Complexity in String Theory Effective Actions

Abstract

The theory of mathematical complexity, in the form of sharp or effective o-minimality, is explored and used to assign a well-defined notion of complexity to the effective actions arising from F-theory compactifications. In particular, we focus on the effective o-minimal structure of Log-Noetherian functions (R_LN) and its Pfaffian closure (R_LNPF), in which all period maps are definable. These period mappings, which are objects from algebraic geometry and Hodge theory, are used to describe the scalar potentials that arise from F-theory and string theory flux compactifications. We take a close look at our notion of complexity, and note that there are some potential issues with the current definitions regarding their growth rate. We then explicitly calculate the complexity of the period map corresponding to some elliptic curves (tori), and also assign a complexity to the corresponding scalar potentials for a flux compactification on one of these elliptic curves (times a rigid Calabi-Yau threefold). We then generalize to arbitrary Calabi-Yau fourfolds, and note the difficulty that arises when trying to account for the symmetry by monodromy transformations when doing so.