Effectively Closed Infinite-Genus Surfaces and the String Coupling

TL;DR

This paper explores the properties of infinite-genus surfaces in string theory, showing their inclusion in a specific category and analyzing the behavior of superstring amplitudes and string coupling in the large genus limit.

Contribution

It introduces the class of effectively closed infinite-genus surfaces within the context of string perturbation theory and analyzes their impact on string amplitudes and coupling constants.

Findings

The maximal set of endpoints has cardinality 2^N.

Superstring amplitude coefficients grow exponentially with genus g.

String coupling aligns with configurations dominating finite vacuum amplitudes.

Abstract

The class of effectively closed infinite-genus surfaces, defining the completion of the domain of string perturbation theory, can be included in the category $O_G$, which is characterized by the vanishing capacity of the ideal boundary. The cardinality of the maximal set of endpoints is shown to be $2^{\mit N}$. The product of the coefficient of the genus-g superstring amplitude in four dimensions by $2^g$ in the $g\to \infty$ limit is an exponential function of the genus with a base comparable in magnitude to the unified gauge coupling. The value of the string coupling is consistent with the characteristics of configurations which provide a dominant contribution to a finite vacuum amplitude.