If both of you want to work on the marriage, she has to tell you what is going on. There has to be a clear plan such as going to marital counseling, discussing what each one wants/expects of one another, improve communication, etc.
Even if you've indicated that you do not want separation, she has to want to work on the marriage, which means she has to halt communicating with the alleged partner.
You will have to be able to trust her even if you've decided to forgive her. Do what is best for you; set expectations of her and a deadline.