Topics at the interface of algebraic geometry and string theory

Abstract

In this thesis, we study three general themes at the interface of algebraic geometry and string theory, focusing on statistics of the F-theory landscape, string junctions in F-theory on singular spaces, and applications of Bridgeland stability conditions. We present an algorithmic construction of 4/3×2.96×10^755 4d F-theory geometries amenable to a systematic study of geometric gauge sectors. For example, non-Higgsable seven-brane clusters occur with probability above 1 − 1.01 × 10^−755 and the geometric gauge group rank is above 160 with probability .999995. We derive sufficient geometric conditions for the existence of a weakly coupled type IIB limit in a general F-theory compactification. In the ensemble of O(10^755) geometries, we demonstrate that the Sen limit exists with probability N_Sen/N_total ≤ 3.0 × 10^−391. Finally, we quantitatively study the possibility that gauge coupling unification can be faked in F-theory compactifications. Within a representative set of gauge sectors in the ensemble, we find that ≈ 77.12% pairs of gauge sectors have equal gauge couplings at codimension one in Kähler moduli space, while ≈ 3.22% pairs never have equal gauge couplings. We study the 6d localized charged matter spectrum in F-theory directly on singular elliptic Calabi-Yau threefolds without resolutions or deformations. We propose a general prescription to determine the charged matter spectrum localized at intersections of seven- branes using string junctions and find agreement with predicted results from 6d anomaly cancellation in all cases considered. Examples include generic Weierstrass models with intersections of arbitrary Kodaira fibers and a residual I1 as well as intersections of non- abelian seven-branes. We prove sharp bounds on the discriminants of stable rank two sheaves on surfaces in three-dimensional projective space. They key technique involve studying them as tor- sion sheaves in projective space via tilit stability in the derived category and proving a Bogomolov inequality for semistable rank two sheaves on integral surfaces in three dimensional projective space. We broadly study the implications of Bridgeland stability conditions on general triangulated categories. We prove non-existence results for a class of stability conditions invariant under the Serre functor. We prove geometric reconstruction results for a class of noncommutative curves, demonstrating for example, that any 1-Calabi-Yau geometric noncommutative scheme with a stability condition is equivalent to the derived category of an elliptic curve. Finally, we systematically study a category associated with the Bondal quiver, compute its Serre functor, and demonstrate that its resolution functor of the nodal cubic is spherical. --Author's abstract