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Název práce: Studium efektivních Lagrangianů a jejich aplikace Autor: Jaroslav Trnka Katedra: Ústav částicové a jaderné fyziky Vedoucí disertační práce: RNDr. Jiří Novotný, CSc., ÚČJF Abstrakt: V této práci studujeme různé aspekty efektivních teorií pole pro kvan- tovou chromodynamiku (QCD). V prvních dvou kapitolách se zaměříme na efek- tivní teorii pro resonance, která interpoluje mezi nízkoenergetickou efektivní teorií (Chirální poruchová teorie) a vysokoenergetickou QCD. V rámci této teorie studu- jeme jednosmyčkovou renormalizaci, jak z pohledu výpočetního pomocí SS-PP korelátoru, tak i čistě koncepčního studiem dynamicky generovaných stupňů vol- nosti. Ve čtvrté kapitole studujeme amplitudy rozptylu v rámci nelineárního sig- ma modelu, který představuje vedoucí člen nízkoenergetické efektivní teorie pro QCD. V návaznosti na nedávné objevy v rámci Yang-Mills teorie se nám podaří v rámci tohoto modelu zkonstruovat rekurzivní relace pro stromové amplitudy. Kromě čistě teoretické důležitosti tohoto faktu představuje tato metoda efektivní výpočetní nástroj nezávislý na formulaci amplitud pomocí Feynmanovských dia- gramů. Klíčová slova: efektivní teorie pole, kvantová chromodynamika, nelineární sigma model...

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.

This thesis is devoted to a thorough study of chiral rings in two-dimensional (0,2) theories. We first discuss properties of chiral operators in general two-dimensional (0,2) nonlinear sigma models, both in theories twistable to the A/2 or B/2 model, as well as in non-twistable theories. As a special case, we study the quantum sheaf cohomology of Grassmannians as a deformation of the usual quantum cohomology. The deformation corresponds to a (0,2) deformation of the nonabelian gauged linear sigma model whose geometric phase is associated with the Grassmannian. Combined with the classical result, the quantum ring structure is derived from the one-loop effective potential. Supersymmetric localization is also applicable in this case, which proves to be efficient in computing A/2 correlation functions. We then compute chiral operators in general (0,2) nonlinear sigma models, and apply them to the Gadde-Gukov-Putrov triality proposal, which says that certain triples of (0,2) GLSMs should RG flow to nontrivial IR fixed points. As another application, we extend previous works to construct (0,2) Toda-like mirrors to the sigma model engineering Grassmannians. / Ph. D. / This thesis studies a mathematical concept called the chiral ring, which emerges from string theory. String theory is a conjectured theory that potentially unifies the existing fundamental physical laws. It has connections with many branches of mathematics, especially geometry. Spacetime is ten-dimensional in string theory, of which four dimensions are visible, and the other six are hidden at ordinary energy levels. The chiral ring encodes many geometric properties of the hidden part of spacetime. These properties can in turn affect the visible universe even at low energies. Research on chiral rings has primarily focused on a special class of geometries which have large symmetries and so are easier to handle. In order to tackle more general scenarios, we analyze the chiral rings corresponding to theories with only half the symmetry and give several new results and applications.

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 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.

In this thesis, we investigate supersymmetric quantum mechanics (SUSYQM) and its relation to index theorems and equivariant cohomology. We define some basic constructions on super vector spaces in order to set the language for the rest of the thesis. The path integral in quantum mechanics is reviewed together with some related calculational methods and we give a path integral expression for the Witten index. Thereafter, we discuss the structure of SUSYQM in general. One shows that the Witten index can be taken to be the difference in dimension of the bosonic and fermionic zero energy eigenspaces. In the subsequent section, we derive index theorems. The models investigated are the supersymmetric non-linear sigma models with one or two supercharges. The former produces the index theorem for the spin-complex and the latter the Chern-Gauss-Bonnet Theorem. We then generalise to the case when a group action (by a compact connected Lie group) is included and want to consider the orbit space as the underlying space, in which case equivariant cohomology is introduced. In particular, the Weil and Cartan models are investigated and SUSYQM Lagrangians are derived using the obtained differentials. The goal was to relate this to gauge quantum mechanics, which was unfortunately not successful. However, what was shown was that the Euler characteristics of a closed oriented manifold and its homotopy quotient by U(1)n coincide.