Taking squares or square roots of differential forms?

Question

Reading the recent paper Loop Integrands from the Riemann Sphere by Yvonne Geyer, Lionel Mason, Ricardo Monteiro and Piotr Tourkine I noticed that the authors occasionally seem to take squares and square roots of differential forms. For instance, see the paragraph right before eq. (5), or see the first equation on page 4 in the upper left corner. I have never seen differential forms appear squared or under square roots. It would be great if someone could explain to me how this makes sense and how to work with them. Maybe you have a reference explaining this? Thanks for any suggestion.

This post imported from StackExchange Physics at 2015-08-09 20:01 (UTC), posted by SE-user Kagaratsch

Answer 1

Hi. Square is just the direct product, like in the metric, not the wedge product. Square root is formally defined, the point being that the complete pfaffian has no square roots because each sqrt{dz} appears twice. So you may think about it as a device to represent the result in terms of the pfaffian of a matrix.