This is a tricky situation and a lot depends on the ability of you and your ex-boyfriend to negotiate in a civil manner over this matter. As a general rule, the mere fact of being given access to an initial piece of research by another person does not preclude you from then getting motivated to do research in that topic. However, it may be that the work you shared with him already made enough of a contribution to his research that authorship credit is warranted. Irrespective of this, I think that most people would also be sympathetic to the idea that you shouldn't be put in an adverse position by reason of having shared your work with a partner during a romantic relationship, and it may raise questions relating to research ethics.
As a first step (unless there is some good reason to the contrary), I recommend you contact your ex-boyfriend and a friendly and civil manner, note your concerns, and see if he is open to negotiating a way forward that is equitable for all of you. You might consider the following important questions:
What is the nature of his interest in the topic? What does he want to work on, to achieve, etc.? Is he specifically interested in the research questions that were part of the work you shared, or is he interested in other aspects of that general topic?
Would he be willing to drop this research project (or focus it on some other aspect of the field that does not overlap with your own research) to allow you to have exclusive work on it? (Depending on his thoughts on the matter, this might be something he would consider out of a sense of loyalty to his past relationship and the fact that you gave him access to your research during that relationship.)
Alternatively, would any of you (you, he, his new girlfriend) feel comfortable/uncomfortable collaborating on the research together? Does your existing research contribution (the one you sent to him) warrant coauthorship credit on the work he is doing?
Alternatively, is it possible for all of you to come up with an appropriate demarcation of topics and research questions that you will each pursue, such that you are not competing over publication of the same research topics? (In this case, you would presumably even cite each others related work.)
I recommend you start with this initial approach and see if you can all work something out between you that you can all live with. If things are too acrimonious to negotiate, you could asking to negotiate with the aid of a counsellor or a neutral senior academic acting as a third-party. If no agreement can be reached, and there is a fundamental disagreement over the legitimacy of the research/authorship, or other aspects of the research ethics of the situation, it would then be reasonable to ask both of the universities if they can review the matter together and make an independent determination of what should occur in this case. That is something that can be done by universities in cases of authorship disputes and other matters bearing on research ethics that arise between their academics.
