Sasakian quiver gauge theories and instantons on cones over lens 5-spaces

Abstract

We consider SU(3)-equivariant dimensional reduction of Yang-Mills theory over certain cyclic orbifolds of the 5-sphere which are Sasaki-Einstein manifolds. We obtain new quiver gauge theories extending those induced via reduction over the leaf spaces of the characteristic foliation of the Sasaki-Einstein structure, which are projective planes. We describe the Higgs branches of these quiver gauge theories as moduli spaces of spherically symmetric instantons which are SU(3)-equivariant solutions to the Hermitian Yang-Mills equations on the associated Calabi-Yau cones, and further compare them to moduli spaces of translationally-invariant instantons on the cones. We provide an explicit unified construction of these moduli spaces as Kähler quotients and show that they have the same cyclic orbifold singularities as the cones over the lens 5-spaces.

Sasakian quiver gauge theories and instantons on cones over round and squashed seven-spheres

“…In this section we consider quiver gauge theory on S 7 ∼ = SU(4)/SU(3), regarded as a Sasaki-Einstein manifold. Since the canonical connection, the structure equations and the instanton equations of any particular odd-dimensional sphere can be easily generalized to all odd-dimensional spheres, the exposition will closely follow that of the five-sphere in [10]. We start by describing the geometry of the homogeneous space and orbifolds thereof, including the canonical connection with respect to the Sasaki-Einstein structure.…”

“…Orbifolds. We conclude by briefly describing the corresponding geometry of the orbifold S 7 /Z k , closely following the treatment of [10]. For compatibility with the bundle structure of the homogeneous space S 7 = SU(4)/SU(3), the cyclic group Z k is embedded in G = SU(4) in a way that it commutes with H = SU(3), i.e.…”

“…This Sasakian quiver gauge theory [9] is motivated by the close relation between Kähler and Sasakian geometry, as well as by the prominent role of Sasaki-Einstein manifolds in the AdS/CFT correspondence. The examples covered so far include the sphere S 5 (and orbifolds thereof) [10] and the conifold T 1,1 [11] in five dimensions, and the Aloff-Wallach space X 1,1 [12] in seven dimensions which also involves aspects of its 3-Sasakian structure. The aim of this paper is to compare Sasakian and 3-Sasakian quiver gauge theories on the round and squashed sphere S 7 in seven dimensions, realized respectively as the homogeneous spaces SU(4)/SU (3) and Sp(2)/Sp (1).…”

Instantons on Calabi-Yau and hyper-Kähler cones

“…This approach has been pursued in the constructions of instantons in various settings, see for instance [18][19][20]22]. Solving the equivariance condition more generally leads to quiver gauge theories [35][36][37][38][39][40] that depend on the chosen manifold. For the moment, we suppose that one has implemented the equivariance conditions and is left with the relevant instanton equations.…”

“…Inspired from the explicit calculations for SU(3)/SU(2) in [38] and SU(4)/SU (3) This is the same rescaling as employed in the definition of the torsion components of the canonical connection Γ P of [17]. See also [38] for an explicit example in n = 2. From now on denote E j := E e j −e n+1 , F j := E −(e j −e n+1 ) , (A.…”