Well, it's a tricky situation, but my first thought was... you're not going to like it, but here goes:
Your wife is polyamorous. You have trouble with the emotional aspect, not the sexual one. How is closing the relationship going to help?
Think about it: she will STILL be in love with the other guy, which is what hurts you. And if she finally gets over him, she will fall in love with others. Won't that hurt you just as much? And won't it hurt you more to see her miserable because she can't do anything about it? It seems to me, if you need emotional monogamy, you need someone who is able to provide it to you. Another monogamous partner. You'd need to break up with her.
If you want to stay with her, which I assume is the case, I don't think telling her she can't see the people she loves is going to help you much. You'll always know she loves them. If you ask her not to tell you about who she has feelings for, you'll keep wondering if she's in love with so and so. She'll be sad about not being with them, she'll feel guilty that she has feelings for them in the first place and that it's hurting you, and you'll probably feel just as bad as you do now.
If your problem was with sexual non-exclusivity, then it would be easier. No sex, and she can love whoever she wants, you wouldn't care, and maybe you could do things in the bedroom that would make it easier for her not to have sex with others. But here, it's feelings, not actions, that bother you. And feelings are not something she can control.
Now, if you think about what bothers you more specifically, maybe you can come up with actions, not feelings, that make you uncomfortable. And maybe you can work with her so that these actions happen less or stop happening.
Alternately, you might come up with actions that she can do with you so that you feel better. Spend more time with you, have regular dates, go on weekends or vacations together... I don't know, but there might be things that would help.
But ultimately, if the problem is that it hurts you to know she loves others, well you already know she does, and chances are she always will, whether she has boyfriends or not.