Heterotic sigma models on $T^8$ and the Borcherds automorphic form ${\mathrm{\Phi }}_{12}$

We consider the spectrum of BPS states of the heterotic sigma model with $(0,8)$ supersymmetry and $T^8$ target, as well as its second-quantized counterpart. We show that the counting function for such states is intimately related to Borcherds’ automorphic form ${\mathrm{\Phi }}_{12}$, a modular form which exhibits automorphy for $O(2,26;\mathbb{Z})$. We comment on possible implications for Umbral moonshine and theories of AdS${}_3$ gravity.