The Eleven-Dimensional Uplift of Four-Dimensional Supersymmetric RG Flow

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The squashed and stretched 7-dimensional internal metric preserving U(1) x U(1) x U(1)ᵣ symmetry possesses an Einstein-Kahler 2-fold which is a base manifold of 5-dimensional Sasaki-Einstein Lᵖ, q, ʳ space. The r(transverse to the domain wall)-dependence of the two 4-dimensional supergravity fields, that play the role of geometric parameters for squashing and stretching, makes the 11-dimensional Einstein-Maxwell equations consistent not only at the two critical points but also along the whole N=2 supersymmetric RG flow connecting them. The Ricci tensor of the solution has common feature with the previous three 11-dimensional solutions. The 4-forms preserve only U(1)ᵣ symmetry for other generic parameters of the metric. We find an exact solution to the 11-dimensional Einstein-Maxwell equations corresponding to the lift of the 4-dimensional supersymmetric RG flow.