Pour connecter la théorie des cordes à la physique observable, il est essentiel de comprendre des vides supersymmétriques à flux non triviaux. Dans cette thèse, ils sont étudiés en deux cadres mathématiques : les SU(n)-structures et la géométrie complexe généralisée. Les variétés équipées de SU(n)-structures sont des généralisations de variétés de Calabi-Yau. La géométrie complexe généralisée est un cadre géométrique qui regroupe les géométries complexe et symplectique. On donne des classes de vide à flux de supergravité de type II et de théorie-M sur des variétés équipées de SU(4)-structures. Des vides explicites sont donnés sur l'espace de Stenzel, un Calabi-Yau non-compact. Ensuite, sur cette variété, des familles de SU(4)-structures sont construites. À l'aide de celles-ci, on trouve des vides à flux sur des variétés non-symplectiques. Il est démontré que les conditions permettant une supersymétrie à d = 2, N = (2,0) de type IIB peut être reformulées dans le langage de la géométrie complexe généralisée, partiellement interprétables en termes de conditions d'intégrabilité de structures presque complexes généralisées. Enfin, la théorie de type II euclidienne est examinée sur des variétés équipées de SU(5)-structures, donnant des équations généralisées qui sont nécessaires mais pas suffisantes pour satisfaire les équations de supersymétrie. Des classes de solutions explicites sont également donnés / Understanding supersymmetric flux vacua is essential in order to connect string theory to observable physics. In this thesis, flux vacua are studied by making use of two mathematical frameworks: SU(n)-structures and generalised complex geometry. Manifolds with $SU(n)$ structure are generalisations of Calabi-Yau manifolds. Generalised complex geometry is a geometrical framework that simultaneously generalises complex and symplectic geometry. Classes of flux vacua of type II supergravity and M-theory are given on manifolds with SU(4) structure. The N = (1,1) type IIA vacua uplift to N=1 M-theory vacua, with four-flux that need not be (2,2) and primitive. Explicit vacua are given on Stenzel space, a non-compact Calabi Yau. These are then generalised by constructing families of non-CY SU(4)-structures to find vacua on non-symplectic SU(4)-deformed Stenzel spaces. It is shown that the supersymmetry conditions for N = (2,0) type IIB can be rephrased in the language of generalised complex geometry, partially in terms of integrability conditions of generalised almost complex structures. This rephrasing for d=2 goes beyond the calibration equations, in contrast to d=4,6 where the calibration equations are equivalent to supersymmetry. Finally, Euclidean type II theory is examined on SU(5)-structure manifolds, where generalised equations are found which are necessary but not sufficient to satisfy the supersymmetry equations. Explicit classes of solutions are provided here as well. Contact with Lorentzian physics can be made by uplifting such solutions to d=1, N = 1 M-theory

This Dissertation covers three aspects of General Relativity: inequivalent Einstein metrics on Lie Group Manifolds, proving the Hoop Conjecture for Black Rings, and investigating ergoregions in magnetised black hole spacetimes. A number of analytical and numerical techniques are employed to that end. It is known that every compact simple Lie Group admits a bi-invariant homogeneous Einstein metric. We use two ansatze to probe the existence of additional inequivalent Einstein metrics on the Lie Group SU (n). We provide an explicit construction of 2k + 1 and 2k inequivalent Einstein metrics on SU (2k) and SU (2k + 1) respectively. We prove the Hoop Conjecture for neutral and charged, singly and doubly rotating black rings. This allows one to determine whether a rotating mass distribution has an event horizon, that it is in fact a black ring. We investigate ergoregions in magnetised black hole spacetimes. We show that, in general, rotating charged black holes (Kerr-Newman) immersed in an external magnetic field have ergoregions that extend to infinity near the central axis unless we restrict the charge to q = amB and keep B below a maximal value. Additionally, we show that as B is increased from zero the ergoregion adjacent to the event horizon shrinks, vanishing altogether at a critical value, before reappearing and growing until it is no longer bounded as B becomes greater than the maximal value.

In this thesis I will outline various quantum phase transitions in 2D models of magnets that are amenable to simulation with quantum Monte Carlo techniques. The key player in this work is the theory of deconfined criticality, which generically allows for zero temperature quantum phase transitions between phases that break distinct global symmetries. I will describe models with different symmetries including SU(N), SO(N), and "easy-plane" SU(N) and I will demonstrate how the presence or absence of continuous transitions in these models fits together with the theory of deconfined criticality.

In this PhD thesis, we investigate instances of how quenched disorder, charge fluctuations, and reduced spatial symmetries can alter the nature of phases and/or phase transitions in many-body systems with active -- and often frustrated -- magnetic degrees of freedom. The thesis is divided into three parts, each of which focuses on the impact of one of the aforementioned perturbations on a different type of physical system. Part I is concerned with the influence of quenched disorder on spiral spin liquids, i.e., correlated states of matter in which a frustrated magnet evades order by fluctuating between a set of degenerate (or quasi-degenerate) coplanar spin spirals. Using the classical J1-J2 Heisenberg model on the honeycomb lattice as a prototype, we analyze effects driven both by (i) isolated impurities and (ii) nonvanishing concentrations of defects at zero temperature. We address (i) by employing perturbative analytical techniques to identify different order-by-quenched-disorder mechanisms and characterize the impurity-induced spin textures. Besides demonstrating that systems hosting spiral spin liquids are exceptionally susceptible to long-range deformations, we prove that the textures generally acquire an oscillatory out-of-plane component which carries direct information about the ground-state manifold and, remarkably, constitutes a bosonic analogue of Friedel oscillations in metals. We investigate (ii) by performing extensive numerical simulations as a means to characterize the zero-temperature phases realized for different types of disorder. Our results show that the competition between incompatible order-by-quenched-disorder mechanisms can, already at a low to moderate concentrations of defects, destabilize long-range order and induce 'spiral spin-glass' states, in which spins are frozen despite displaying spatial correlations akin to those of a spiral spin liquid. We interpret this finding in the light of effective low-energy theories, which allow us to make concrete statements about the fate of the system at nonzero temperatures. Finally, we discuss extensions of our conclusions to three-dimensional models, as well as their applications to experiments. In Part II, we examine how strong valence fluctuations affect quantum phase transitions marked by the breakdown of coherent Kondo screening in metals with lattices of local magnetic moments. To this end, we study a generalized Anderson lattice model via a parton mean-field theory that describes various Kondo-screened phases in addition to a fractionalized Fermi liquid -- an unscreened phase where electron-like quasiparticles from the conduction sea coexist with fractionalized excitations of a quantum spin liquid. Our results indicate that, at fixed chemical potential, abrupt valence changes can coincide with Kondo breakdown transition and render it discontinuous. We also show that, at a fixed total filling, this translates into phase-separation tendencies which, upon the inclusion of long-range Coulomb interactions, give rise to inhomogeneous states where Kondo-screened and unscreened regions coexist on a mesoscopic scale. We conclude by connecting our findings to experiments in heavy-fermion compounds and other condensed-matter systems. Finally, Part III discusses how the reduction of the spatial symmetries of an SU(2) or generalized SU(N) antiferromagnet can induce different types of magnetic order. This thread includes two separate projects. First, we study the precise means through which a collinear SU(2) Heisenberg antiferromagnet becomes a ferrimagnet once all symmetries between its magnetic sublattices are eliminated. Using general symmetry principles, we elucidate that this process is associated with the splitting of degenerate magnon bands and that the unequal thermal population of the latter is ultimately what generates a uniform magnetization. We thus establish that, in the systems under consideration, ferrimagnetism is induced by thermal fluctuations and that the resulting magnetization curve has a nonmonotonic temperature dependence. This prediction is then supported by an explicit linear spin-wave calculation for a layered variant of a J1-J2 Heisenberg model on a square lattice, which allows us to analytically derive the low-temperature behavior of the uniform magnetization. We show that the same T^4 power law and general qualitative behavior is obtained for a layered and distorted version of the Shastry-Sutherland model. Finally, we complement our results by positively identifying fluctuation-induced ferrimagnetism in Mn2Mo3O8 and discussing the prospects for its emergence in the high-pressure phases of SrCu2(BO3)2. Second, we demonstrate that the concept of altermagnetism, a type of magnetic order that has attracted great interest lately as a potential gateway to unconventional phases of matter and novel technological applications, can be extended to SU(N) magnets with N>2. To begin, we indicate how simple models for such generalized altermagnets can be constructed by reducing the spatial symmetries of SU(N) Heisenberg Hamiltonians with N-color antiferromagnetic ground states. We then apply this procedure to concrete examples with N<5 and conduct comparative analyses of our N=2 and N=3 models via linear spin-wave and flavor-wave calculations. As a result, we find that both systems share the same characteristic behavior of insulating altermagnets, namely a symmetric splitting between magnon bands with different sets of quantum numbers and definite chiralities. Finally, we show that the analogy between the models persists when they are supplemented with charge carriers to become metallic. Much like its SU(2) counterpart, the SU(3) altermagnet exhibits flavor-split electronic bands with crossings determined by a combination of spin and lattice symmetries.:1 Introduction I Magnetic frustration, quenched disorder, and their interplay 2 Elements of frustrated magnetism 2.1 Heisenberg magnetism in Mott insulators 2.2 Frustration and classical ground-state degeneracies 2.3 Order by disorder 2.3.1 Thermal order by disorder 2.3.2 Quantum order by disorder 2.3.3 Conceptual differences and possible competition between thermal and quantum order by disorder 2.3.4 Pseudo-Goldstone modes and experimental realizations 2.4 Classical spin liquids 2.5 Quantum spin liquids 3 Quenched disorder in magnetic systems 3.1 Types of disorder and their consequences 3.2 Random-field disorder: Stability of long-range order and the Imry-Ma argument 3.3 Random-mass disorder: Rounding of first-order phase transitions and the Harris criterion 3.4 Spin glasses: Basic concepts and experimental signatures 3.4.1 The spin glass transition 4 Spiral spin liquids 4.1 J1-J2 Heisenberg model on the honeycomb lattice 4.1.1 Teaser: Enhanced ground-state degeneracy at J2/J1 = 1/2 4.2 Classical ground states and the spiral contour 4.3 Influence of thermal fluctuations 4.3.1 Low temperatures: Thermal order by disorder and nematic order 4.3.2 Intermediate temperatures: Spin liquid regimes 4.4 Linear spin-wave theory and quantum order by disorder 5 Disorder effects in spiral spin liquids 5.1 Disordered J1-J2 Heisenberg model on the honeycomb lattice 5.2 Isolated impurities 5.2.1 Linear response theory: Spin textures and energy corrections 5.2.2 Order by quenched disorder 5.2.3 Friedel-like oscillations 5.2.4 In-plane textures and their impact on long-range order 5.3 Numerics on clean finite systems 5.4 Nonvanishing concentration of impurities 5.4.1 Bond defects of the same type and orientation 5.4.2 Bond defects of random orientation: Emergent glassiness from competing order-by-quenched-disorder mechanisms 5.4.3 Randomly distributed vacancies 5.5 Effects or thermal fluctuations and additional couplings 5.5.1 J1-J2 honeycomb model 5.5.2 Spiral spin liquids in three dimensions 5.6 Connections to experiments 5.7 Summary and outlook II Quantum phase transitions in local-moment metals 6 Single-impurity and lattice Kondo effects: Experimemtal signatures, theory basics, and Kondo breakdown 6.1 Dilute magnetic impurities in metals and the Kondo effect 6.1.1 The Anderson impurity model and local moment formation 6.1.2 From the Anderson to the Kondo model 6.1.3 Kondo temperature and the onset of screening 6.1.4 Breakdown of the Kondo effect in quantum impurity models 6.2 Heavy-fermion metals and Kondo lattices 6.2.1 Anderson and Kondo lattice models 6.2.2 Heavy Fermi liquid 6.2.3 RKKY interaction 6.2.4 Doniach picture and quantum criticality in heavy-fermion systems 6.2.5 Breakdown of screening in Kondo lattices 7 Kondo-breakdown transitions and phase-separation tendencies in valence-fluctuating heavy-fermion metals 7.1 Anderson-Heisenberg lattice model and relevant limits 7.2 Parton mean-field theory 7.2.1 Mean-field equations 7.2.2 Implementation: Lattice, numerics, and temperature regime 7.3 Mean-field phases and phase diagrams 7.3.1 Mean-field phases 7.3.2 Phase diagram and Kondo breakdown at a fixed chemical potential 7.3.3 Phase diagram and Kondo breakdown at a fixed filling 7.4 Discussion: Fluctuation effects, frustrated phase separation, and inhomogeneous states 7.5 Summary and outlook III Modifying magnetic order by symmetry reduction 8 Classification of nonrelativistic collinear magnetic states 8.1 Spin-space and spin point groups 8.1.1 Four classes of collinear magnetism 8.2 Application to collinear magnets in metallic systems 8.3 Application to collinear magnets in insulating systems 8.3.1 Linear spin-wave theory for a collinear Heisenberg magnet 8.3.2 Symmetries of the linear spin-wave spectrum 9 Fluctuation-induced ferrimagnetism 9.1 Ferrimagnetism from thermal fluctuations in a toy model 9.1.1 The toy model, its symmetries, and its ground state 9.1.2 Ground-state antiferromagnetism 9.1.3 Spin-wave spectrum 9.1.4 Uniform magnetization at T>0 9.1.5 Discussion: Nonmonotonic magnetization curves and the influence of magnetic anisotropies 9.2 Ferrimagnetism in a layered and distorted Shastry-Sutherland model 9.2.1 Background: The Shastry-Sutherland model 9.2.2 The layered and distorted Shastry-Sutherland model 9.2.3 Spin-wave theory: Magnon spectrum and ferrimagnetism 9.2.4 Application to SrCu(BO3)2 under pressure 9.3 Summary and outlook 10 From SU(2) to SU(N) altermagnetism 10.1 Prelude: Elements of the theory of su(N) algebras, their representations, and applications to magnetism 10.1.1 The {M,0} irreps of SU(N): Schwinger bosons and the Holstein-Primakoff transformation 10.2 Heisenberg models for SU(N) altermagnetism 10.2.1 Symmetries and good quantum numbers 10.2.2 SU(2) Heisenberg model on the checkerboard lattice 10.2.3 SU(3) Heisenberg model on the hexatriangular lattice 10.2.4 SU(4) Heisenberg model on the cross-cubic lattice 10.3 Spin-wave and flavor-wave theories: Magnon spectra and their symmetries 10.3.1 SU(2) altermagnet 10.3.2 SU(3) altermagnet 10.4 Magnon chirality 10.4.1 SU(2) altermagnet 10.4.2 SU(3) altermagnet 10.5 Electronic band structure: Spin- and flavor-split bands 10.5.1 SU(2) checkerboard Kondo lattice model 10.5.2 SU(3) hexatriangular Kondo lattice model 10.6 Summary and outlook Appendices A Ground-state degeneracy at J2/J1 = 1/2 (Chapter 4) B Iterative minimization algorithm (Chapter 5) C Details on derivations from Chapter 5 C.1 Derivation of Eq. (5.10) C.2 Derivation of Eq. (5.16) C.3 Asymptotic behavior of Friedel-like oscillations for a circular spiral contour D In-plane component of impurity-induced textures: A case study (Chapter 5) D.1 Implementation D.2 Nearest-neighbor bond defect D.3 Next-nearest-neighbor bond defect D.4 Vacancy D.4.1 Changes in textures with increasing J2/J1 E Different parton mean-field theories (Chapter 7) E.1 Fermionic SU(N) and Sp(2N) theories E.1.1 SU(N) large-N limit E.1.2 Sp(2N) large-N limit E.2 Compressibility of different homogeneous spin-liquid states Bibliography