Note that I am not licensed to practice in your state.
Identity theft, which is what she did, is a crime. It is a crime that is punishable by time in prison. With that said, you probably don't want to call the cops on your wife but you should seriously consider what her actions imply about your marriage and relationship. It is highly disrespectful at least on her part to steal your identity and oen a card in your name. She is in essence, stealing money from you and not telling you about it. You need to have a serious sit-down talk with her about the status of your relationship putting all legal and financial matters aside.
Like I said, legally you can have her arrested. Financially, you are not responsible for that debt but only if you claim that you were a victim of identity theft which would require you to file a police report on your wife. The best thing to do going forward, is to constantly monitor your own credit report. There are 3 major credit bureaus and you are entitled to a free credit report from each of the credit bureaus once a year. Therefore, every 4 months, you should get a free credit report from one of the credit bureaus. That way, you can go the entire year while getting free credit reports.
Next, you should put a credit alert on all your credit reports. This is an alert that you tell each of the credit bureaus to put on so that they will alert you each time a new credit application is requested. You will have to authorize each move and that way, you can monitor your credit constantly.