On Gauge Couplings in String Theory

TL;DR

This work analyzes how gauge couplings in four-dimensional heterotic string vacua depend on the dilaton and moduli, establishing that the dilaton contribution to gauge couplings is universal at one loop while moduli-dependent threshold corrections are tightly constrained by holomorphy and anomaly considerations. It develops two complementary views—Wilsonian couplings that are holomorphic and protected beyond one loop, and physical running couplings that include low-energy effects—showing their consistency and linking the universal dilaton piece to Green–Schwarz mixing via $V^{(1)}$. For orbifold and $(2,2)$ vacua, the authors derive explicit forms for the moduli-dependent threshold corrections, relate them to string-theoretic thresholds, and demonstrate that the modular properties and anomaly constraints fix the holomorphic functions $f_a^{(1)}(M)$ up to constants; in factorizable orbifolds these reduce to sums of logarithms of Dedekind eta functions, $ ext{log}\eta(iM^i)$, with remaining constants fixed by large-radius growth considerations. The quintic Calabi–Yau example illustrates the large-radius and conifold behaviors of the threshold corrections, showing that the $E_8$ coupling has a fixed large-radius logarithmic growth while the $E_6$ coupling can exhibit logarithmic or more severe behavior depending on possible 27+27̄ multiplets becoming massless; overall, the paper provides a coherent framework linking EFT constraints, string threshold data, and geometrical/topological information to determine gauge-coupling dependence in string vacua.

Abstract

We investigate the field dependence of the gauge couplings of $N=1$ string vacua from the point of view of the low energy effective quantum field theory. We find that field-theoretical considerations severely constrain the form of the string loop corrections; in particular, the dilaton dependence of the gauge couplings is completely universal at the one-loop level. The moduli dependence of the string threshold corrections is also constrained, and we illustrate the power of such constraints with a detailed discussion of the orbifold vacua and the $(2,2)$ (Calabi-Yau) vacua of the heterotic string.