Why must this DIY lab bench power supply be powered by an AC supply?
C2, D5, D6 form a charge pump to produce the negative voltage required to allow the opamps to get down to the negative rail, it is a fairly popular trick when you want a supply that can vary right the way down to 0V (Which is otherwise a surprisingly hard thing to pull off). 

Charge pumps require the input to be AC (or at least pulsating DC).