Fermionic determinant for SU(N) caloron with nontrivial holonomy

In the finite-temperature Yang-Mills theory we calculate the functional determinant for fermions in the fundamental representation of the SU(N) in the background of an instanton with non-trivial values of the Polyakov line at spatial infinity. This object, called the Kraan--van Baal -- Lee--Lu caloron, can be viewed as composed of N Bogomolny--Prasad--Sommerfeld monopoles (or dyons). We compute analytically two leading terms of the fermionic determinant at large separations between dyons.