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On a structure of the one-loop divergences in $4D, {\cal N}=2$ supersymmetric sigma-model

We study the quantum structure of four-dimensional ${\cal N}=2$ superfield sigma-model formulated in harmonic superspace in terms of the omega-hypermultiplet superfield $ω$. The model is described by harmonic superfield sigma-model metric $g_{ab}(ω)$ and two potential-like superfields $L^{++}_{a}(ω)$ and $L^{(+4)}(ω)$. In bosonic component sector this model describes some hyper-Kähler manifold. The manifestly ${\cal N}=2$ supersymmetric covariant background-quantum splitting is constructed and the superfield proper-time technique is developed to calculate the one-loop effective action. The one-loop divergences of the superfield effective action are found for arbitrary $g_{ab}(ω), L^{++}_{a}(ω), L^{(+4)}(ω)$, where some specific analogy between the algebra of covariant derivatives in the sigma-model and the corresponding algebra in the ${\cal N}=2$ SYM theory is used. The component structure of divergences in the bosonic sector is discussed.