I'm a complete beginner in Riemann geometry and trying to read Langer & Singer's 2007 paper, and I need some help getting familiar with some notations on quadratic differentials.
For $\Gamma$ be an algebraic curve in $\mathbb{C}\mathbb{P}^2$ and $G=\Gamma\cap\mathbb{C}\mathbb{A}$, they introduced the isotropic coordinates $(R,B),\; R=X+iY, B=X-iY$ for $(X,Y)\in G$, with corresponding projections $\rho(R,B)=R$, $\beta(R,B)=B$.
Here's where I started to get lost. They defined the fundamental form on $\mathbb{C}^2$ to be $Q=dX^2+dY^2=dRdB$. I think I can do the computation $$Q=d(X^2+Y^2)=d(RB)=d(R)d(B) $$ but I have no idea whether this algebraic operation is valid, and why. Moreover, I have no intuition on what $dX^2, dY^2, dRdB$ even are. I know a little about differential forms, but I don't think this is connected to differential forms because the algebraic operations above seems outlandish?
Is the notation d just a dummy variable? and I might be over-thinking things.. because it seems to me the quadratic differential isn't really that different from the product of two 1-form.