Supersymmetry in Four Dimensions

We study four dimensional chiral N = 1 supersymmetric theories on which the scalar manifold is described by Kähler geometry and further, it can be viewed as Kähler-Ricci soliton. All couplings and solutions, namely BPS domain walls and their supersymmetric Lorentz invariant vacuums turn out to be evolved with respect to the soliton. Two models are discussed, namely N = 1 theory on Kähler-Einstein manifold and U(n) symmetric Kähler-Ricci soliton with positive definite metric. In the first example we find that the evolution of the soliton causes topological change and correspondingly, modifies the Morse index of nondegenerate vacuums realized in the parity transformation of the Hessian matrix of the scalar potential after hitting singularity which is natural in the global theory and for nondegenerate Minkowskian vacuums of the local theory. However, such situation is not trivial in anti de Sitter (AdS) vacuums. In an explicit model, we find that this geometric (Kähler-Ricci) flow can also change the index of vacuum before and after singularity. Finally, in the last example since around the origin the metric is diffeomorphic to l CP, we have then to consider it in asymptotic region. Our analysis shows that no index modification of vacuums is present in both global and local theories. email: [email protected], [email protected]