Sigma-model limit of Yang–Mills instantons in higher dimensions

We consider the Hermitian Yang–Mills (instanton) equations for connections on vector bundles over a 2n-dimensional Kähler manifold X   which is a product Y×ZY×Z of p- and q-dimensional Riemannian manifold Y and Z   with p+q=2np+q=2n. We show that in the adiabatic limit, when the metric in the Z   direction is scaled down, the gauge instanton equations on Y×ZY×Z become sigma-model instanton equations for maps from Y   to the moduli space MM (target space) of gauge instantons on Z   if q≥4q≥4. For q<4q<4 we get maps from Y   to the moduli space MM of flat connections on Z  . Thus, the Yang–Mills instantons on Y×ZY×Z converge to sigma-model instantons on Y while Z shrinks to a point. Put differently, for small volume of Z, sigma-model instantons on Y   with target space MM approximate Yang–Mills instantons on Y×ZY×Z.