Covariant derivative expansion of the Yang-Mills effective action at high temperatures

Integrating out fast varying quantum fluctuations about Yang-Mills fields ${A}_{i}$ and ${A}_{4},$ we arrive at the effective action for those fields at high temperatures. Assuming that the fields ${A}_{i}$ and ${A}_{4}$ are slowly varying but that the amplitude of ${A}_{4}$ is arbitrary, we find a nontrivial effective gauge invariant action both in the electric and magnetic sectors. Our results can be used for studying correlation functions at high temperatures beyond the dimensional reduction approximation, as well as for estimating quantum weights of classical static configurations such as dyons.