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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.

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    $\begingroup$ R. L. Bryant and S. M. Salamon, On the construction of some complete metrics with exceptional holonomy, Duke Math. J. 58 (1989), no. 3, 829–850. M $\endgroup$ – user21574 Nov 29 '17 at 3:17
  • $\begingroup$ The left invariant 1-forms are the 1-forms which are invariant when you think of the 3-sphere as $SU(2)$ (or as the unit quaternions), and you left translate on $SU(2)$ (or the unit quaternions) as a Lie group. $\endgroup$ – Ben McKay Nov 29 '17 at 7:08
  • $\begingroup$ There are indeed two different meanings of the letter $a$ in the notation of the paper, one as a cube root of volume of a particular 3-sphere, and the other as an index for a basis of the Lie algebra of $SU(2)$. $\endgroup$ – Ben McKay Nov 29 '17 at 7:13
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If you are unfamiliar with left invariant 1-forms, you might read Stillwell, Naive Lie Theory. The left invariant 1-forms are the 1-forms which are invariant when you think of the 3-sphere as $SU(2)$ (or as the unit quaternions), and you left translate on $SU(2)$ (or the unit quaternions) as a Lie group. To be specific, take a variable quaternion $q$ of unit norm, and consider the expression $q^{-1} dq$. This is a 1-form valued in the imaginary quaternions (because $|q|=1$), and so has three components $q^{-1} dq = \omega^1 i +\omega^2 j +\omega^3 k$, and these $\omega^a$ are the three left-invariant 1-forms on $S^3$.

There are indeed two different meanings of the letter $a$ in the notation of the paper, one as a cube root of volume of a particular 3-sphere, and the other as an index for a basis of the Lie algebra of $SU(2)$.

As Hassan Jolany pointed out above, you might find it helpful to read R. L. Bryant and S. M. Salamon, On the construction of some complete metrics with exceptional holonomy, Duke Math. J. 58 (1989), no. 3, 829–850; the authors of the paper you are reading certainly did.

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