Non-Perturbative Heterotic Duals of M-Theory on $G_2$ Orbifolds

Abstract

By fibering the duality between the $E_8\times E_8$ heterotic string on $T^3$ and M-theory on K3, we study heterotic duals of M-theory compactified on $G_2$ orbifolds of the form $T^7/{\mathbb{Z}}_2^3$. While the heterotic compactification space is straightforward, the description of the gauge bundle is subtle, involving the physics of point-like instantons on orbifold singularities. By comparing the gauge groups of the dual theories, we deduce behavior of a "half-$G_2$" limit, which is the M-theory analog of the stable degeneration limit of F-theory. The heterotic backgrounds exhibit point-like instantons that are localized on pairs of orbifold loci, similar to the "gauge-locking" phenomenon seen in Hořava-Witten compactifications. In this way, the geometry of the $G_2$ orbifold is translated to bundle data in the heterotic background. While the instanton configuration looks surprising from the perspective of the $E_8\times E_8$ heterotic string, it may be understood as T-dual $\text{Spin}(32)/{\mathbb{Z}}_2$ instantons along with winding shifts originating in a dual Type I compactification.

42 pages, 4 figures; v2: Typos corrected