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 X which is a product Y x Z of p- and q-dimensional Riemannian manifold Y and Z with p+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 x Z become sigma-model instanton equations for maps from Y to the moduli space M (target space) of gauge instantons on Z if q>= 4. For q<4 we get maps from Y to the moduli space M of flat connections on Z. Thus, the Yang-Mills instantons on Y x 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 M approximate Yang-Mills instantons on Y x Z.

arXiv comment1+14 pages; v2: 2 footnotes and 3 refs. added, published version; v3: gauge-fixing on 3-torus corrected, 4 more refs. added

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• Sasakian quiver gauge theories and instantons on cones over lens 5-spaces
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• Non-Abelian sigma models from Yang-Mills theory compactified on a circle
• Supermembrane limit of Yang-Mills theory
• Skyrme and Faddeev models in the low-energy limit of 4d Yang-Mills-Higgs theories
• Skyrme-Faddeev model from 5d super-Yang-Mills
• Scattering of instantons, monopoles and vortices in higher dimensions
• Instantons on Calabi-Yau and hyper-Kähler cones
• Dual infrared limits of 6d $\mathcal{N}$=(2,0) theory