I was notified by the dean in writing that a student from my class made the complaint and that the dean is investigating.
That doesn't answer my question, unless you're implying that you don't know the answer. In any event, your statement that "the dean
is investigating" implies that the investigation is ongoing. Until it's concluded, any consideration of legal action would be premature.
From a non-legal standpoint, do you really think suing a student would be at all productive?
Also, in your original post, you wrote that, as a result of this allegation, "the administration of the college [tried] to discredit [you] as an instructor," but now you're saying that "the dean is investigating." Investigating an allegation and "trying to discredit" you are not at all the same thing.
A written complaint to the school over which something the student believes to be true, even if later proven false, is not libel. . . . Unless you can prove it was knowingly false with intent to harm you and you can prove actual damages then it is not with worth considering legal action.
Except for the part about needing to prove actual damages, that's not necessarily correct. Whether or not the student subjectively believes the statement to be true, if he/she is negligent as to the truth or falsity of the statement, he/she could be held liable. It is not necessary to prove that the defendant knew of the statement's falsity and had "intent to harm."
See DeAngelis v. Hill, 847 A.2d 1261, 1267-68 (N.J. 2004).