Convergence of the infinite beta-function series in string theory

Establish the convergence or precise summability properties of the full infinite alpha′-expansion of the beta functions that govern background-field dynamics in string theory, thereby clarifying whether the resulting infinite-order system admits a well-defined Cauchy problem and controlled solutions.

Background

The authors examine Penrose’s concern that the full string background equations arise from vanishing beta functions expressed as an infinite series in the string constant α′, leading to an infinite system of unbounded differential order.

They argue that well-posedness may be maintained if the series converges, but note that, to date, the convergence issue is unaddressed in the literature and that beta functions often do not converge, making the problem pressing.

Clarifying the convergence behavior is essential both for the mathematical consistency of the background-field dynamics and for assessing criticisms based on alleged excessive functional freedom or ill-posed initial data.