Affine Algebras at Infinite Distance Limits in the Heterotic String

We analyze the boundaries of the moduli spaces of compactifications of the heterotic string on $T^d$, making particular emphasis on $d=2$ and its F-theory dual. We compute the OPE algebras as we approach all the infinite distance limits that correspond to (possibly partial) decompactification limits in some dual frame. When decompactifying $k$ directions, we find infinite towers of states becoming light that enhance the algebra arising at a given point in the moduli space of the $T^{d-k}$ compactification to its $k$-loop version, where the central extensions are given by the $k$ KK vectors. For $T^2$ compactifications, we reproduce all the affine algebras that arise in the F-theory dual, and show all the towers explicitly, including some that are not manifest in the F-theory counterparts. Furthermore, we construct the affine $SO(32)$ algebra arising in the full decompactification limit, both in the heterotic and in the F-theory sides, showing that not only affine algebras of exceptional type arise in the latter.