ON TOPOLOGICALLY COMPLEX COMPACTIFICATIONS OF STRING THEORY

Abstract

This dissertation studies compactifications of string theory on Calabi-Yau manifolds with large Hodge numbers. We develop efficient algorithms to construct Calabi-Yau manifolds as hypersurfaces in toric varieties, compute relevant topological data, and study the corresponding compactifications of critical superstring theories, with an emphasis on type IIB flux compactifications. In chapter 2, we prove rigorous upper bounds on the number of fine, regular, star triangulations of polytopes in the Kreuzer-Skarke list, $N_{FRST} < 1.53 \times 10^{928}$, and on the number of topologically inequivalent hypersurfaces, $N_{CY} < 1.65 \times 10^{428}$. We introduce efficient algorithms for constructing representative ensembles of Calabi-Yau hypersurfaces, including the extremal case $h^{1,1}=491$, and we study the distributions of topological and physical data therein. Finally, we demonstrate that neural networks can accurately predict these data once the triangulation is encoded in terms of the secondary polytope. In chapter 3, we show that the K\"ahler cones of Calabi-Yau hypersurfaces are very narrow at large $h^{1,1}$, and as a consequence, control of the $\alpha^{\prime}$ expansion in string compactifications on these hypersurfaces is correlated with the presence of ultralight axions. If every effective curve has volume $\ge 1$ in string units, then the typical volumes of irreducible effective curves and divisors, and of the hypersurface itself, scale as $(h^{1,1})^p$, with $3\lesssim p \lesssim 7$ depending on the type of cycle in question. Instantons from branes wrapping these cycles are thus highly suppressed, giving rise to ultralight axions. In chapter 4, we perform an extensive analysis of the statistics of axion masses and interactions in compactifications of type IIB string theory, and we show that black hole superradiance excludes some regions of Calabi-Yau moduli space. Regardless of the cosmological model, a theory with an axion whose mass falls in a superradiant band can be probed by the measured properties of astrophysical black holes, unless the axion self-interaction is large enough to disrupt formation of a condensate. We study a large ensemble of compactifications on Calabi-Yau hypersurfaces, with $1 \le h^{1,1} \le 491$ closed string axions, and determine whether the superradiance conditions on the masses and self-interactions are fulfilled. The axion mass spectrum is largely determined by the K\"ahler parameters, for mild assumptions about the contributing instantons, and takes a nearly-universal form when $h^{1,1} \gg 1$. When the K\"ahler moduli are taken at the tip of the stretched K\"ahler cone, the fraction of geometries excluded initially grows with $h^{1,1}$, to a maximum of $\approx 0.5$ at $h^{1,1} \approx 160$, and then falls for larger $h^{1,1}$. Further inside the K\"ahler cone, the superradiance constraints are far weaker, but for $h^{1,1} \gg 100$ the decay constants are so small that these geometries may be in tension with astrophysical bounds, depending on the realization of the Standard Model.