My husband is having an "emotional" affair with another woman

I don’t think the statute in New York defines adultery as acts that include “emotional affairs.” It seems to me you would have to prove that your spouse is actively engaged in “adulterous” relations¬†with another person in order to make out a cause of action for adultery (and you know exactly what I mean by “adulterous”). There’s got to be physical proof.
In order to get a divorce on the basis of adultery, you have to provide the court with proof which might include photographs, emails, audio, phone records, text messages, admissions from your spouse, a pregnancy or existence of extra marital children, and a host of other things, including affidavit from friends, acquaintances and third parties who know about the affair.
I would think an “emotional affair” would be more difficult to prove. But as I think about it, it could probably be proven with electronic evidence such as emails and the like.
But it may be wise to also allege cruel and inhuman treatment in your complaint, and depending on the circumstances, you may also allege constructive abandonment just in case you are not able to make out a prima facie adultery cause of action.