Fermionic determinant for dyons and instantons with nontrivial holonomy

TL;DR

This paper computes the exact fermionic functional determinant in the background of periodic instantons with nontrivial holonomy, revealing its dependence on holonomy, temperature, and monopole separation, with both analytical and numerical results.

Contribution

It provides the first exact calculation of the fermionic determinant for SU(2) instantons with nontrivial holonomy, including analytical expressions and numerical evaluations.

Findings

Derived compact expressions for small and large monopole separation.

Numerically computed the determinant for arbitrary parameters.

Showed the determinant's dependence on holonomy and temperature.

Abstract

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 r_12, 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 r_12 and compute the determinant numerically for arbitrary r_12 and v.