Semiclassical quantization of the mixed-flux AdS${}_3$ giant magnon

We obtain explicit formulas for the eight bosonic and eight fermionic fluctuations around the mixed-flux generalization of the Hofman-Maldacena giant magnon on AdS${}_3\times$S${}^3\times$T${}^4$ and AdS${}_3\times$S${}^3\times$S$^3 \times$S$^1$. As a check of our results, we confirm that the semiclassical quantization of these fluctuations leads to a vanishing one-loop correction to the magnon energy, as expected from symmetry based arguments.