Could you elaborate on the cost to the authors? Thanks.

There is no cost as such--the negative paymens are virtual rather than actual. What publishers do is potentially give an author an advance on royalties. Then as the book sells you get a statement (usually semi-annually or quarterly) that shows what the sales have been, what your royalties on those sales are, and what they've paid you so far. If they've paid you more than the sales so far, the royalty statement shows negative.

You can get a negative statement either because you've not sold enough copies to cover the advance, or if the returns exceeded sales for the royalty period. (Which does happen--publishers do take back unsold books in a number of circumstances)

You don't ever have to mail your publisher a check or anything, you just won't get a check from them. (Potentially ever, I suppose, if sales never cover the advance)