What is "$D$ equal to $\frac{1}{2}$ everywhere" in original GAN paper
$\mathbf{D}$ is the Discriminator network, but it really is being modelled as a probability distribution - the discriminator:


  estimates the probability that a sample came from the training data rather than G


where $\mathbf{G}$ is the Generator network.

In the context of a two-player min-max game, the authors say a unique solution exists in the case that the Generator perfectly learns the underlying distribution of the dataset. When this point is reached, the Discriminator's estimates will not be better than random guesses.

This means that the accuracy of the estimates will just be 50% (assuming a 50-50 split of real versus generated samples), so it doesn't know if the Generator created the image (which would correspond to an estimate of 100%) or definitely did not create that image (an estimate of 0%).

So $D = 0.5$ everywhere means the Generator has perfectly learned the data and can produce images that are indistinguishable from the real images... so the Discriminator is basically clueless and is flipping a coin.