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In this thesis we study the framework of asymptotic Hodge theory and its applications in both the string landscape and the landscape of 2d integrable field theories. We show how this mathematical framework allows for a general characterization of the asymptotic behaviour of physical couplings in low-energy effective theories coming from string theory, and apply this knowledge to investigate the finiteness and geometric structure of the string landscape landscape. At the same time, we find that the defining equations of variations of Hodge structure also arise in the context of certain integrable field theories, which opens the way to finding new classes of very general solutions to said models. Part I reviews the relevant aspects of type IIB / F-theory flux compactifications and the resulting landscape of 4d low-energy effective $\mathcal{N}=1$ supergravity theories. Part II provides an in-depth discussion on asymptotic Hodge theory, including detailed explanations on the nilpotent orbit theorem of Schmid, and the multi-variable Sl(2)-orbit theorem of Cattani, Kaplan, and Schmid. This part of the thesis also contains new results regarding the multi-variable bulk reconstruction procedure, which have not appeared in the author's previous publications. Part III concerns the application of the aforementioned results to study the finiteness of the F-theory flux landscape. Additionally, motivated by recent advances in the field of o-minimal geometry and the theory of unlikely intersections, we propose three conjectures which aim to address finer features of the flux landscape. Part IV investigates two corners of the landscape of 2d integrable non-linear sigma-models, namely the $\lambda$-deformed gauged WZW model and the critical bi-Yang-Baxter model. Notably, it is shown that asymptotic Hodge theory can be used to find broad classes of solutions these models., Comment: PhD thesis, 358 pages; based on arXiv:2103.12746, arXiv:2112.00031, arXiv:2212.03893, arXiv:2311.09295

In this paper, we explore the string theory landscape obtained from type IIB and F-theory flux compactifications. We first give a comprehensive introduction to a number of mathematical finiteness theorems, indicate how they have been obtained, and clarify their implications for the structure of the locus of flux vacua. Subsequently, in order to address finer details of the locus of flux vacua, we propose three mathematically precise conjectures on the expected number of connected components, geometric complexity, and dimensionality of the vacuum locus. With the recent breakthroughs on the tameness of Hodge theory, we believe that they are attainable to rigorous mathematical tools and can be successfully addressed in the near future. The remainder of the paper is concerned with more technical aspects of the finiteness theorems. In particular, we investigate their local implications and explain how infinite tails of disconnected vacua approaching the boundaries of the moduli space are forbidden. To make this precise, we present new results on asymptotic expansions of Hodge inner products near arbitrary boundaries of the complex structure moduli space., Comment: 47 pages + appendices, 11 figures; v2: formulation of generalized tadpole conjecture improved; v3: refined conjecture 1

We study integrable deformations of two-dimensional non-linear sigma-models and present a new class of classical solutions to critical bi-Yang-Baxter models for general groups. For the simplest example, namely the SL(2,R) bi-Yang-Baxter model, we show that our solutions can be mapped to the known complex uniton solutions of the SU(2) bi-Yang-Baxter model. In general, our solutions are constructed from so-called Sl(2)-orbits that play a central role in the study of asymptotic Hodge theory. This provides further evidence for a close relation between integrable non-linear sigma-models and the mathematical principles underlying Hodge theory. We have also included a basic introduction to the relevant aspects of asymptotic Hodge theory and have provided some simple examples., Comment: 35 pages, 1 figure

We investigate a relationship between a particular class of two-dimensional integrable non-linear $\sigma$-models and variations of Hodge structures. Concretely, our aim is to study the classical dynamics of the $\lambda$-deformed $G/G$ model and show that a special class of solutions to its equations of motion precisely describes a one-parameter variation of Hodge structures. We find that this special class is obtained by identifying the group-valued field of the $\sigma$-model with the Weil operator of the Hodge structure. In this way, the study of strings on classifying spaces of Hodge structures suggests an interesting connection between the broad field of integrable models and the mathematical study of period mappings., Comment: 34 pages

It was recently suggested that certain UV-completable supersymmetric actions can be characterized by the solutions to an auxiliary non-linear sigma-model with special asymptotic boundary conditions. The space-time of this sigma-model is the scalar field space of these effective theories while the target space is a coset space. We study this sigma-model without any reference to a potentially underlying geometric description. Using a holographic approach reminiscent of the bulk reconstruction in the AdS/CFT correspondence, we then derive its near-boundary solutions for a two-dimensional space-time. Specifying a set of $ Sl(2,\mathbb{R})$ boundary data we show that the near-boundary solutions are uniquely fixed after imposing a single bulk-boundary matching condition. The reconstruction exploits an elaborate set of recursion relations introduced by Cattani, Kaplan, and Schmid in the proof of the $Sl(2)$-orbit theorem. We explicitly solve these recursion relations for three sets of simple boundary data and show that they model asymptotic periods of a Calabi--Yau threefold near the conifold point, the large complex structure point, and the Tyurin degeneration., Comment: 44 pages plus appendices, 1 figure