Supersymmetric Sigma Model

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. The Lagrangian and variation from the book is:

$$L=\frac{1}{2} g_{I J} \dot{\phi}^I \dot{\phi}^J+\frac{i}{2} g_{I J}\left(\bar{\psi}^I D_t \psi^J-D_t \bar{\psi}^I \psi^J\right)-\frac{1}{2} R_{I J K L} \psi^I \bar{\psi}^J \psi^K \bar{\psi}^L,$$ where $$D_t \psi^I=\partial_t \psi^I+\Gamma_{J K}^I \partial_t \phi^J \psi^K,$$ \begin{aligned} \delta \phi^I & =\epsilon \bar{\psi}^I-\bar{\epsilon} \psi^I, \\ \delta \psi^I & =\epsilon\left(i \dot{\phi}^I-\Gamma_{J K}^I \bar{\psi}^J \psi^K\right), \\ \delta \bar{\psi}^I & =\bar{\epsilon}\left(-i \dot{\phi}^I-\Gamma_{J K}^I \bar{\psi}^J \psi^K\right), \end{aligned}

• Related question: physics.stackexchange.com/q/52462/2451 Commented Feb 1, 2013 at 17:29
• Please type equations instead of pasting images. Images make editing impossible, suffer the possibility of link rot, are harder to read, and cannot be searched. Commented Nov 14, 2015 at 6:47
• @DanielSank : My apologies, I shall enter the equations from now on. Commented Nov 18, 2015 at 6:54
• Hint: Start with flat space, and confirm the invariance of this noninteracting limit. Now promote to a hypersphere, the primitive hyperspherical σ model. The metric, Christoffels, and curvature are very simple function you may manipulate directly. Only then, bother with the full Riemannian manifold, and recall the standard identities applicable. Commented Jan 28, 2017 at 23:17