What do double angle brackets mean? E.g., $\langle \langle \exp (-G(A))\rangle \rangle$.

Question

I came across this in an academic paper on the mass gap, and it had no explanation of the double angle bracket notation: $$\langle \langle \exp (-G(A))\rangle \rangle = (2\pi/e)^{-|\Lambda|}\int_C \exp(-G(A+d\theta))\mathscr D\theta_\Lambda.$$

Anyone know what the double angle brackets mean? Thanks.

Answer 1

The text in the paper immediately preceding this equation says:

We define the compact gauge average of $\exp(-G)$ by

So the double angle brackets apparently means "compact gauge average" (in analogy to single angle brackets often being used for some kind of average) and the left side is defined to be the right side.