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I am sorry to hear about this situation.
To sue in a state court, one needs to have a "cause of action." There are numerous causes of action, such as "breach of contract," "negligence," "fraud," "unjust enrichment," etc., as well as causes of action rooted in statutory law. Every state has their own although they are very similar to each other in every state because they all stem from the same common law.
In California, there is no cause of action for "catching someone cheating" if you are not married. This is not technically adultery. So I am afraid that you cannot sue for something like this.
The best that someone in your situation can do is request palimony (see HERE) from them, in a lawsuit.
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