I want to create a Browser extension that reproduce the SO navbar (inbox and achievements):

I need for that to authenticate the current user using the Authentication API.

For this step, I need to provide a redirect_uri parameter that must be under an apps registered domain. It is used to redirected the user when he logged in.

My issue is that browser extensions don't have URL (it's just like a local web app). So how can I log a user in a browser extension using the API?