$\DeclareMathOperator\SU{SU}$I am reading Jason Lotay and Goncalo Oliveira's paper $\SU(2)^2$ invariant $G_2$-instantons, and have a few questions from the same.

If we consider the space $M = S^3 \times S^3$. Then the cone metric can be written as $g = dt^2 + g_t$, where $g_t = \sum_{i=1}^3(2A_i)^2\eta_i^+ \wedge \eta_i^+ + (2B_i)^2\eta_i^- \wedge \eta_i^-$ is the compatible metric given by the $\SU(2)\times\SU(2)$ invariant $\SU(3)$ structure on $\{t\}\times M$.

Here $\eta_i^\pm$ are the standard coframe of 1-forms, while the functions $A_i(t)$, $B_i(t)$ specify the deformation of the cone singularity. I have the following questions:

  1. How can we see that the metric $g_t$ (and in general the $\SU(3)$ structure as given) is $\SU(2)\times\SU(2)$ invariant?

  2. Why does an extra $\operatorname U(1)$ symmetry force $A_2=A_3$, $B_2=B_3$?

  3. Why does an extra $\SU(2)$ symmetry force $A_1=A_2=A_3$, $B_1=B_2=B_3$?


$\DeclareMathOperator\SU{SU}$For the first one, you should note that the six $1$-forms $\eta^{\pm}_i$ are all defined using the left-invariant $1$-forms on $S^3 \times S^3 \cong \SU(2) \times \SU(2)$, hence they are $\SU(2)^2$-invariant. Since $g_t$, $\omega$ and $\Omega_j$ are all defined using $\eta^{\pm}_i$, it follows that the $\SU(3)$-structure is invariant by $\SU(2)^2$.

For the second one, the $\operatorname U(1)$ action is generated by the vector field $T_1^+$. If you impose that $\mathcal{L}_{T_1^+}g_t=0$ then you should find that $A_2=A_3$ and $B_2=B_3$. For the third one, just repeat the same argument with $T_2^+$ and $T_3^+$ as well.


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