Sasakian quiver gauge theories and instantons on the conifold

Abstract

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

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

“…In this reduction, M-theory on AdS 4 × S 7 /Z k becomes IIA string theory on AdS 4 × CP 3 [29], which in the 't Hooft limit is dual to N = 6 superconformal Chern-Simons theories with matter fields [15] to which our constructions apply. Similar reductions to the underlying Kähler leaf spaces have been carried out for the Sasaki-Einstein manifolds considered in [9][10][11].…”

“…For the fundamental representation of SU(4), the generator of Z k is chosen based on the decomposition (3.30) and the quantum numbers with respect toÎ 8 as As expected, this is the higher-dimensional analogue of the quiver for the fundamental representation of SU(3) in [10]. The holomorphicity condition Y α , Y β = 0 is trivially satisfied, while the stability condition (5.7) yields 11) where ξ i are the components of the decomposition of Ξ according to the action γ(h k ).…”

“…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.…”

Sasakian quiver gauge theory on the Aloff–Wallach space X1,1

“…The odd-dimensional counterparts of Kähler spaces are Sasaki manifolds [17], and among them Sasaki-Einstein manifolds [18] are of particular interest for compactifications in string theory because, by definition, their metric cones are Calabi-Yau [19,20]. In the literature, Sasakian quiver gauge theory has been studied on the orbifold S 3 /Γ [22], on orbifolds S 5 /Z q+1 of the five-sphere [23] and on the space T 1,1 [24], the base space of the conifold. The five-dimensional Sasaki-Einstein coset spaces as well as the new examples [25,26] are of interest for versions of the AdS/CFT correspondence.…”