Is there a belt that would cause the ac to freeze up if broken?

Yes, the blower belt. If the blower belt, the belt that turns the blower wheel to produce air breaks on a DX unit the refirgerant system will continue to pump through the evaportator coil. With no air to remove the cooling it will begin to freeze on the coil. Over time the compressor will more than likely shut down on low pressure or overloads.