Melvin Models and Diophantine Approximation

TL;DR

This paper investigates Melvin models with irrational twist parameters, revealing their connection to arithmetic properties like Lyapunov exponents and showing that the regularized dimension of localized states often behaves as a random variable.

Contribution

It introduces a novel analysis of the regularized dimension in Melvin models, linking it to number-theoretic properties and probabilistic behavior for generic twist parameters.

Findings

Regularized dimension relates to arithmetic properties of the twist parameter.

For almost all twist parameters, the regularized dimension is a random variable.

The behavior of localized states is influenced by Lyapunov exponents.

Abstract

Melvin models with irrational twist parameter provide an interesting example of conformal field theories with non-compact target space, and localized states which are arbitrarily close to being delocalized. We study the torus partition sum of these models, focusing on the properties of the regularized dimension of the space of localized states. We show that its behavior is related to interesting arithmetic properties of the twist parameter $\gamma$, such as the Lyapunov exponent. Moreover, for $\gamma$ in a set of measure one the regularized dimension is in fact not a well-defined number but must be considered as a random variable in a probability distribution.