Flat BPS Domain Walls on 2d K\"ahler-Ricci Soliton

In this paper we address several aspects of flat Bogomolnyi-Prasad-Sommerfeld (BPS) domain walls together with their Lorentz invariant vacua of 4d N=1 supergravity coupled to a chiral multiplet. The scalar field spans a one-parameter family of 2d K"ahler manifolds satisfying a K"ahler-Ricci flow equation. We find that BPS equations and the scalar potential deform with respect to the real parameter related to the K"ahler-Ricci soliton. In addition, the analysis using gradient and renormalization group flows are carried out to ensure the existence of Lorentz invariant vacua related to Anti de Sitter/Conformal Field Theory (AdS/CFT) correspondence.