Tameness and Complexity in flux compactifications
Summary
The Swampland program aims to identify constraints on the effective field theories that can be consistently coupled to quantum gravity. Evidence supporting a generalized finiteness principle underlying the Swampland criteria motivates the introduction of a `tame' or moderate geometry---one suitable for describing physics while inherently incorporating finiteness properties at the most fundamental level. Within this framework, a natural notion of complexity can be rigorously introduced, establishing a way to upper bound the information required to describe tame sets and functions. We explore the recently discovered definability of the period map within an effectively o-minimal structure, and derive an upper bound on its complexity. This constitutes a first step towards understanding the reduction in complexity experienced by the periods of a Calabi-Yau threefold upon breaking discrete symmetries by turning on fluxes. This also turns out to be relevant when discussing the complexity of physical quantities. Finally, we examine a lattice point counting theorem that may shed light on how the enormous number of flux vacua arising from string compactifications relates to the complexity of the corresponding moduli space and flux lattice, where the inclusion or exclusion of dualities plays a crucial role.