Yes, you will be issued a Subpoena, and the presiding judge for the case can sentence you with Contempt of Court if you fail to testify. The idea is to keep you in jail under Contempt charges until you do whatever the judge wants.

The longest sentence for contempt of court is H. Beatty Chadwick (1994 - 2009), which was over a divorce settlement. Susan McDouglas went to jail over not answering questions about Bill Clinton in civil case, and was pardoned by him later. https://en.wikipedia.org/wiki/Susan_McDougal