Fermionic determinant for dyons and instantons with nontrivial holonomy

We calculate exactly the functional determinant for fermions in fundamental representation of $SU(2)$ in the background of periodic instanton with nontrivial value of the Polyakov line at spatial infinity. The determinant depends on the value of the holonomy $\mathrm{v}$, the temperature, and the parameter ${r}_{12}$, which at large values can be treated as separation between the Bogomol'nyi-Prasad-Sommerfield monopoles monopoles (or dyons) which constitute the periodic instanton. We find a compact expression for small and large ${r}_{12}$ and compute the determinant numerically for arbitrary ${r}_{12}$ and $\mathrm{v}$.