Sticking with your requirements of around 12mm or so focal length, rectilinear, autofocus and less than around $400, the simple answer to this is "no, you can't have that" even if you're prepared to compromise on other things like speed, optical quality and (lack of) zoom. If you're prepared to give up on autofocus (which generally isn't too much of a problem for ultra-wide lenses because the hyperfocal distance is only a couple of metres), there are a couple of options from Samyang/Rokinon:

• Their 8mm fisheye mentioned in Lars's answer.
• Their 14mm f/2.8 rectilinear lens. As I write this (December 2013), this is nominally available, but I'm not aware of any reviews yet. The "SLR" version of the lens retails for around $400, so I'd expect the price of the E-mount version to settle around about there. Note that this lens is the same design as the "SLR" version, just with an adapter tube permanently attached so isn't as small as a native design for E-mount could be.

The same applies to other lens mounts as well, so this isn't a case of there being a hole in the E-mount line-up: it's just that wide-angle lenses are expensive.