I am reading this paper and am having a very hard time understanding the metric given on page six. The manifold we are working in "has the same topology as $\mathbb{R}^4 \times S^3$." The metric is given by $$ds^2 = \alpha^2 dr^2 + \gamma^2 (\omega'^a)^2 + \beta^2(\omega^a - \frac{1}{2}\omega'^a)^2$$

with $$\alpha^{-2} = 1 - \frac{a^3}{r^3}$$ $$\beta^2 = \frac{r^2}{9}(1-\frac{a^3}{r^3})$$ $$\gamma^2 = \frac{r^2}{12}$$

and where $\omega'^a$ and $\omega^a$ are the left invariant one forms on $S'^3$ and $S^3$ respectively.

Specifically, I am confused about what "a" in the superscript of the one forms means since it seems like it's an index, but then later it's used to define $\alpha, \beta,$ and $\gamma$ as if it's another coordinate. Also what exactly are the "left invariant one forms"?

Any references that might help with this would be greatly appreciated as well. It might be useful to know that I am aiming to calculate the Killing vector fields of this metric, and am generally not very familiar with physics notation.