It probably runs under Wine, looking at the binaries it seems to be built against dx9 and gl ES, so it hopefully would default to gl ES if any problems come up.

the container application is based off CEF3 ( http://code.google.com/p/chromiumembedded/ ) , here is the repository https://github.com/adobe/brackets-shell

Whilst Wine is a good stop gap, it makes things ugly as sin and clunky feeling and is is nowhere near as good as a native experience. Why would anyone switch to a program you have to run in Wine when there are already excellent editors out there that are native, such as Sublime Text 2?