Fingerprinting New Physics with Effective Field Theories

Abstract
Given the absence of direct evidence for new resonances beyond the Standard Model (BSM) at the Large Hadron Collider (LHC) so far, a complementary strategy to search for new physics in an indirect way is provided by the Standard Model Effective Field Theory (SMEFT). As the low-energy limit of a generic ultraviolet (UV) completion of the SM, the SMEFT provides a powerful theoretical framework to correlate deviations from the SM between different processes, offering experimental sensitivity to a plethora of SM extensions. This thesis presents a state-of-the-art SMEFT interpretation of the top, Higgs, diboson and electroweak sectors, taking into account data collected at the Large Electron-Positron Collider (LEP), the SLAC Large Detector (SLD) and the LHC. We also include the effect of the upcoming High-Luminosity LHC (HL-LHC) upgrade and demonstrate the unprecedented impact on the SMEFT parameter space of two proposed electron-positron colliders: the electron-positron Future Circular Collider (FCC-ee) and the Circular Electron Positron Collider (CEPC). We present constraints both in terms of Wilson coefficients, and couplings and masses of a wide range of UV-complete models through a newly developed automatised limit-setting procedure. We further present novel methodological advances through the development of unbinned multivariate likelihoods specialised to the SMEFT that provide maximal sensitivity to new physics using classification and regression techniques from Machine Learning. Our results provide an extensive characterisation of the SMEFT parameter space as probed both by current and future colliders, providing timely input to the upcoming European Strategy for Particle Physics Update.

Topological states in aperiodic, non Hermitian and electronically correlated systems

Abstract
This thesis explores how topological concepts, usually associated with abstract mathematics, provide insights into complex physical systems. Physicists use topology to study systems that exhibit robust properties unaffected by minor changes. For example, topological insulators are materials that conduct electricity only at their boundaries and could improve the efficiency of current electronics due to their dissipationless transport properties. Similarly, topological superconductors might enable scalable quantum computing by protecting delicate quantum states from environmental interference. This thesis focuses on multiple aspects where topology can arise. One of those is on non-Hermitian systems, which typically describe systems coupled to environmental interactions. Using graph theory and the concept of latent symmetries, we identified new topological phases within seemingly complicated systems. We also constructed a continuum approximation framework to study disorder at phase transitions in non-Hermitian systems, providing insights into topological stability. Beyond non-Hermitian systems, we also developed, along similar lines, continuum approximations to study a topological insulator in the presence of electron-electron interactions, showing that they do not alter the principal mechanism behind a topological phase. A second focus is on aperiodic structures like quasicrystals, which lack traditional periodic order but exhibit long-range correlations. Quasicrystals can be constructed by projecting higher-dimensional crystals into lower dimensions, revealing hidden fractal structures. The thesis explores how impurities disrupt quasicrystalline order and examines topological phenomena in these aperiodic systems, such as topological charge pumping and boundary states protected by inversion symmetry. To summarize, this work reveals hidden topological structures in non-traditional systems, offering a foundation for future experimental and theoretical research in topological phases and aperiodic materials.

Asymptotic Hodge Theory in String Compactifications and Integrable Systems

Abstract
String theory is a proposal for a description of nature on the smallest length scales, where both quantum mechanics and gravity are expected to play an essential role. A curious feature of the theory is that its basic building blocks – one-dimensional strings – behave as if they were moving in nine-dimensional space, as opposed to the three-dimensional space we are familiar with. String theory proposes that the six additional dimensions are curled up into extremely small sizes, in a process called “compactification”. Importantly, what happens in these hidden six dimensions has an enormous influence on the physics we observe at larger length scales, including the strength of the interactions between elementary particles and the value of the cosmological constant. An intriguing aspect of string theory is that there are countless possible ways to compactify these extra dimensions, resulting in a vast “landscape” of potential universes, each with distinct physical properties. In a beautiful interplay of physics and mathematics, the physical characteristics of these universes are intricately linked to the geometric properties of the internal six-dimensional space. The main aim of this thesis is to study the latter using the sophisticated mathematical framework of asymptotic Hodge theory. One of the central results is that one can obtain a good approximation of numerous physical observables by studying the allowed singularities of the six-dimensional internal space. In accordance with some fundamental theorems in the field of algebraic geometry, these can be classified in great generality and reveal intriguing underlying structures. In particular, this abstract approach leads to a comprehensive, algorithmic method for calculating physical observables in string compactifications. In this way, we obtain a very general characterization of universes that can be constructed in string theory and find that they are in fact finite in number. Additionally, based on recent advances in the field of tame geometry it is suggested that this number might be much smaller than previously expected. Finally, we observe that the same mathematical structures applied above in the context of string compactifications arise in a completely different corner of physics: the study of integrable non-linear sigma models. In particular, the same computational methods used to compute physical observables in string theory can be used to find previously unknown solutions for certain classes of such models. Our results open up a new avenue of research, bridging two distinct areas of theoretical physics.