For starters, DO NOT sign anything unless you have carefully read it.

When you say they want you to sign an "affirmation", I am hoping you got the word wrong.

By definition an "affirmation" is "...a written statement under oath attesting to the veracity or accuracy of facts in the statement or in relation to some other document or assertion.". If you are not careful, you may end up signing a form confirming the debt.

What you want is an "affidavit". That means "...a written statement, sworn to be true by the person who makes it, for use as evidence in a court of law."

It looks the same but is quite different. You don't know any facts in this case...only what you currently know. You want to only make a statement that you don't know anything about these charges, not affirm something.

You can get an ID Theft affidavit by going to the FTC website at http://www.ftc.gov/bcp/conline/pubs/credit/affidavit.pdf#search='identity%20theft%20affidavid '

OK, to answer your question....from a legal point of view they didn't steal anything from you. They robbed the airline or whoever they used your credit card on. You are a victim of ID theft only. The will not share any information about this with you. The sad fact is that there is so much ID theft going on, it's unlikely they will even investigate this unless there is a very good paper trail to find them.

For now, worry about fixing the damage and protecting yourself from future damage.