Understanding math notation in infoGAN paper
The notation $D_\mathrm{KL}(P \| Q)$ is pretty much standard for the Kullback-Leibler divergence. There is an interesting discussion on Mathematics SE on the reasons for the fairly unusual notation used for divergences.

In general, $x \sim G(z, c)$ means &quot;$x$ is a random variable distributed according to $G(z, c)$&quot;. In the subscript the author means to take the expectation with respect to the random variable $x$, supposing that $x$ has that distribution.

If you have a variable, such as $a$, it is not unusual to call a related variable $a'$ (usually pronounced &quot;a prime&quot;). In this case $c'$ is just a random variable distributed according to $P(c\mid x)$, with the name $c'$ probably meant to remind you that the variable is in some way connected to $c$. In cases where derivatives are obviously not the right thing, this is probably what a prime symbol means.