the gauge group. Last, we have constructed hybrid Landau-Ginzburg models that RG flow to a new family of non-compact Calabi-Yau threefolds, constructed as fiber products of genus g curves and noncompact Kahler threefolds. We only considered curves given as branched double covers of P^1. Our construction utilizes nonperturbative constructions of the genus g curves, and so provides a new set of exotic UV theories that should RG flow to sigma models on Calabi-Yau manifolds, in which the Calabi-Yau is not realized simply as the critical locus of a superpotential. / Doctor of Philosophy / This thesis is devoted to a study of vacua of supersymmetric string theory (superstring theory) by gauged linear sigma models. String theory is best known as the candidate to unify Einstein’s general relativity and quantum field theory. We are interested in theories with a symmetry exchanging bosons and fermions, known as supersymmetry. The study of superstring vacua makes it possible to connect string theory to the real world, and describe the Standard model as a low energy effective theory. Gauged linear sigma models are one of the most successful models to study superstring vacua by, for example, providing insights into the global structure of their moduli spaces. We will use gauged linear sigma models to study mirror symmetry and its heterotic generalization “(0, 2) mirror symmetry.” They are both world-sheet dualities relating different interpretations of the same (internal) superstring vacua. Mirror symmetry is a very powerful duality which exchanges classical and quantum effects. By studying mirror symmetry and (0, 2) mirror symmetry, we gain more knowledge of the properties of superstring vacua.

Dans cette thèse on étudie certaines constructions géométriques qui apparaissent naturellement dans le contexte des modèles sigma, leur jaugeage et supersymétrisation. La thèse comprend trois parties. La première partie (chapitres 1 et 2) contient des faits issus de la géométrie différentielle classique et de la géométrie graduée nécessaires pour comprendre les résultats clés de la thèse. On survole la géométrie liée aux variétés de Poisson et variétés symplectiques. On généralise ces notions aux variétés de Dirac et variétés n-plectiques, et établit leur liens avec les algebroïdes de Courant. Le langage principal utilisé dans la thèse pour la description mathématique des modèles sigma – c'est la géométrie graduée – on définit donc des bases de calcul sur les supervariétés et variétés graduées ainsi que les notions des Q-structures et des variétés multigraduées. La deuxième partie (chapitres 3 et 4) a pour but d’interpréter géométriquement l'invariance de jauge de certains modèles sigma. On établit la relation entre les symétries de modèle sigma de Dirac, et comme cas particulier de modèle sigma de Poisson (tordu), avec les sous-algèbres des sections d'algebroïde de Courant. On généralise la notion de cohomologie équivariante, ce qui permet d'obtenir les modèles sigma avec le groupe des symétries prescrit, en particulier on construit les groupes nécessaires pour les modèles sigma mentionnés. La troisième partie (chapitre 5) adresse l'extension graduée des modèles sigma (comme en supersymétrisation). Ceci est en fait lié auxstructures géométriques qui peuvent être définies sur l'espace des applications entre les variétés multigraduées / In this thesis we study some geometric constructions appearing naturally in the context of sigma models, their gauging and supersymmetrization. The thesis consists of three parts. The first part (chapters 1 and 2) contains facts coming from classical differential geometry and graded geometry, they are needed to understand the main results of the thesis. We review the geometric constructions related to Poisson and symplectic manifolds. We generalize these notions to Dirac and n-plectic manifolds and establish the links with Courant algebroids. The main language used in the thesis for mathematical description of the sigma models is the graded geometry - we thus define the basis of calculus on supermanifolds and graded manifolds, as well as describe the notions of Q-structures and multigraded manifolds. The main goal of the second part (chapters 3 and 4) is to interpret geometrically the gauge invariance of some sigma models. We establish the relation of the symmetries of the Dirac sigma model, and as a particular case of the (twisted) Poisson sigma model, with the subalgebra of sections of Courant algebroid. We generalize the notion of equivariant cohomology, that permits to recover the sigma models with a prescribed group of gauge symmetries. In particular we construct the necessary groups for the mentioned sigma models. The third part (chapter 5) addresses the graded extension of the sigma models (like in supersymmetrization). It is in fact related to the geometric structures that can be defined on the space of maps between multigraded manifolds.

The interplay of thermodynamic properties of strongly interacting matter and its emission of photons is investigated. For this purpose the Lagrangian of the quark meson model (in the literature also dubbed "linear sigma model" or "linear sigma model with quarks") is extended by an electromagnetic sector. Based on this extended Lagrangian both the grand-canonical potential and the generating functional of correlation functions are calculated in a consistent manner. From the former, the phase structure and various thermodynamical properties are determined. Especially, the dependence of certain landmarks (critical point, intersections of the phase boundary with the coordinate axes, etc.) of the phase diagram with respect to the model parameters is investigated in detail. With the help of the generating functional in turn, the photon propagator can be computed whose imaginary part is connected to the emission rate of photons. The leading order of the result with respect to the number of participating particles and the power of the quark-meson coupling is expressed in terms of tree level diagrams, which are calculated likewise. On this basis, the photon emissivity with respect to temperature, chemical potential and photon frequency is calculated and analyzed addressing various questions. The dependence of the particle masses with respect to temperature and chemical potential leaves notable imprints on the emissivities of the individual production processes. Especially a first-order phase transition can easily be identified, since, there, the emissivity may jump - depending on the temperature - by a factor of about ten. Contrarily, within our analysis, we do not find signatures in the photon emissivities that specifically mark a critical end point. Moreover, it is investigated on which parameters the photon emission rate depends in the low- and high-frequency regions. With these results the behavior of the emissivity with respect to temperature and chemical potential can be understood and many peculiarities of the emissivities can be explained. / Das Zusammenspiel der thermodynamischen Eigenschaften von stark wechselwirkender Materie und deren Emission von Photonen wird untersucht. Dazu wird die Lagrangedichte des Quark-Meson-Modells (auch: Linear-Sigma-Modell oder Linear-Sigma-Modell mit Quarks) um einen elektromagnetischen Sektor erweitert. Aus der so erweiterten Lagrangedichte werden auf konsistente Weise sowohl das großkanonische Potential als auch das erzeugende Funktional der Korrelationsfunktionen ermittelt. Aus ersterem werden die Phasenstruktur des Modells sowie zahlreiche thermodynamische Eigenschaften berechnet. Insbesondere wird die Abhänigkeit einiger Orientierungspunkte (kritischer Punkt, Schnittpunkte der Phasengrenze mit den Koordinatenachsen usw.) des Phasendiagramms von den Modellparametern detailiert untersucht. Mit Hilfe des erzeugenden Funktionals wiederum kann der Photonenpropagator bestimmt werden, dessen Imaginärteil mit der Emissionsrate von Photonen zusammenhängt. Die führende Ordnung in einer Entwicklung nach der Anzahl der beteiligten Teilchen und der Potenz der Quark-Meson-Kopplung lässt sich durch Baumgraphen-Diagramme darstellen, die ebenfalls berechnet werden. Auf dieser Basis wird die Photon-Emissivität in Abhängigkeit von Temperatur, chemischem Potential und Photon-Frequenz berechnet und unter verschiedenen Gesichtspunkten analysiert. Die Abhängigkeit der Teilchenmassen von Temperatur und chemischem Potential hinterlässt teilweise ausgeprägte Signaturen in den Emissivitäten der einzelnen sub-Prozesse. Insbesondere ein Phasenübergang erster Ordnung zeigt sich deutlich, da an diesem die Emissivität - abhänging von der Temperatur - um einen Faktor der Größenordnung zehn springen kann. Jedoch finden wir im Rahmen dieser Analyse keine spezifischen Signaturen in den Photonen-Emissivitäten, die einen kritischen Punkt auszeichnen. Des weiteren wird untersucht von welchen Parametern die Photonen-Emissionsrate in den Bereichen niedriger oder hoher Photonen-Frequenzen abhängt. Mit diesen Ergebnissen kann das Verhalten der Emissivität in Abhängigkeit von Temperatur und chemischem Potential gut verstanden und zahlreiche Auffälligkeiten in den Emissivitäten erklärt werden.

The master thesis is focused on efficiency evaluation of company IN-EKO TEAM, s. r. o. The theoretical part contains the relevant information from professional publications, basic concepts are explained here and models supporting business management, business process modeling models, and models emphasizing self-evaluation are presented. The second part is an efficiency evaluation by using the selected START Plus model. The key part is the suggestion part in which measures will be proposed to improve the efficiency of examined subject.

In der vorliegenden Arbeit werden versteckte Symmetrien innnerhalb der N=4 supersymmetrischen Yang--Mills Theorie oder der nach der AdS/CFT Korrespondenz dualen Beschreibung durch eine String-Theorie in AdS5 x S5 besprochen. Dabei betrachten wir die Maldacena--Wilson Schleife, die sich für diese Untersuchungen besonders eignet, da ihr Vakuum-Erwartungswert für glatte Kurven nicht divergiert und die vermutete Dualität zu Streuamplituden wenigstens konzeptionell eine Möglichkeit bietet, etwaige Symmetrien zu anderen Observablen zu übertragen. Ihre Beschreibung durch Minimalflächen in AdS5 erlaubt es, Symmetrien mithilfe der Integrabilität der zugrunde liegenden klassischen String-Theorie zu konstruieren. Dieser Zugang wurde bereits in der Herleitung der Yang'schen Symmetrie der Maldacena--Wilson Schleife bei starker Kopplung sowie in der Beschreibung von Deformationen gleiches Flächeninhalts von Minimalflächen in AdS3 verwendet. Diese beiden Ergebnisse werden in der vorliegenden Arbeit miteinander verbunden und erweitert. Im Sinne einer systematischen Herangehensweise besprechen wir zunächst die Symmetriestruktur der zugrunde liegenden String-Theorie. Diese Diskussion lässt sich auf die Diskussion von String-Theorien in symmetrischen Räumen verallgemeinern. Dabei zeigt sich, dass die Symmetrie, welche die Deformationen gleiches Flächeninhalts in AdS3 erzeugt, in der Symmetriestruktur dieser Modelle eine zentrale Rolle einnimmt: Sie wirkt als Aufsteige-Operator auf den unendlich vielen erhalten Ladungen und generiert somit den Spektralparameter. Weiterhin lässt sie sich anwenden, um ausgehend von der globalen Symmetrie sämtliche Symmetrien des zugrunde liegenden Modells zu konstruieren. Sie wird daher als die Master-Symmetrie dieser Modelle bezeichnet. Zusätzlich wird die Algebra der Symmetrie-Variationen sowie der erhaltenen Ladungen ausgearbeitet. Für den konkreten Fall von Minimalflächen in AdS5 diskutieren wir die Deformation der Minimalflächenlösung für den Fall eines lichtartigen Vierecks. Diese liefert die duale Beschreibung der Streuamplitude für vier Gluonen. Damit unternehmen wir einen ersten Schritt zur Übertragung der Master-Symmetrie auf Streuamplituden. Weiterhin berechnen wir die Variation der Randkurven der Minimalflächen unter der Master- und Yang'schen Symmetrie für allgemeine, glatte Randkurven. Das Ergebnis dieser Rechnung führt auf eine Verallgemeinerung der Master-Symmetrie zu einer Variation, die von der Kopplungskonstanten abhängt und für beliebige Werte der Kopplungskonstanten eine Symmetrie der Maldacena--Wilson Schleife darstellt. Unsere Diskussion erklärt das Scheitern vorheriger Versuche, die entsprechende Symmetrie im Spezialfall von Minimalflächen in AdS3 zu schwacher Kopplung zu übertragen. Wir besprechen verschiedene Ansätze, die Yang'sche Symmetrie zu schwacher oder beliebiger Kopplung zu übertragen, schlussfolgern aber letztendlich, dass eine Yang'sche Symmetrie der Maldacena--Wilson Schleife nicht vorzuliegen scheint. Die Situation ändert sich, wenn wir Wilson Schleifen in Superräumen betrachten. Diese sind die natürlichen supersymmetrischen Erweiterungen der Maldacena--Wilson Schleife. Für die Yang'sche Invarianz ihres Vakuum-Erwartungswerts wurden wichtige Anhaltspunkte gefunden und sowohl die Beschreibung dieser Operatoren als auch der Beweis der Yang'schen Invarianz bei schwacher Kopplung wurden parallel zur Arbeit an der vorliegenden Dissertation vervollständigt. Wir diskutieren das Gegenstück zu diesem Ergebnis bei starker Kopplung. Dort wird die Wilson Schleife durch eine Minimalfläche beschrieben, welche im Superraum der Superstring-Theorie vom Typ IIB in AdS5 x S5 liegt. Der Vergleich der bei starken Kopplung etablierten Invarianz mit den entsprechenden Generatoren bei schwacher Kopplung zeigt, dass die Symmetrie-Generatoren einen lokalen Anteil enthalten, der auf nicht-triviale Weise vom Wert der Kopplungskonstanten abhängt. Zusätzlich finden wir sogenannte Bonus-Symmetrien. Diese sind die analogen Generatoren in den höheren Ordnungen zum Hyperladungs-Generator, der selbst keine Symmetrie darstellt. Wir zeigen, dass diese Symmetrien in allen höheren Ordnungen der Yang'schen Algebra vorliegen. / This thesis discusses hidden symmetries within N=4 supersymmetric Yang--Mills theory or its AdS/CFT dual, string theory in AdS5 x S5. Here, we focus on the Maldacena--Wilson loop, which is a suitable object for this study since its vacuum expectation value is finite for smooth contours and the conjectured duality to scattering amplitudes provides a conceptual path to transfer its symmetries to other observables. Its strong-coupling description via minimal surfaces in AdS5 allows to construct the symmetries from the integrability of the underlying classical string theory. This approach has been utilized before to derive a strong-coupling Yangian symmetry of the Maldacena--Wilson loop and describe equiareal deformations of minimal surfaces in AdS3. These two findings are connected and extended in the present thesis. In order to discuss the symmetries systematically, we first discuss the symmetry structure of the underlying string model. The discussion can be generalized to the discussion of generic symmetric space models. For these, we find that the symmetry which generates the equiareal deformations of minimal surfaces in AdS3 has a central role in the symmetry structure of the model: It acts as a raising operator on the infinite tower of conserved charges, thus generating the spectral parameter, and can be employed to construct all symmetry variations from the global symmetry of the model. It is thus referred to as the master symmetry of symmetric space models. Additionally, the algebra of the symmetry variations and the conserved charges is worked out. For the concrete case of minimal surfaces in AdS5, we discuss the deformation of the four-cusp solution, which provides the dual description of the four-gluon scattering amplitude. This marks the first step toward transferring the master symmetry to scattering amplitudes. Moreover, we compute the master and Yangian symmetry variations of generic, smooth boundary curves. The results leads to a coupling-dependent generalization of the master symmetry, which constitutes a symmetry of the Maldacena--Wilson loop at any value of the coupling constant. Our discussion clarifies why previous attempts to transfer the deformations of minimal surfaces in AdS3 to weak coupling were unsuccessful. We discuss several attempts to transfer the Yangian symmetry to weak or arbitrary coupling, but ultimately conclude that a Yangian symmetry of the Maldacena--Wilson loop seems not to be present. The situation changes when we consider Wilson loops in superspace, which are the natural supersymmetric generalizations of the Maldacena--Wilson loop. Substantial evidence for the Yangian invariance of their vacuum expectation value has been provided at weak coupling and the description of the operator as well as its weak-coupling Yangian invariance were subsequently established in parallel to the work on this thesis. We discuss the strong-coupling counterpart of this finding, where the Wilson loop in superspace is described by minimal surfaces in the superspace of type IIB superstring theory in AdS5 x S5. The comparison of the strong-coupling invariance derived here with the respective generators at weak coupling shows that the generators contain a local term, which depends on the coupling in a non-trivial way. Additionally, we find so-called bonus symmetry generators. These are the higher-level recurrences of the superconformal hypercharge generator, which does not provide a symmetry itself. We show that these symmetries are present in all higher levels of the Yangian.