On the computation of non-perturbative effective potentials in the string theory landscape -- IIB/F-theory perspective

TL;DR

This paper explores the computational complexity of determining non-perturbative effects in string theory landscapes, highlighting the potential for undecidability and identifying special cases where systematic solutions are feasible.

Contribution

It introduces a framework linking string theory computations to Diophantine equations, discusses undecidability issues, and develops methods for computing line bundle cohomology in specific Calabi-Yau cases.

Findings

Systematic computation of non-perturbative effects may be undecidable in general.

Certain Calabi-Yau manifolds allow linear Diophantine equations, enabling systematic solutions.

Developed technology for computing equivariant line bundle cohomology on toric varieties.

Abstract

We discuss a number of issues arising when computing non-perturbative effects systematically across the string theory landscape. In particular, we cast the study of fairly generic physical properties into the language of computability/number theory and show that this amounts to solving systems of diophantine equations. In analogy to the negative solution to Hilbert's 10th problem, we argue that in such systematic studies there may be no algorithm by which one can determine all physical effects. We take large volume type IIB compactifications as an example, with the physical property of interest being the low-energy non-perturbative F-terms of a generic compactification. A similar analysis is expected to hold for other kinds of string vacua, and we discuss in particular the extension of our ideas to F-theory. While these results imply that it may not be possible to answer systematically certain physical questions about generic type IIB compactifications, we identify particular Calabi-Yau manifolds in which the diophantine equations become linear, and thus can be systematically solved. As part of the study of the required systematics of F-terms, we develop technology for computing Z_2 equivariant line bundle cohomology on toric varieties, which determines the presence of particular instanton zero modes via the Koszul complex. This is of general interest for realistic IIB model building on complete intersections in toric ambient spaces.