I am familar with some far off ideas concerning negative energy like wormhole creation or fasther than light travel by manipulating spacetime curvature in front and back of a theoretical spaceship (all these are more or less scifi too) but I've never heard of negative energy being used with the concept of anti-gravity in a serious scientific discussion by physisists. I may be wrong though, maybe some of the more knowlegable members (physisists) can help you out on this one.
If you watched the video, you now know that gravity is actualy spacetime curvature - the more mass, the more curved space-time is.
As for anti-gravity I'd use this analogy... We have heat but can we have "anti-heat" ? We have cold but we cannot refer to cold (lack of heat) as "anti-heat" - theres no such thing as "anti-heat" There is just heat or there is less heat and thats it.