Does one owe expiation for having sworn on the Qur'an to be faithful to a man, then leaving him?

Swearing by Allah to continue a relationship with a non-mahram man is a sin. Placing one's hand on the Quran while swearing is merely for emphasis. You must abandon this sin and repent from it. You are also required to offer an expiation for breaking an oath (kaffarat yameen). So, whoever swears to commit a sin, he must expiate for his oath and not commit that sin.