Abstract
We study gravity duals to a broad class of N=2 supersymmetric gauge theories defined on a general class of three-manifold geometries. The gravity backgrounds are based on Euclidean self-dual solutions to four-dimensional gauged supergravity. As well as constructing new examples, we prove in general that for solutions defined on the four-ball the gravitational free energy depends only on the supersymmetric Killing vector, finding a simple closed formula when the solution has U(1) x U(1) symmetry. Our result agrees with the large N limit of the free energy of the dual gauge theory, computed using localization. This constitutes an exact check of the gauge/gravity correspondence for a very broad class of gauge theories with a large N limit, defined on a general class of background three-manifold geometries.
Highlights
Exact results in quantum field theories are rare and for some time the gauge/gravity duality [1,2,3] has been a main tool for obtaining such results in a growing variety of situations
We study gravity duals to a broad class of N = 2 supersymmetric gauge theories defined on a general class of three-manifold geometries
As well as constructing new examples, we prove in general that for solutions defined on the four-ball the gravitational free energy depends only on the supersymmetric Killing vector, finding a simple closed formula when the solution has U(1) × U(1) symmetry
Summary
Exact results in quantum field theories are rare and for some time the gauge/gravity duality [1,2,3] has been a main tool for obtaining such results in a growing variety of situations. We will show that given a (non-singular) anti-self-dual metric on the ball with U(1) isometry, and a choice of an arbitrary Killing vector therein, we can construct a (non-singular) instanton configuration, such that together these give a smooth supersymmetric solution of minimal gauged supergravity Assuming this metric is asymptotically locally (Euclidean) AdS, we will show that on the conformal boundary the four-dimensional solution reduces to a three-dimensional geometry solving the rigid Killing spinor equations of [24, 25], in the form presented in [20]. Appendices A and B contain details about the geometry, while in appendix C we present a unified view of all the examples, arising as particular cases of the m-pole metrics [26]
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