Advances in Theoretical and Mathematical Physics

Volume 20 (2016)

Number 4

Sasakian quiver gauge theories and instantons on Calabi–Yau cones

Pages: 821 – 882

DOI: http://dx.doi.org/10.4310/ATMP.2016.v20.n4.a4

Authors

Olaf Lechtenfeld (Institut für Theoretische Physik & Riemann Center for Geometry and Physics, Leibniz Universität, Hannover, Germany)

Alexander D. Popov (Institut für Theoretische Physik & Riemann Center for Geometry and Physics, Leibniz Universität, Hannover, Germany)

Richard J. Szabo (Department of Mathematics, Heriot-Watt University, Edinburgh, Scotland, United Kingdom; Maxwell Institute for Mathematical Sciences, Edinburgh; and The Higgs Centre for Theoretical Physics, Edinburgh)

Abstract

We consider $\mathrm{SU}(2)$-equivariant dimensional reduction of Yang–Mills theory on manifolds of the form $M \times S^3 / \Gamma$, where $M$ is a smooth manifold and $S^3 / \Gamma$ is a three-dimensional Sasaki–Einstein orbifold. We obtain new quiver gauge theories on $M$ whose quiver bundles are based on the affine ADE Dynkin diagram associated to $\Gamma$. We relate them to those arising through translationally-invariant dimensional reduction over the associated Calabi–Yau cones $C (S^3 / \Gamma)$ which are based on McKay quivers and ADHM matrix models, and to those arising through $\mathrm{SU}(2)$-equivariant dimensional reduction over the leaf spaces of the characteristic foliations of $S^3 / \Gamma$ which are Kähler orbifolds of $\mathbb{C}P^1$ whose quiver bundles are based on the unextended Dynkin diagram corresponding to $\Gamma$. We use Nahm equations to describe the vacua of $\mathrm{SU}(2)$-equivariant quiver gauge theories on the cones as moduli spaces of spherically symmetric instantons. We relate them to the Nakajima quiver varieties which can be realized as Higgs branches of the worldvolume quiver gauge theories on $\mathrm{D}p$-branes probing $\mathrm{D}(p + 4)$-branes which wrap an ALE space, and to the moduli spaces of spherically symmetric solutions in putative non-abelian generalizations of two-dimensional affine Toda field theories.

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Published 4 November 2016