Your thread title: "How to regain the trust? DO I walk or stay and try and work it out?"
The short answer. In order to regain the trust, HE has to EARN IT. That decision and "trust" is not up to you alone. What has he done, if anything, to regain your trust? Has he given you details? Do you know the whole picture? Can you make an informed decision? Apparently you know nothing about the AP, in that you don't know if she is involved with someone else and the name. There is information out there that your WS is not telling you, therefore it is logical NOT to trust him.
Does he want you to stay? Why? If he is not willing to be completely open and honest with you and deal with your hurt, why does he want you there? To be a doormat? Because he is "comfortable"? Or because he truly loves you and intends to be faithful? Do his actions meet his words?
Of course, the final decision is yours. But if you don't trust him, please don't waste your life and your future with someone that doesn't deserve it. If he earns your trust again, fine. If he does not, then you are merely accepting his behavior on his terms, not yours. Think about it.
I wish you well.