Can I sue my husband's mistress for the break up of my marriage?
I think that what you are thinking of is something known as "alienation of affection" (also known as a "homewrecker law"). This is a cause of action based on any willful or malicious interference with marriage relations by a third party. However, not all states recognize the right to bring an alienation of affections lawsuit. In fact only a handful of states still allow them but MN is not one of them.
Note: Additionally, in the event of a divorce, your husband's adultery more than likely will not be relevant in an award of spousal support or custody.