I was working through the Mirror Symmetry book by Clay Math Institute. It deals with supersymmetric sigma model in 10.4 section. It doesn't derive how the action is invariant under the variation. I am trying hard, but stuck at few places. Does any one have any references for such a calculations ? The Lagrangian and variation is given in the following image.
Showing the supersymmetry invariance of SUSY nonlinear sigma models requires the use of various identities from differential geometry; in particular, the (target space) derivative of the metric is related to the Christoffel symbol, and the the derivative of the Christoffel symbol is related to the Riemann curvature tensor. Besides these, there are various other useful geometrical identities, refer to Chapter 7 and Chapter 8 of Nakahara's Geometry, Topology and Physics for these. For example, it is quite common for the supersymmetry variation of the action to give quantities which are symmetric under the exchange of two covariant indices (due to some differential geometric identitiy), and when these indices are contracted with the contravariant indices of two identical fermionic fields, the resulting quantity is zero, due to the antisymmetry of the fermionic fields.
The OP has mentioned the case of a 1d SUSY sigma model, also called supersymmetric quantum mechanics, but points similar to those in the previous paragraph should hold for all SUSY sigma models.