Heterotic/Heterotic Duality in D=6,4

TL;DR

This work broadens the scope of heterotic/heterotic duality beyond symmetric $E_8\times E_8$ embeddings by showing that asymmetric embeddings can yield consistent dual pairs, with spontaneous symmetry breaking addressing negative kinetic-term signals in six dimensions. Upon further toroidal compactification to $D=4$, the dual theories organize according to the $N=2$ beta-functions via $\beta^{N=2}_\alpha = 12\left(1+\frac{\tilde{v}_\alpha}{v_\alpha}\right)$, yielding two regimes: $\beta^{N=2}_\alpha=12$ for symmetric embeddings and $\beta^{N=2}_\alpha=24$ for asymmetric embeddings, with large-$T$ behavior $f_\alpha \to S+\left(\frac{\tilde{v}_\alpha}{v_\alpha}\right)T$ and duals under $S$/$T$ exchange. Non-perturbative gauge groups are possible for the $12$ case, while the $24$ case aligns with perturbative physics; freely acting twists then generate $N=1$, $D=4$ duals, preserving $S$-$T$-$U$ permutation and linking to heterotic/type II dual chains via Higgsing sequences. The results illuminate a rich network of dual descriptions across dimensions and supersymmetries, with explicit examples connecting anomaly structures, gauge-kinetic functions, and moduli dynamics.

Abstract

We consider $E_8\times E_8$ heterotic compactifications on $K3$ and $K3\times T^2$. The idea of heterotic/heterotic duality in $D=6$ has difficulties for generic compactifications since for large dilaton values some gauge groups acquire negative kinetic terms. Recently Duff, Minasian and Witten (DMW) suggested a solution to this problem which only works if the compactification is performed assuming the presence of symmetric gauge embeddings on both $E_8$'s. We consider an alternative in which asymmetric embeddings are possible and the wrong sign of kinetic terms for large dilaton value is a signal of spontaneous symmetry breaking. Upon further toroidal compactification to $D=4$, we find that the duals in the DMW case correspond to $N=2$ models in which the $β$-function of the different group factors verify ${β}_α=12$, whereas the asymmetric solutions that we propose have ${β}_α=24$. We check the consistency of these dualities by studying the different large $T,S$ limits of the gauge kinetic function. Dual $N=1$, $D=4$ models can also be obtained by the operation of appropriate freely acting twists, as shown in specific examples.