We consider a 4D theory with a chiral anomaly, on $R^3 \times S^1$. From the 3D perspective, it seems at first that the anomaly is lost since odd dimensions do not allow for local anomalies. However the anomaly cannot simply disappear, and by choosing a regulator that preserves the symmetries of the UV (4D lorentz invariance in this case) in order to integrate out the KK-modes, we show that field-dependant Chern-Simons terms are generated at one loop. These are not gauge invariant and in fact capture the whole 4D anomaly, in a 3D language. We further extend these results to 6D anomalies and comment on the implications for F-theory compactifications.
Integrating out the KK-modes also leads infinite distance in radius modulus space. We explain how this relates to the Swampland Distance Conjecture and the idea of emergence. We then apply this to the F-theory circle and explain how it relates to infinite distances in the Kaehler moduli space of the Calabi-Yau three-fold on which F-theory is compactified. These singularities can be analyzed from their monodromy matrix, which depends on the intersection number of the CY. This suggests that some topological data of the CY are emergent.