Abstract

Diophantine equations are in general undecidable, yet appear readily in string theory. We demonstrate that numerous classes of Diophantine equations arising in string theory are decidable and propose that decidability may propagate through networks of string vacua due to additional structure in the theory. Diophantine equations arising in index computations relevant for D3-instanton corrections to the superpotential exhibit propagation of decidability, with new and existing solutions propagating through networks of geometries related by topological transitions. In the geometries we consider, most divisor classes appear in at least one solution, significantly improving prospects for Kahler moduli stabilization across large ensembles of string compactifications.

Highlights

  • Many problems arising in string theory are computationally hard, which could be dynamically relevant

  • Since these data fix some physical observables of the string theory compactification, decision problems involving Diophantine equations abound in studies of the string landscape

  • We studied Diophantine equations that arise in string theory with regard to their decidability and physics

Read more

Summary

INTRODUCTION

Many problems arising in string theory are computationally hard, which could be dynamically relevant. The way in which Diophantine equations arise in string compactifications is by the appearance of integer topological data, such as Chern classes, that describe the compactification space, internal gauge fluxes, and the cycles wrapped by branes Since these data fix some physical observables of the string theory compactification, decision problems involving Diophantine equations abound in studies of the string landscape. Given a class of seemingly intractable Diophantine equations that arise for a physical problem in string theory, does the additional structure of the theory render them decidable? This applies to the largest known ensembles of such bases, which exhibit similar physical features and have 2.96 × 10755 and Oð103000Þ geometries [24,25] These extremely large (but finite) networks are intractable by brute force techniques, but propagation of decidability allows for concrete statements about instanton solutions.

PHYSICS IMPLICATIONS OF KNOWN DIOPHANTINE RESULTS
Cubic Diophantine equations in string theory
PROPAGATION OF INSTANTON SOLUTIONS THROUGH NETWORKS
Concrete solutions to SEQ-E3-INDEX
D2 D3 χC0
IMPLICATIONS OF PROPAGATION FOR KÄHLER MODULI STABILIZATION
Findings
OF RESULTS
Full Text
Published version (Free)

Talk to us

Join us for a 30 min session where you can share your feedback and ask us any queries you have

Schedule a call