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

Abstract

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 .

Instantons on Calabi-Yau and hyper-Kähler cones

“…However, satisfying the holomorphicity condition (3.11) for any J already implies the stability-like conditions, as it is shown e.g. in Section 4.5 of [35]. It can be also seen in the explicit instanton equations we use for the discussion in Section 5.1.…”

“…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.…”

“…The exact description of the equivariant connections depends on the concrete 3-Sasakian manifold taken into account, but we review the example of the squashed seven-sphere, as studied in [35]. Its equivariance condition requires I j , X µ = f ν jµ X ν , where j = 8, 9, 10 labels the generators of the Sp (5.…”

“…in[35], we can replace H αn ∈ m by a new elementH such that adH (E e i −e n+1 ) = (n + 1)E e i −e n+1 ∀i = 1, . .…”

Sasaki–Einstein metrics on a class of 7-manifolds

“…Sasaki-Einstein metrics on 7-manifolds continue to play an important role in Mtheory as well as black hole physics [Spa11, GLPS17,MT18]. An important reason for this is that Sasaki-Einstein manifolds admit supersymmetry and are used in the AdS-CFT correspondence.…”