Finiteness Theorems and Counting Conjectures for the Flux Landscape

TL;DR

This paper investigates the mathematical structure of flux vacua in string theory, proposing conjectures on their distribution and analyzing finiteness theorems with recent advances in Hodge theory to understand the landscape's complexity.

Contribution

It introduces conjectures on the flux vacua landscape and advances the understanding of finiteness theorems using recent developments in Hodge theory and asymptotic analysis.

Findings

Finiteness theorems restrict the structure of flux vacua.

Proposed conjectures on the number and complexity of vacua.

New results on asymptotic behavior near moduli space boundaries.

Abstract

In this paper, we explore the string theory landscape obtained from type IIB and F-theory flux compactifications. We first give a comprehensive introduction to a number of mathematical finiteness theorems, indicate how they have been obtained, and clarify their implications for the structure of the locus of flux vacua. Subsequently, in order to address finer details of the locus of flux vacua, we propose three mathematically precise conjectures on the expected number of connected components, geometric complexity, and dimensionality of the vacuum locus. With the recent breakthroughs on the tameness of Hodge theory, we believe that they are attainable to rigorous mathematical tools and can be successfully addressed in the near future. The remainder of the paper is concerned with more technical aspects of the finiteness theorems. In particular, we investigate their local implications and explain how infinite tails of disconnected vacua approaching the boundaries of the moduli space are forbidden. To make this precise, we present new results on asymptotic expansions of Hodge inner products near arbitrary boundaries of the complex structure moduli space.