Currently the father of my child & I have no custody arrangement. He is on the birth certificate and has signed paternity papers. He see's our daughter once or twice a week, but only because I need him to watch her while I work. Half the time he sleeps and his mother watches her.
I want, so badly, to move to NC. My sister lives down there & we would be a great help to each other. I need help so I can finally finish my degree and she would be more than happy to help me. I can barely make rent here in CT, and I'm sick of living paycheck to paycheck. & my daughter could be with her cousins!
Even though he basically sucks, (Sorry, don't know how else to put it), He would not want us to go. Can I leave without his permission since we do not have a custody arrangement of any kind?

Absent a court order you can move. However, there is a possibility a court will order you or the child to return.