Threshold Corrections and Symmetry Enhancement in String Compactifications

TL;DR

This work derives moduli-dependent threshold corrections for Abelian orbifold compactifications by summing the massive string spectrum into target-space duality invariant automorphic functions. It systematically treats continuous Wilson lines and points in moduli space where extra massless states appear, yielding logarithmic singularities tied to the stringy Higgs mechanism and to gravitational couplings, with explicit constructions for $SO(p+2,2)$ and $SU(m+1,1)$ cosets. The authors develop a four-step framework to obtain untwisted moduli, solve twist constraints, and express the mass formula as $|{ m M}|^2/Y$ with a Kähler potential $K=- ext{log }Y$, leading to automorphic threshold functions that involve $j(T)$ and $oldsymbol{ ext{η}}$-functions. They also discuss Green-Schwarz effects and the implications for perturbative gauge/gravitational thresholds, potential non-perturbative dynamics, and domain-wall phenomena in string compactifications.

Abstract

We present the computation of threshold functions for Abelian orbifold compactifications. Specifically, starting from the massive, moduli-dependent string spectrum after compactification, we derive the threshold functions as target space duality invariant free energies (sum over massive string states). In particular we work out the dependence on the continuous Wilson line moduli fields. In addition we concentrate on the physically interesting effect that at certain critical points in the orbifold moduli spaces additional massless states appear in the string spectrum leading to logarithmic singularities in the threshold functions. We discuss this effect for the gauge coupling threshold corrections; here the appearance of additional massless states is directly related to the Higgs effect in string theory. In addition the singularities in the threshold functions are relevant for the loop corrections to the gravitational coupling constants.