Abstract
We find an explicit supergravity background dual to the $\Omega$-deformation of a four-dimensional $\mathcal{N}=2$ SCFT on $\mathbb{R}^4$. The solution can be constructed in the five-dimensional ${\cal N}=4^+$ gauged supergravity and has a nontrivial self-dual 2-form. When uplifted to type IIB supergravity the background is a deformation of AdS$_5\times S^5$ which preserves 16 supercharges. We also discuss generalizations of this solution corresponding to turning on a vacuum expectation value for a scalar operator in the dual SCFT.
Highlights
The Ω-deformation was introduced by Nekrasov in [1] as a tool to calculate the path integral of four-dimensional N 1⁄4 2 gauge theories via supersymmetric localization
We find an explicit supergravity background dual to the Ω-deformation of a four-dimensional N 1⁄4 2 superconformal field theory (SCFT) on R4
We presented a supergravity background which describes holographically a four-dimensional N 1⁄4 2 SCFT subject to an Ω-deformation
Summary
The Ω-deformation was introduced by Nekrasov in [1] as a tool to calculate the path integral of four-dimensional N 1⁄4 2 gauge theories via supersymmetric localization. The dots stand for possible dependence of the path integral on various deformation parameters, like Coulomb branch vacuum expectation values (vevs) and superpotential mass terms, compatible with supersymmetry Rather than viewing the Ω-deformation as a modification of the path integral of the supersymmetric QFT one can consider it as a background of rigid four-dimensional N 1⁄4 2 supergravity; see for example [12,13]. In addition the supergravity approximation requires that we work in the planar limit and at large ’t Hooft coupling In this context the holographic dual of the Ω-background is a simple modification of the well-known vacuum AdS5 solution of five-dimensional N 1⁄4 4þ gauged supergravity [14]. We present this solution explicitly and show how to embed it in type IIB supergravity where it describes holographically the Ω-background for the N 1⁄4 4 supersymmetric Yang-Mills (SYM) theory
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