E9 symmetry in the heterotic string on S1 and the weak gravity conjecture

We show that compactifications of the heterotic string on a circle exhibit at boundary moduli space ($R\to 0$, or equivalently decompactification limit $R \to \infty$) tower winding momentum modes enhance $E_8 \times E_8$ $SO(32)$ gauge symmetry to affine algebras $(E_9 \oplus E_9)/\sim$ (the identification means two copies $E_9$ share same central extension) and $\hat{D}_{16}$, respectively. also prove these towers satisfy lattice Weak Gravity Repulsive Force Conjectures.