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 non-trivial value of the Polyakov line at spatial infinity. The determinant depends on the value of the holonomy v, the temperature, and the parameter r12, which at large values can be treated as separation between the Bogomolny–Prasad– Sommerfeld monopoles (or dyons) which constitute the periodic instanton. We find a compact expression for small and large r12 and compute the determinant numerically for arbitrary r12 and v.