The sorority whose members loudly sang Kanye West's "Gold Digger" and included the n-word ... will NOT be punished ... TMZ has learned. 

A rep from the University of New Hampshire tells us the Alpha Phi sisters showed poor judgment in the video of them using the n-word, but UNH is not investigating the matter. The rep says the initial report claiming UNH had opened a probe was incorrect -- and there will be no discipline for the sorority.

We're told the women of Alpha Phi have apologized.

The university rep also tells us UNH has a strong commitment to the First Amendment, but also has a duty to help students understand how their actions and words can affect people.