Asymptotic String Compactifications: Periods, flux potentials, and the swampland

Abstract

In this thesis we have studied various applications of asymptotic Hodge theory in string compactifications. This mathematical framework captures how physical couplings of the resulting effective theories behave near boundaries in the scalar field space where the internal Calabi-Yau manifold degenerates. Here we conclude by giving a summary of each of the three parts in which this thesis is divided. Part I introduced the techniques from asymptotic Hodge theory we used throughout this thesis. We reviewed the results of the nilpotent orbit theorem of Schmid and the multi-variable sl(2)-orbit theorem of Cattani, Kaplan and Schmid. This discussion was tailored to applications in the study of string compactifications, explaining how to describe important physical couplings such as Kähler potentials and flux superpotentials near boundaries in moduli spaces. The nilpotent orbit approximation yields an asymptotic expansion divided into leading polynomial terms with exponential corrections. In turn, the sl(2)-orbit approximation implements a hierarchy in the large field limit towards this boundary, enabling the use of algebraic structures such as sl(2,R)-triples to describe asymptotic behavior. Part II discussed a geometrical application of asymptotic Hodge theory with the construction of general models for asymptotic periods. To be precise, we studied the (3,0)-form periods of the Calabi-Yau manifold near boundaries in complex structure moduli space. These periods determine for instance the N=2 vector sector of Type IIB compactifications and encode part of the N=1 Kähler potential and flux superpotential in Type IIB orientifolds. Taking the constraints imposed by asymptotic Hodge theory as consistency principles, we developed new methods for constructing these periods. We explicitly carried out our program for all possible one- and two-moduli boundaries and constructed general models for their asymptotic periods. Part III discussed two applications of asymptotic Hodge theory in string compactifications. Chapter 7 focused on bounds put by the Weak Gravity Conjecture, which predicts the existence of states whose charge must larger than or equal to its mass compared to the black hole extremality bound. We studied the charge-to-mass ratios of BPS black holes in four-dimensional N=2 supergravity theories arising from Type IIB Calabi-Yau threefold compactifications. Geometrically these states arise from D3-branes wrapped on certain three-cycles of the internal geometry. We computed the asymptotic charge-to-mass ratios for a particular set of sl(2)-elementary states that couple to the asymptotic graviphoton. In turn, we determined the radii of the ellipsoid that forms the extremality region of electric BPS black holes, thereby giving us precise order one bounds. Chapter 8 studied moduli stabilization in asymptotic regimes in complex structure moduli space. We compared two equivalent sets of extremization conditions: a self-duality for the fluxes or demanding vanishing F-terms. The former is formulated via the Hodge star of the underlying geometry, while the latter are computed via the (D, 0)-form periods. By the nilpotent orbit approximation the self-duality condition (or equivalently the scalar potential) then admits a leading polynomial approximation where all exponential corrections can be consistently dropped. In contrast, the F-terms (or equivalently the superpotential) required us to take metric-essential exponential corrections into account.