How to Solve a Hodge Theory Equation?

So i'm going to decide that i have a zeta field, which is an omega one tenser spinners. And a field new, which always strikes me as a ginnish field now, is a mega zero tenser s. Okay, what equation would they solve if we were doing hodges tory? Relative to this complex? The equation would look something like this. There'd be one equation that was very simple, and then there'd be one equations that would be, like, really hard to guess.