Toroidal Compactification of Heterotic 6D Non-Critical Strings Down to Four Dimensions

TL;DR

This work shows that toroidal compactification of a 6D heterotic non-critical (tensionless) string theory with $N=1$ SUSY and $E_8$ current algebra yields a 4D $N=2$ theory described by a Seiberg-Witten curve that depends on the torus modulus $\sigma$ and the $E_8$ Wilson lines $w_i$. By analyzing large-$u$ behavior and performing scaling limits, the authors reproduce the SW curves of $SU(2)$ QCD with matter and reveal sectors with enhanced $E_{6,7,8}$ global symmetry; the coefficients of the curve are fixed by the $E_8$ lattice and its relation to almost Del Pezzo surfaces/F-theory. The results connect the microscopic HTS dynamics to 4D $N=2$ Coulomb branches, provide a bridge to F-theory geometry near $E_8$ singularities, and suggest generalizations to higher-rank theories and other tensionless string constructions. This framework advances understanding of how tensionless strings encode 4D gauge dynamics and may illuminate microscopic structure via the interpreted $u$-variable and its instanton expansions.

Abstract

The low-energy limit of the 6D non-critical string theory with $N=1$ SUSY and $E_8$ chiral current algebra compactified on $T^2$ is generically an $N=2$ $U(1)$ vector multiplet. We study the analog of the Seiberg-Witten solution for the low-energy effective action as a function of $E_8$ Wilson lines on the compactified torus and the complex modulus of that torus. The moduli space includes regions where the Seiberg-Witten curves for $SU(2)$ QCD are recovered as well as regions where the newly discovered 4D theories with enhanced $E_{6,7,8}$ global symmetries appear.