We classify N = 2 Minkowski 4 solutions of IIB supergravity with an SU (2) R symmetry geometrically realized by an S 2 -foliation in the remaining six dimensions. For the various cases of the classification, we reduce the supersymmetric system of equations to PDEs. These cases often accommodate systems of intersecting branes and half-maximally supersymmetric AdS 5,6,7 solutions when they exist. As an example, we analyze the AdS 6 case in more detail, reducing the supersymmetry equations to a single cylindrical Laplace equation. We also recover an already known linear dilaton background dual to the (1, 1) Little String Theory (LST) living on NS5-branes, and we find a new Minkowski 5 linear dilaton solution from brane intersections. Finally, we also discuss some simple Minkowski 4 solutions based on compact conformal Calabi-Yau manifolds.2 This is not the only way to realise SU(2), indeed it is possible to decompose M 6 as a fibration of S 3 over some M 3 in terms of the Maurer-Cartan of SU(2). It is however unclear whether anything beyond the SU(2) × U(1) preserving squashed 3-sphere is compatible with SU(2) R . When it is compatible one can always T-dualise on the Hopf fibre and end up in a class with a round S 2 factor. Thus up to T-duality the combined results of this work and [26] cover such cases

We consider Spin(4)-equivariant dimensional reduction of Yang-Mills theory on manifolds of the form $M^d \times T^{1,1}$, where $M^d$ is a smooth manifold and $T^{1,1}$ is a five-dimensional Sasaki-Einstein manifold Spin(4)/U(1). We obtain new quiver gauge theories on $M^d$ extending those induced via reduction over the leaf spaces $\mathbb{C}P^1 \times \mathbb{C}P^1$ in $T^{1,1}$. We describe the Higgs branches of these quiver gauge theories as moduli spaces of Spin(4)-equivariant instantons on the conifold which is realized as the metric cone over $T^{1,1}$. We give an explicit construction of these moduli spaces as K\"ahler quotients.Comment: 30 page

We study quiver gauge theories on the round and squashed seven-spheres, and orbifolds thereof. They arise by imposing G-equivariance on the homogeneous space G/H = SU(4)/SU(3) endowed with its Sasaki-Einstein structure, and G/H = Sp(2)/Sp(1) as a 3-Sasakian manifold. In both cases we describe the equivariance conditions and the resulting quivers. We further study the moduli spaces of instantons on the metric cones over these spaces by using the known description for Hermitian Yang-Mills instantons on Calabi-Yau cones. It is shown that the moduli space of instantons on the hyper-Kähler cone can be described as the intersection of three Hermitian Yang-Mills moduli spaces. We also study moduli spaces of translationally invariant instantons on the metric cone R 8 /Z k over S 7 /Z k .

We consider the SU(3)-equivariant dimensional reduction of gauge theories on spaces of the form M d × X 1,1 with d-dimensional Riemannian manifold M d and the Aloff-Wallach space X 1,1 = SU(3)/U(1) endowed with its Sasaki-Einstein structure. The condition of SU(3)-equivariance of vector bundles, which has already occurred in the studies of Spin(7)-instantons on cones over Aloff-Wallach spaces, is interpreted in terms of quiver diagrams, and we construct the corresponding quiver bundles, using (parts of) the weight diagram of SU(3). We consider three examples thereof explicitly and then compare the results with the quiver gauge theory on Q 3 = SU(3)/(U(1) × U(1)), the leaf space underlying the Sasaki-Einstein manifold X 1,1 . Moreover, we study instanton solutions on the metric cone C (X 1,1 ) by evaluating the Hermitian Yang-Mills equation. We briefly discuss some features of the moduli space thereof, following the main ideas of a treatment of Hermitian Yang-Mills instantons on cones over generic Sasaki-Einstein manifolds in the literature.

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The instanton equations on vector bundles over Calabi-Yau and hyper-Kähler cones can be reduced to matrix equations resembling Nahm's equations. We complement the discussion of Hermitian Yang-Mills (HYM) equations on Calabi-Yau cones, based on regular semi-simple elements, by a new set of (singular) boundary conditions which have a known instanton solution in one direction. This approach extends the classic results of Kronheimer by probing a relation between generalised Nahm's equations and nilpotent pairs/tuples. Moreover, we consider quaternionic instantons on hyper-Kähler cones over generic 3-Sasakian manifolds and study the HYM moduli spaces arising in this set-up, using the fact that their analysis can be traced back to the intersection of three Hermitian Yang-Mills conditions.