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

Abstract

We consider the Hermitian Yang–Mills (instanton) equations for connections on vector bundles over a 2n-dimensional Kähler manifold Xwhich is a product Y×Zof p-and q-dimensional Riemannian manifold Yand Zwith p+q=2n. We show that in the adiabatic limit, when the metric in the Zdirection is scaled down, the gauge instanton equations on Y×Zbecome sigma-model instanton equations for maps from Yto the moduli space M(target space) of gauge instantons on Zif q≥4. For q<4we get maps from Yto the moduli space Mof flat connections on Z. Thus, the Yang–Mills instantons on Y×Zconverge to sigma-model instantons on Ywhile Zshrinks to a point. Put differently, for small volume of Z, sigma-model instantons on Ywith target space Mapproximate Yang–Mills instantons on Y×Z.