Melvin Models and Diophantine Approximation

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.