Sorry to hear you're seeing it hung there.
You can find some mentions of "Connecting to authentication server" on the old developer forums, the old community forums, and Stack Overflow, but generally those are now resolved issues. This was a much more common problem before we added proxy support to Airspace Home.
So I don't think you're seeing the identical issue as one of those other posts, or at least we can't confirm that without some more detail.
In Airspace Home 1.2, there is still the shortcoming that a DNS error or other connection issue will print an error in the log, but will continue showing "Connecting to authentication server" to the user. Could you post the end of your log to see if there are any clues there? It's under %LocalAppData%\Airspace\log.txt on Windows and ~/Library/Application Data/Airspace/log.txt on OS X.
If the log indicates that it is purely hanging trying to reach the server, then please provide more info about your home network (are you behind a proxy?), the country are you connecting from, and can you reach https://central.leapmotion.com in your browser?