can my husband sue my boyfriends business because he thinks we had sex before I left him?

There are many variables that are not present in your question.  I see you left your husband in August, but you don't say where you last lived with your husband, so I am assuming it is in South Carolina.  You don't say where your boyfriend's business is, so I am also going to assume it is in South Carolina.  If those are correct, then I am not sure why your husband wants to sue your boyfriend's business, and not your boyfriend.  In any event, you also don't say what he wants to sue him for, so I am assuming it is something like alienation of affection.  If all that is correct, then he will most likely not be successful, because I am not aware of a court that has upheld an alienation of affection lawsuit here in South Carolina in the recent past.  If he wants to sue for some other reason, the answer could be different depending on the facts.