Relations between propagators and Green functions of Hamiltonians which are SUSY partners have been obtained. New exact propagators for the family of multi-well, time-dependent and non-hermitian potentials have been calculated. Non-conservative SUSY transformation has been studied in the case of multichannel Schrodinger equation with different thresholds. Spectrum (bound/virtual states and resonances) of the non-conservative SUSY partner of zero potential has been founded. Exactly solvable model of the magnetic induced Feshbach resonance has been constructed. This model was tested in the case of Rb$^{85}$. Conservative SUSY transformations of the first and the second order have been studied in the case of multichannel Schrodinger equation with equal thresholds. Transformations which introduce non-trivial coupling between scattering channels have been founded. The first order SUSY transformation which preserves $S$-matrix eigen-phase shifts and modifies mixing parameter has been founded in the case of two channel scattering with partial waves of different parities. In the case of coinciding parities we have found the second order SUSY transformation which preserves $S$-matrix eigen-phase shifts and modifies mixing parameter. Phenomenological two channel $^3S_1-^3D_1$ neutron-proton potential has been obtained by using single channel, phase equivalent and coupling SUSY transformations applied to zero potential.

Inflationary cosmology is the leading candidate for explaining the homogeneity, isotropy and spatial flatness of the universe whilst also providing the mechanism for the seeding of large scale structure. The central theme of inflationary dynamics involves the evolution of a scalar field, called the inflaton, such that its potential drives an accelerated expansion. Warm inflation is the dynamical realization in which interactions between the inflaton and other fields can lead to dissipation of inflaton energy to other dynamical degrees of freedom. Heavy fields coupled to the inflaton mediate the transfer of inflaton energy to light degrees of freedom which thermalize and heat the universe. This damps the inflaton’s motion and allows for the potential formation of a thermal bath during the inflationary period. Hybrid inflation models are a natural way in which warm inflation can be realized, with dissipation of inflaton energy mediated by the waterfall fields to fields in the light sector. In this thesis I outline the dynamics and observational predictions of supersymmetric hybrid inflation driven by radiative corrections in the warm regime. As in the standard cold inflationary scenario inflation ends when the effective mass squared of the waterfall field becomes negative, with the tachyonic instability driving the system to a global minimum in a process called the waterfall transition. I present the effect of including thermal mass corrections to the waterfall fields, and SUSY mass splittings on the quantum effective potential and the resulting dissipation coefficient. I show that including dissipative effects can significantly prolong the inflationary period to produce 50-60 e-folds of inflation with an observationally consistent primordial spectrum. Inflation still requires a microphysical description within a fundamental theory of quantum gravity. This has prompted the search for inflaton candidates within the superabundance of scalar fields present in string theory compactifications, with brane-antibrane inflation in particular emerging as a concrete implementation of SUSY hybrid inflation in a UV complete particle physics model. Inflation proceeds in a brane-antibrane system through the movement of a stack of branes towards a stack of antibranes, with the inflaton field being the interbrane distance. Warm inflation can be implemented in a brane-antibrane system with dissipation of inflaton energy mediated by fields corresponding to strings stretched between the brane and antibrane stacks. It has been shown that this dissipation of inflaton energy in warm inflation can greatly alleviate the η-problem in brane-antibrane scenarios. Whilst these strings mediating dissipation have end points fixed on to both the D3 and D3 stacks, the compact nature of the geometry within which the system is constructed allows these strings to have different winding modes. We investigated how strings with increasing winding number can provide an enhancement to the dissipation coefficient, allowing a significant reduction in the number of branes and antibranes in the warm inflation system, whilst also modifying the inflationary dynamics by reducing the speed at which the system evolves. This may go some way to alleviating the η-problem associated with some constructions of brane-antibrane inflation whilst also potentially providing the best way to motivate the large field multiplicities associated with warm inflation models.

This thesis studies N=(2,2) gauged linear sigma models (GLSMs) and three-dimensional N=2 Chern-Simons-matter theories and their mathematical applications. After a brief review of GLSMs, we systematically study nonabelian GLSMs for symplectic and orthogonal Grassmannians, following up a proposal in the math community. As consistency checks, we have compared global symmetries, Witten indices, and Calabi-Yau conditions to geometric expectations. We also compute their nonabelian mirrors following the recently developed nonabelian mirror symmetry. In addition, for symplectic Grassmannians, we use the effective twisted superpotential on the Coulomb branch of the GLSM to calculate the ordinary and equivariant quantum cohomology of the space, matching results in the math literature. Then we discuss 3d gauge theories with Chern-Simons terms. We propose a complementary method to derive the quantum K-theory relations of projective spaces and Grassmannians from the corresponding 3d gauge theory with a suitable choice of the Chern-Simons levels. In the derivation, we compare to standard presentations in terms of Schubert cycles, and also propose a new description in terms of shifted Wilson lines, which can be generalized to symplectic Grassmannians. Using this method, we are able to obtain quantum K-theory relations, which match known math results, as well as make predictions. / Doctor of Philosophy / In this thesis, we study two specific models of supersymmetric gauge theories, namely two-dimensional N=(2,2) gauged linear sigma models (GLSMs) and three-dimensional N=2 Chern-Simons-matter theories. These models have played an important role in quantum field theory and string theory for decades, and generated many fruitful results, improving our understanding of Nature by drawing on many branches in mathematics, such as complex differential geometry, intersection theory, quantum cohomology/quantum K-theory, enumerative geometry, and many others. Specifically, this thesis is devoted to studying their applications in quantum cohomology and quantum K-theory. In the first part of this thesis, we systematically study two-dimensional GLSMs for symplectic and orthogonal Grassmannians, generalizing the study for ordinary Grassmannians. By analyzing the Coulomb vacua structure of the GLSMs for symplectic Grassmannians, we are able to obtain the ordinary and equivariant quantum cohomology for these spaces. A similar methodology applies to 3d Chern-Simons-matter theories and quantum K-theory, for which we propose a new description in terms of shifted Wilson lines.

Diese Dissertation beschäftigt sich mit der Suche nach dem skalaren Top Quark (stop) und dem skalaren Bottom Quark (sbottom) innerhalb des Minimal Supersymmetric Standard Model (MSSM) unter der Annahme der R-Paritätserhaltung. Suchen nach den folgenden Zerfallsmoden des Stop-Quark wurden durchgeführt: stop -> c neutralino_1, stop -> b l sneutrino (wobei l mit gleichen Wahrscheinlichkeiten entweder electron, muon oder tau-lepton ist) und stop -> b tau sneutrino (nur das Tau-Lepton wird berücksichtigt). Zusätzlich wurde der Dreikörperzerfall stop -> b W neutralino_1 im erlaubten Massenbereich M_stop > M_b + M_W + M_neutralino1 >= 86 GeV gesucht. Für das Sbottom-Quark wurde der Zerfall sbottom -> b neutralino_1 studiert. Jede dieser Zerfallsmoden wurde voneinander unabhängig unter der Annahme eines 100 %-igen Verzweigungsverhältnisses untersucht. Für diese Suche wurden Daten aus electron-positron-Kollisionen bei Schwerpunktsenergien im Bereich von 202-208 GeV benutzt. Die Daten wurden im Jahr 2000 von dem L3 Detektor am Large Electron Positron Collider (LEP) am CERN aufgenommen. Ferner wurden die Resultate der Datenanalyse aus dem Jahr 2000 mit Resultaten der Squark-Suche kombiniert, die die L3 Kollaboration in vorhergehenden Jahren bei Schwerpunktsenergien von 161 bis 202 GeV durchgeführt hat. Die untersuchten Squark Zerfallskanäle bestimmen die Topologie der für uns interessanten Ereignisse: 2 Jets (oder b-Jets) + fehlende Energie (+ 2 Leptonen für die Stop-Dreikörperzerfälle). Die stop -> b W neutralino_1 Zerfallstopologie hängt signifikant von den weiteren Zerfällen des W-Bosons ab und kann bis zu 6 Jets im Endzustand haben. Die Annahme der R-Paritätserhaltung impliziert die Stabilität des leichtesten supersymmetrischen Teilchens (des LSP), das das leichteste Neutralino ist. Das LSP wechselwirkt nur schwach und entweicht deswegen unentdeckt. Ein besonderes Merkmal der Signal-Ereignisse ist somit eine erhebliche Menge fehlender Energie. Die sichtbare Energie ist in etwa proportional zu der Massendifferenz zwischen dem Squark und dem LSP. Weil die Standardmodell-Untergrundzusammensetzung vom Anteil der sichtbaren Energie abhängt, hängt die Analyse auch vom Wert von der Massendifferenz ab. Abhängig von der Menge fehlender Energie kann der Standardmodell-Untergrund in drei Kategorien eingeteilt werden: - die zwei-Fermion-Prozesse sind e e -> e e, e e -> mu mu, e e -> tau tau und e e -> e e q q; - die vier-Fermion-Kategorie besteht aus e e -> W W, e e -> W e nu, e e -> Z Z und e e -> Z e e Prozessen; - die zwei-Photon-Untergrundprozesse sind e e -> e e e e, e e -> e e mu mu, e e -> e e tau tau und e e -> e e q q. Der letzte Prozess, e e -> e e q q, trägt den grössten Anteil zu den SM-Untergrundprozessen bei (wegen sehr hohem und stark schwankendem E_miss und dem grössten Wirkungsquerschnitt). Im ersten Schritt der Analyse wurden Events mit der gewünschten Topologie (2 Jets und hohes E_miss) vorselektiert. Die Selektion von Stop- und Sbottom-Ereignissen wurde durch die Minimierung der mit 95 % Confidence Level (C.L.) erwarteten oberen Grenze des Squark-Wirkungsquerschnitts - berechnet aus MC-Vorhersagen - optimiert, wobei der kleine theoretisch vorhergesagte Produktionswirkungsquerschnitt des Squarks berücksichtigt wurde. In allen für den jeweiligen Squark Zerfallskanal optimierten Selektionen stimmt die Anzahl von Daten Events mit der erwarteten Anzahl von Standardmodellprozessen überein: - für den stop -> c neutralino_1 Zerfall wurden 29 Daten-Events beobachtet, wobei 26.5 +- 2.7 Events von den SM-Prozessen erwartet wurden; - für den Dreikörperzerfall stop -> b l sneutrino, wurden 4 Daten-Events selektiert bei einer Standardmodell-Erwartung von 4.0 +- 1.0 Events; - für den Zerfall stop -> b tau sneutrino sind die Daten- und SM-Eventzahlen 5 bzw. 3.9 +- 1.0; - in der Selektion für stop -> b W neutralino_1, wurden 184 Daten Events beobachtet und 181.6 +- 3.0 Events wurden vom Standardmodell vorhergesagt; - für den Bottom Squark Zerfall sbottom -> b neutralino_1 entsprachen die beobachteten 6 Events der SM-Erwartung von 7.7 +- 1.3 Events. Es wurden keine MSSM-Skalar-Quarks in den Daten des Experiments beobachtet und das Resultat der Suche ist negativ. Die modellunabhängige 95 % C.L. obere Grenze für den Squark-Produktionswirkungsquerschnitt wurde aus der gemessenen Anzahl von Daten-Events und der aus dem Standardmodell erwarteten Eventanzahl berechnet. Für die Berechnung der oberen Grenzen der Produktionswirkungsquerschnitte wurden die Resultate der Squark-Suchen aus den L3-Daten bei Schwerpunktsenergien von c.m.s. Energie 202 - 208 GeV mit den Resultaten aus vorherigen Suchen der L3-Kollaboration bei 161 GeV - 202 GeV kombiniert. Eine neue Methode wurde entwickelt, um die kombinierten Grenzen zu berechnen. Die Methode berücksichtigt die statistische Unabhängigkeit jeder Messung und die Abhängigkeit des Squark-Produktionswirkungsquerschnittes von der Schwerpunktsenergie. In der Berechnung wurde den systematischen Unsicherheiten in der Standardmodell-Untergrundabschätzung und der Signal-Selektionseffizienz Rechnung getragen. Für die hier betrachteten Squark-Zerfälle werden typisch folgende oberen Grenzen mit 95 % C.L. für den Squark Produktionswirkungsquerschnitt erhalten: ~ 0.05-0.2 pb (für stop) und ~ 0.05-0.1 pb (für sbottom). Bei den Suchen nach dem Stop-Dreikörperzerfall stop -> b W neutralino_1 wurden die Produktionswirkungsquerschnitte über 0.7-1.0 pb mit 95 % C.L. ausgeschlossen. Innerhalb des Minimal Supersymmetrischen Standard Modells mit R-Paritätserhaltung wurden die unabhängigen Wirkungsquerschnittsgrenzen für den Ausschluss von MSSM Parametern benutzt, insbesondere für die Stop- und Sbottom-Massen. Die Squark-Massen wurden für jeden betrachteten Zerfallskanal in zwei möglichen Szenarien ausgeschlossen: für den maximalen und den (näherungsweise) minimalen theoretischen Wirkungsquerschnitt. Der erste Fall korrespondiert zur maximalen Mischung zwischen den links- und rechtshändigen Squark-Eigenzuständen, $\cos\theta_{LR}$ = 1; der zweite Fall ist definiert durch den Wert von $\cos\theta_{LR}$, bei dem die Squarks vom $Z^0$ Boson entkoppeln. Abhängig vom Wert $\Delta M$ wurden die Squark Massen mit 95 % C.L. bis zu den folgenden Werten ausgeschlossen: - für stop -> c neutralino_1: M_stop < 90-93 GeV (min. Wirkungsquerschnitt), M_stop < 95-96 GeV (max. Wirkungsquerschnitt), - für stop -> b l sneutrino: M_stop < 87-89 GeV (min. Wirkungsquerschnitt), M_stop < 90-91 GeV (max. Wirkungsquerschnitt), - für stop -> b tau sneutrino: M_stop < 83-88 GeV (min. Wirkungsquerschnitt), M_stop < 88-91 GeV (max. Wirkungsquerschnitt), - für sbottom -> b neutralino_1: M_stop < 76-83 GeV (min. Wirkungsquerschnitt), M_stop < 94-97 GeV (max. Wirkungsquerschnitt), In beiden Fällen werden die experimentell beobachteten 95 % C.L. Massen Ausschlussgrenzen mit den aus Monte Carlo Simulationen ohne SUSY Teilchen erwarteten verglichen. Die experimentallen Ausschlussgrenzen Sind verträglich mit den erwarteten. Die mit 95 % C.L. erhaltene obere Grenze für den Stop-Produktionquerschnitt ist im Zerfall stop -> b W neutralino_1 grösser als die zugehörige theoretische Vorhersage. Der Ausschluss mit 95 % C.L. auf Massen war mit dem zur Verfügung stehenden Datensatz aus diesen Grund nicht möglich. Unter der Annahme, dass die Zerfallstopologie der skalaren Quarks der ersten zwei Generationen ähnlich dem Zweikörperzerfall des Stop ist, wurden die Resultate der Suche nach dem Zerfall stop -> c neutralino_1 auch für die Berechnung der Massenausschlussgrenzen für die Squarks der ersten beiden Familien benutzt. Zwei Möglichkeiten wurden hier in Erwägung gezogen: die Massenentartung zwischen vier (scalar u, d, c, s) und fünf (sbottom zusätzlich) Squarks. Die Ausschlussgrenzen mit 95 % C.L. auf die massenentarteten skalaren Quarks in den Fällen der "nur-rechts" oder "links-und-rechts" Eigenzustände sind die folgenden: - für die Massenentartung zwischen vier Squarks: M_squark < 95-96 GeV ("nur-rechts"), M_squark < 99-100 GeV ("links-und-rechts"); - für die Massenentartung zwischen fünf Squarks: M_squark < 96-97 GeV ("nur-rechts"), M_squark < 99-101 GeV ("links-und-rechts"); Mit der Annahme der Gaugino-Vereinigung an der GUT-Skala im MSSM wurden die Grenzen für die vierfach massenentarteten Squarks erneut in der Squark-Gluino Ebene interpretiert. Ferner wurde das absolute Limit auf den MSSM-Parameter M_2, der für tan(beta) = 4 aus anderen L3-SUSY-Suchen (für Chargino, Neutralino und skalare Leptonen) ermittelt worden ist, in ein Gluino-Massenlimit übersetzt. Die mit 95 % C.L. erhaltenen Ausschlussgrenzen in der Squark-Gluino Massenebene sind - M_gluino > 267-314 GeV, - M_squark > 99-100 GeV. / This thesis is devoted to searches for the scalar top and the scalar bottom quarks within the framework of the Minimal Supersymmetric Standard Model (MSSM) with the assumption of R-parity conservation. Searches for the following decay modes of the stop quark have been performed: stop -> c neutralino_1, stop -> b l sneutrino, (where l is either electron, muon or tau-lepton with equal probabilities) and stop -> b tau sneutrino (where only the tau-lepton is considered). In addition, a three body decay stop -> b W neutralino_1 has been searched for in the allowed mass region of M_stop > M_b + M_W + M_neutralino1 >= 86 GeV. For the sbottom quark the decay sbottom -> b neutralino_1 was considered. Each of these decay modes was considered independently assuming a branching ratio of 100 %. For this search, the experimental data of electron-positron collisions at center-of-mass energies (c.m.s.) in the range of 202-208 GeV have been used. These data were collected in the year 2000 by the L3 detector at the Large Electron Positron Collider (LEP) at CERN. The results of the year 2000 data analysis were also combined with results of the squark searches performed by the L3 Collaboration in previous years at center-of-mass energies from 161 up to 202 GeV. The analyzed squark decay channels determine the topology of the events of our interest: 2 jets (or b-jets) + missing energy (+ 2 leptons for stop three body decays). The stop -> b W neutralino_1 decay topology depends significantly on the further decay of the W boson and can have up to 6 jets in the final state. The assumed conservation of R-parity implies stability of the lightest supersymmetric particle (the LSP), which is the lightest neutralino. The LSP interacts only weakly and thus escapes undetected. This leads to a large missing energy as a feature of the signal events. The visible energy is roughly proportional to the difference between the masses of the squark and the LSP, and since the Standard Model background composition depends on the visible energy fraction, the whole analysis depends also on the value of this mass difference. Depending on the magnitude of visible energy, the Standard Model background can be grouped into three categories: - the two-fermion processes are e e -> e e, e e -> mu mu, e e -> tau tau and e e -> q q; - the four-fermion category is composed of e e -> W W, e e -> W e nu, e e -> Z Z and e e -> Z e e processes; - the two-photon background processes are e e -> e e e e, e e -> e e mu mu, e e -> e e tau tau and e e -> e e q q. The last process, e e -> e e q q, constitutes the largest fraction of all SM background processes (due to very high and highly fluctuating missing energy and the highest cross section). At the very first step of the analysis, only the events of interesting topology (with 2 jets and high missing energy) were preselected. Then, taking into account the small value of the theoretically predicted production cross section of the scalar quarks, the selection of stop and sbottom events was optimized by minimization of the 95 % confidence level expected upper limit on the squark cross section using calculated Monte Carlo events. In all selections optimized for each particular squark decay channel, the number of selected data events statistically agrees with the number of events expected from the Standard Model processes: - for stop -> c neutralino_1 decay, 29 data evens were observed, while 26.5 +- 2.7 were expected from the SM processes; - for the three body decay stop -> b l sneutrino, 4 data events were selected and the expectation from the Standard Model is 4.0 +- 1.0 events; - for the decay stop -> b tau sneutrino, the data and SM event numbers are 5 and 3.9 +- 1.0, respectively; - in the selection for stop -> b W neutralino_1, 184 data events were observed and 181.6 +- 3.0 were expected from the Standard Model; - for the bottom squark decay sbottom -> b neutralino_1 the observed 6 events correspond to the SM expectation of 7.7 +- 1.3. Thus, the MSSM scalar quarks were not observed in the experimental data and the search results are negative. The model independent 95 % C.L. upper limits on the squark production cross section have been derived from the numbers of the observed data events and numbers of events expected from the Standard Model. For calculation of the upper cross section limits, the results of the squark searches performed in the L3 data of c.m.s. energy 202 - 208 GeV were combined with results of searches performed by the L3 Collaboration previously in the data of c.m.s. energy from 161 up to 202 GeV. A new method has been developed for calculating such combined limits. This method takes into account the statistical independence of each measurement and the dependency of the squark production cross section on the center-of-mass energy. In this calculation, the systematic uncertainties in the Standard Model background estimation and in the signal selection efficiency have been also accounted for. For the considered squark decays, the typical obtained 95 % C.L. upper limits on the squark production cross section are: ~ 0.05-0.2 pb (for stop) and ~ 0.05-0.1 pb (for sbottom). In the searches for the stop three body decay stop -> b W neutralino_1, the cross sections above 0.7-1.0 pb have been excluded at 95 % C.L. Within the framework of MSSM with conserved R-parity, the experimental model independent cross section limits have been used for exclusion of the MSSM model parameters, in particular, exclusion of the stop and the sbottom masses. For each considered decay channel, the squark masses have been excluded in two possible scenarios: for the maximal and for the (approximately) minimal theoretical cross section. The first case corresponds to the maximal mixing between the left and right squark eigenstates, cos(theta) = 1; the second case is defined by the cos(theta) value, where squarks decouple from the Z boson. Depending on the mass difference between squark and the LSP, the squark masses have been excluded at 95 % C.L. up to the following values: - for stop -> c neutralino_1: M_stop < 90-93 GeV for minimal cross section, M_stop < 95-96 GeV for maximal cross section; - for stop -> b l sneutrino: M_stop < 87-89 GeV for minimal cross section, M_stop < 90-91 GeV for maximal cross section; - for stop -> b tau sneutrino: M_stop < 83-88 GeV for minimal cross section, M_stop < 88-91 GeV for maximal cross section; - for sbottom -> b neutralino_1: M_stop < 76-83 GeV for minimal cross section, M_stop < 94-97 GeV for maximal cross section. For both cases, the experimentally observed 95 \% C.L. mass exclusions are compared to the expected ones, which have been obtained from the Monte-Carlo assuming no SUSY particles. The observed exclusions of the squark masses are at the same level as the expected ones. The obtained 95 % C.L. upper limits on the stop production cross section in the decay stop -> b W neutralino_1 are bigger than the corresponding theoretical predictions, so, the exclusion of masses at 95 % C.L. was not possible with the available data sample. Assuming the topology of decays of the scalar quarks of the first two generations to be similar to the two body decay of the stop, the results of the searches for the decay stop -> c neutralino_1 have been also used for calculation of the mass exclusion limits for the squarks of the first two families. Two possibilities were considered here: the mass degeneracy between four (scalar u, d, c, s) and five (scalar b in addition) squarks. The 95 % C.L. exclusion limits on the mass degenerate scalar quarks for the cases of the "right-only" or "left-and-right" eigenstates are the following: - for the mass degeneration between 4 squarks: M_squark < 95-96 GeV ("right-only"), M_squark < 99-100 GeV ("left-and-right"); - for the mass degeneration between 5 squarks: M_squark < 96-97 GeV ("right-only"), M_squark < 99-101 GeV ("left-and-right"). Using the MSSM assumption about gaugino unification at the GUT scale, the limits on the four mass degenerate squarks have been reinterpreted on the squark-gluino mass plane. Moreover, the absolute limit on the MSSM parameter M_2, obtained for tan(beta) = 4 from other L3 SUSY searches (for chargino, neutralino and scalar leptons), has been translated into a gluino mass limit. The obtained 95 % C.L. exclusions in the squark-gluino mass plane are - M_gluino > 267-314 GeV, - M_squark > 99-100 GeV.

京都大学 / 0048 / 新制・課程博士 / 博士(理学) / 甲第20904号 / 理博第4356号 / 新制||理||1625(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 杉本 茂樹, 教授 川合 光, 准教授 國友 浩 / 学位規則第4条第1項該当 / Doctor of Science / Kyoto University / DFAM

We attack the question of BPS occupancy in a wide class of 4d N = 2 quantum field theories. We first review the Seiberg-Witten approach to finding the low energy Wilsonian effective action actions of such theories. In particular, we analyze the case of Gaiotto theories, which provide a large number of non-trivial examples in a unified framework. We then turn to understanding the massive BPS spectrum of such theories, and in particular their relation to BPS quivers. We present a purely 4d characterization of BPS quivers, and explain how a quiver's representation theory encodes the solution to the BPS occupancy problem. Next, we derive a so called mutation method, based on exploiting quiver dualities, to solve the quiver's representation theory. This method makes previously intractable calculations nearly trivial in many examples. As a particular highlight, we apply our methods to understand strongly coupled chambers in ADE SYM gauge theories with matter. Following this, we turn to the general story of quivers for theories of the Gaiotto class. We present a geometric approach to attaining quivers for the rank 2 theories, leading to a very elegant solution which includes a specification of quiver superpotentials. Finally, we solve these theories by an unrelated method based on gauging flavor symmetries in their various dual weakly coupled Lagrangian descriptions. After seeing that this method agrees in the rank 2 case, we will apply our new approach to the case of rank n. / Physics

The Atiyah-Singer index theorem, the Euler number, and the Hirzebruch signature are derived via the supersymmetric path integral. Concisely, the supersymmetric path integral is a combination of a bosonic and a femionic path integral. The action in the supersymmetric path integral includes here bosonic, fermionic- and isospin fields (backgroundfields), where the cross terms in the Lagrangian are nicely eliminated due to scaling of the fields and using techniques from spontaneous breaking of supersymmetry (that give rise to a mechanism, analogous to the Higgs-mechanism, but here regarding the so called superparticles instead).  Thus, the supersymmetric path integral is a product of three pathintegrals over the three given fields, respectively, that can be evaluated exactly by means of Gaussian integrals. The closely related Witten index is a measure of the failure of spontaneous breaking of supersymmetry. In addition, the basic concepts of supersymmetry breaking are reviewed.

In this thesis we study the connection between conformal symmetry breaking and the the renormalization group. In the first chapter we review the main properties of conformal field theories (CFTs), Wilsonian RG and describe how renormalization induces a flow between different CFTs. The prominent role is given to the trace of energy-momentum tensor (TEMT) as a measure for conformal symmetry violation. Scaling properties of supersymmetric gauge theories are also reviewed . In the second chapter the quantum action principle is introduced as a scheme for renormalizing composite operators. The framework is then applied to derive conditions for UV finiteness of two-point correlators of composite operators with special emphasis on TEMT. We then proceed to discuss the application of the Feynman-Hellmann theorem to evaluate gluon condensates. In the third chapter the basic elements the Trace anomaly on curved space are examined. The finiteness results from Chapter 2 are given physical meaning in relation with the RG flow of the geometrical quantity ~ d (coefficient of □R in the anomaly). The last chapter is dedicated to the a-theorem. First we apply some of the results derived in Chapter 3 to extend the known perturbative calculation for the flow of the central charge βa for gauge theories with Banks-Zaks fixed point. In the last part we review the main ideas of the recent proof of the a-theorem by Komargodski and Schwimmer and apply their formalism to re-derive the known non-perturbative formula for ∆ βa of SUSY conformal window theories.

In this thesis, we investigate four dimensional supersymmetric indices. The motivation for studying such objects lies in the physics of Seiberg's electric-magnetic duality in supersymmetric field theories. In the first chapter, we first define the index and underline its cohomological nature, before giving a first computation based on representation theory of free superconformal field theories. After listing all representations of the superconformal algebra based on shortening conditions, we compute the associated Verma module characters, from which we can extract the index in the appropriate limit. This approach only provides us with the free field theory limit for the index and does not account for the values of the $R$-charges away from free field theories. To circumvent this limitation, we then study a theory on $\mathbb{R}\times S^3$ which allows for a computation of the superconformal index for multiplets with non-canonical $R$-charges. We expand the fields in harmonics and canonically quantise the theory to analyse the set of quantum states, identifying the ones that contribute to the index. To go beyond free field theory on $\mathbb{R}\times S^3$, we then use the localisation principle to compute the index exactly in an interacting theory, regardless of the value of the coupling constant. We then show that the index is independent of a particular geometric deformation of the underlying manifold, by squashing the sphere. In the final chapter, we show how the matching of the index can be used in the large $N$ limit to identify the $R$-charges for all fields of the electric-magnetic theories of the canonical Seiberg duality. We then conclude by outlining potential further work.

Resumo: A supersimetria é um tópico importante na física teórica atual. Em particular, tem-se dedicado grande esforço no estudo das extensões supersimétricas do Modelo Padrão (SM) desde a década de 80. A incorporação da supersimetria no SM resulta em uma grande quantidade de modelos. O modelo com o conteúdo mínimo de partículas assim como de interações é chamado o Modelo Mínimo Supersimétrico (MSSM). DEvido à supersimetria, todos os modelos supersimétricos apresentam diferenças com relação ao SM. A principal delas, além do conteúdo de partículas, está no setor de Higgs. Em particular, o setor de Higgs do modelo MSSM contem cinco graus de liberdade (cinco bósons de Higgs), diferentemente do SM, que contem apenas um bóson de Higgs. Outra diferença importante no caso do MSSM deve-se à mistura dos estados associados pela supersimetria aos bósons de Gauge e aos bósons de Higgs, chamados gauginos e higgsinos respectivamente, cujos autoestados de massa são conhecidos como charginos e neutralinos. Estas partículas desempenham um papel fundamental na possível descoberta da supersimetria na escala de energia de TeV's. / Abstract: Supersymmetry is a fundamental topic in the actual theoretical physics. In particular, since the 80's, huge efforts have been done studying the supersymmetric extensions of the Standard Model (SM). Including supersymmetry in the SM generates a great amount of models. Among all of these, there is one that involves the minimum number of particles and interactions. This model is known as the Minimal Supersymmetric Standard Model (MSSM). Due to the incorporation of supersymmetry, all the extensions have differences in relation with the SM. The most remarkable one, beyond the particles content, lies in the Higgs sector. Particularly, in the MSSM Higg's sector there are five degrees of freedom (five Higgs bosons), in contrast with the SM (just one). Another difference is related wit the higgsino and gaugino mixture. This result in the presence of mass eigenstates known as charginos and neutralinos. The later particles play a fundamental role in the possible test of supersymmetry at the TeV's scales. / Orientador: Fernando Luiz de Campos Carvalho / Coorientador: Rogério Rosenfeld / Mestre