# Notation issue in a paper on loop quantum gravity/geometrodynamics

I was reading a paper and I came across some notation that confused me. This was concerning the extrinsic curvature tensor defined as $K_{ab} = \frac{\mu}{2}K^i_{(a}K^i_{b)}$. I am normally comfortable with tensor indices, but the placement of the brackets seemed strange. This was not a typo, as later on, similar notation was used for another tensor $\mathcal{K}_{ab} = \mu^2K^i_{[a}E^i_{b]}$. Is this a standard notation used in these areas, and what does it refer to?

Yes: parentheses and brackets are standard notation. They mean symmetrisation and anti-symmetrisation: \begin{aligned} A_{(\mu\nu)}\equiv\frac{1}{2}(A_{\mu\nu}+A_{\nu\mu})\\ A_{[\mu\nu]}\equiv\frac{1}{2}(A_{\mu\nu}-A_{\nu\mu}) \end{aligned} and similar relations for higher-order tensors. One should note that the factor $\frac12$ is sometimes omitted by some authors.