Is audio carried over a HDMI cable? Or do I need a separate one?
yes it can be, however if its to carry audio to your amp you need to check if your amp can receive audio over hdmi as some amps with hdmi ports dont accept audio over hdmi.
It also depends on the source - Sky HD for example although it carries sound over its hdm connection, at the moment it's stereo only. So you have to use an optical digital cable in conjunction with it if you want surround sound via an amp.